Högsko!an i Luleå Biblioteket
DOCTORAL THESIS
1 9 7 9 : 0 5 D
I M P A C T A N D O P T I M U M T R A N S M I S S I O N O F W A V E S
SOME T H E O R E T I C A L AND E X P E R I M E N T A L STUDIES
B Y
R A M GUPTA
DIVISION O F S O L I D M E C H A N I C S
UNIVERSITY OF LULEA
I M P A C T A N D O P T I M U M T R A N S M I S S I O N O F W A V E S
SOME T H E O R E T I C A L AND E X P E R I M E N T A L STUDIES
B Y
R A M GUPTA
AKADEMISK AVHANDLING
som med vederbörligt tillstånd av Tekniska Fakultetsnämnden vid Hög- skolan i Luleå för avläggande av teknisk doktorsexamen kommer att offentligen försvaras å Högskolans Aula i Centrumhuset onsdagen den 27 februari 1980, kl 10.00.
DOCTORAL THESIS
1 9 7 9 : 0 5 DI M P A C T A N D O P T I M U M T R A N S M I S S I O N O F W A V E S
SOME T H E O R E T I C A L AND E X P E R I M E N T A L STUDIES
B Y
R A M GUPTA
DIVISION O F S O L I D M E C H A N I C S
UNIVERSITY OF LULEÅ
T h i s d o c t o r a l t h e s i s i n c l u d e s t h e f o l l o w i n g p a p e r s :
A Elastic impact between a finite conical rod and a long cylindrical rod, t o g e t h e r w i t h L. N i l s s o n ,
J o u r n a l o f S o u n d a n d V i b r a t i o n BO ( 4 ) , 5 5 5 - 5 6 3 , ( 1 9 7 8 ) . A l s o p r e s e n t e d a t E u r o m e c h 81 C o l l o q u i u m , L i b l i c e C a s t l e , C z e c h o s l o v a k i a , S e p t . 1 3 - 1 7 , 1 9 7 6 , a n d a t S v e n s k a M e k a n i k d a g a r , L i n k ö p i n g , S w e d e n , D e t . 2 8 - 2 9 , 1 9 7 7 .
B Propagation of elastic waves in rods with variable cross-section, a c c e p t e d f o r p u b l i c a t i o n i n ASME J o u r n a l o f A p p l i e d M e c h a n i c s .
C Optimum transmission of an elastic wave through joints, t o g e t h e r w i t h B. L u n d b e r g a n d L r E . A n d e r s s o n ,
Wave M o t i o n 1_ ( 3 ) , 1 9 3 - 2 0 0 , ( 1 9 7 9 ) . A l s o p r e s e n t e d a t 2 0 t h P o l i s h S o l i d M e c h a n i c s C o n f e r e n c e , P o r a b k a - K o z u b n i k , P o l a n d , S e p t . 3-11, 1978.
D Optimization of wave transmitting joints, U n i v e r s i t y o f L u l e å , T e c h n i c a l R e p o r t No. 1 9 7 9 : 8 0 T . A l s o
p r e s e n t e d a t S v e n s k a M e k a n i k d a g a r , G ö t e b o r g , S w e d e n , May 1 1 - 1 2 , 1 9 7 9 .
E Experiments on optimum wave transmitting joints, U n i v e r s i t y o f L u l e å , T e c h n i c a l R e p o r t No. 19 7 9 : 8 1 7 .
The c o n t e n t s o f p a p e r s • a n d E a r e t o be p u b l i s h e d i n a c o n d e n s e d f o r m .
CONTENTS
I m p a c t a n d o p t i m u m t r a n s m i s s i o n o t w a v e s
I n t r o d u c t i o n
Summary o f a p p e n d e d p a p e r s A c k n o w l e d g e m e n t s
R e f e r e n c e s
P a p e r A: E l a s t i c i m p a c t b e t w e e n a f i n i t e c o n i c a l r o d a n d a l o n g c y l i n d r i c a l r o d
P a p e r B: P r o p a g a t i o n o f e l a s t i c w a v e s I n r o d s w i t h v a r i a b l e c r o s s - s e c t i on
P a p e r C: O p t i m u m t r a n s m i s s i o n o f e l a s t i c w a v e s t h r o u g h j o i n t s
P a p e r 0: O p t i m i z a t i o n o f w a v e t r a n s m i t t i n g j o i n t s . .
P a p e r E: E x p e r i m e n t s on o p t i m u m wave t r a n s m i t t i n g i o i n t s
KEY-WORDS
I m p a c t , o p t i m i z a t i o n , e l a s t i c , w a v e , e n e r g y , t r a n s m i s s i o n , i m p e d a n c e , j o i n t .
1
IMPACT AND OPTIMUM TRANSMISSION OF WAVES
INTRODUCTION
I m p a c t i s s a i d t o o c c u r when t w o b o d i e s c o l l i d e . I n s t a n c e s o f u n d e s i r e d c o l l i s i o n s a r e n u m e r o u s and w e l l k n o w n . I n many e n g i n e e r i n g a p p l i c a t i o n s , h o w e v e r , i m p a c t I s i n t e n d e d and u s e d t o a d v a n t a g e . C e n t r a l l o n g i t u d i n a l i m p a c t o f s l e n d e r b o d i e s has a n u m b e r o f i m p o r t a n t t e c h n i c a l a p p l i c a t i o n s due t o ( i ) t h e l a r g e f o r c e s g e n e r a t e d t h r o u g h I m p a c t and [ i i ) t r a n s f o r m a t i o n o f t h e m e c h a n i c a l e n e r g y p r o d u c e d by t h e i m p a c t . Some e x a m p l e s a r e p i l e d r i v i n g , r i v e t i n g and p e r c u s s i v e r o c k d r i l l i n g .
The c l a s s i c a l t h e o r y o f i m p a c t o f r i g i d m a s s e s i s i n s u f f i c i e n t f o r e x p l a i n i n g t h e v a r i o u s p r o c e s s e s o o c u r i n g i n t h e s e ' a p p l i - c a t i o n s . An I m p r o v e d t h e o r y [ a t r a v e l l i n g w a v e t h e o r y o r p u l s e
t h e o r y ) w h i c h t a k e s i n t o a c c o u n t t h e e l a s t i c i t y o f t h e c o l l i d i n g b o d i e s was g i v e n a b o u t a c e n t u r y ago by Neumann [ 1 ] and de S a i n t - V e n a n t [ 2 , 3 ] . S i n c e t h e n t h e s u b j e c t has b e e n t r e a t e d by a l a r g e n u m b e r o f a u t h e r s , s e e , e . g . , [ 4 - 6 ] . F u r t h e r i m p r o v e - m e n t s h a v e b e e n made w h e r e t h e v a r i o u s t h r e e - d i m e n s i o n a l e f f e c t s a r e t a k e n i n t o a c c o u n t [ 7 , 8 ] . I n s n g i n e e r i n g a p p l i c a t i o n s o f t h e p u l s e t h e o r y , h o w e v e r , t h e a n a l y s e s a r e o f t e n r e s t r i c t e d t o t h e s i m p l e s t c a s e s o f o n e - d i m e n s i o n a l I m p a c t s i t u a t i o n s . The c r o s s - s e c t i o n a l p r o p e r t i e s o f t h e i m p a c t i n g r o d s a r e a s s u m e d t o be e i t h e r c o n s t a n t o r v a r y i n g i n s t e p s a l o n g t h e r o d l e n g t h s as i n [ 9 , 1 0 ] . T h i s i s b e c a u s e o f t h e d i f f i c u l t i e s i n v o l v e d i n t h e a n a l y t i c a l t r e a t m e n t o f t h e c a s e o f a r b i t r a r i l y v a r y i n g c r o s s - s e c t i o n s [ 1 1 ] . The c a s e o f a f i n i t e c o n i c a l r o d has b e e n
2
s t u d i e d i n some d e t a i l i n p a p e r A. F o r d i f f e r e n t d e g r e e s o f c o n i c a l n e s s , c o m p a r i s o n s a r e made b e t w e e n t h e o n e - d i m e n s i on a 1 a n a l y t i c a l , e x p e r i m e n t a l a n d t h r e e - d i m e n s i o n a l f i n i t e e l e m e n t r e s u l t s . The r e s u l t s p r o v i d e i n s i g h t s a b o u t t h e r a n g e o f v a l i d i t y o f t h e o n e - d i m e n s i o n a l m o d e l a n d a b o u t t h e u s e f u l n e s s o f t h e f i n i t e e l e m e n t m e t h o d f o r t r e a t i n g i m p a c t p r o b l e m s .
A s u b s t a n t i a l a m o u n t o f w o r k i s c u r r e n t l y b e i n g u n d e r t a k e n i n t h e f i e l d o f o p t i m i z e d s t r u c t u r a l d e s i g n . T h i s i s e v i d e n t f r o m t h e r e c e n t l i t e r a t u r e . S e e , e . g . , [ 1 2 , 1 3 ] f o r a r e v i e w o f t h e f i e l d . M o r e s p e c i f i c a l l y , i n t h e a r e a o f c o n t i n u o u s e l a s t i c v i b r a t i n g s y s t e m s t w o k i n d s o f p r o b l e m s h a v e r e c e i v e d c o n s i d e - r a b l e a t t e n t i o n , [ i ] M a x i m i z i n g t h e l o w e s t c h a r a c t e r i s t i c v a l u e ( e i g e n v a l u e ) [ 1 4 - 1 6 ] , a n d ( i i ) m i n i m i z i n g t h e d y n a m i c r e s p o n s e f o r v a r i o u s a p p l i e d l o a d i n g s [ 1 7 , 1 8 ] .
Some a p p l i c a t i o n s o f l o n g i t u d i n a l l y v i b r a t i n g r o d s h a v e b e e n m e n t i o n e d e a r l i e r . I n v i e w o f s u c h a p p l i c a t i o n s a n o t h e r p r o b l e m o f i n t e r e s t i s t o m a x i m i z e t h e e f f i c i e n c y o f e n e r g y t r a n s m i s s i o n , i n o t h e r w o r d s t o m i n i m i z e t h e l o s s e s o f e n e r g y d u e t o r e f l e c t i o n s f r o m i n h o m o g e n e i t i e s i n t h e v i b r a t i n g r o d s . T h i s i s t h e s u b j e c t o f p a p e r s C, D a n d E. I n t h e s e p a p e r s t h e t r a n s m i s s i o n o f e l a s t i c w a v e e n e r g y t h r o u g h a j o i n t b e t w e e n t w o u n i f o r m r o d s i s s t u d i e d . The e f f i c i e n c y o f e n e r g y t r a n s m i s s i o n ( d e f i n e d as t h e r a t i o o f t r a n s m i t t e d t o i n c i d e n t w a v e e n e r g y ) i s m a x i m i z e d . The i n t e r e s t i n t h e s e p r o b l e m s i s m a i n l y d u e t o t h e i r a p p l i c a t i o n i n p e r c u s s i v e d r i l l i n g . H o w e v e r , t h e r e s u l t s c a n be d i r e c t l y i n t e r p r e t e d t o some o t h e r f i e l d s l i k e e l e c t r o m a g n e t i c w a v e s i n t r a n s m i s s i o n l i n e s a n d s h a l l o w w a t e r w a v e s . The c o n c e p t o f c h a r a c t e r i s t i c i m p e d a n c e , r e c a p i t u l a t e d i n p a p e r B f a c i l i t a t e s s u c h i n t e r p r e t a - t i o n s .
L i k e t h e i m p a c t p r o b l e m t r e a t e d i n p a p e r A, t h e o p t i m i z a t i o n p r o b l e m s t r e a t e d i n p a p e r s C t o E a l s o c o n c e r n t h e p r o p a g a t i o n o f l o n g i t u d i n a l e l a s t i c w a v e s i n r o d s w i t h v a r i a b l e c r o s s - s e c t i o n s . The m o t i o n o f s u c h r o d s i s g o v e r n e d by t h e W e b s t e r h o r n e q u a t i o n w h i c h i s t h o r o u g h l y d i s c u s s e d i n [ 1 9 ] .
I n p e r c u s s i v e d r i l l i n g [ 2 0 ] , t h e k i n e t i c e n e r g y o f a hammer i s t r a n s f o r m e d t h r o u g h i m p a c t i n t o e l a s t i c s t r e s s wave e n e r g y .
Hammers a n d d r i l l r o d s o f c y l i n d r i c a l s h a p e s a r e c o m m o n l y e m p l o y e d i n m o d e r n d r i l l i n g m a c h i n e s . T h e r e f o r e , a r e c t a n g u l a r s t r e s s p u l s e i s g e n e r a t e d by t h e i m p a c t - . T h i s s t r e s s p u l s e p r o p a g a t e s a l o n g t h e d r i l l r o d s w h i c h a r e o f t e n c o n n e c t e d by
( c y l i n d r i c a l ) j o i n t s . L o s s e s i n t h e e n e r g y t r a n s m i s s i o n o c c u r due t o r e f l e c t i o n s a t t h e r o d - j o i n t i n t e r f a c e s . One way o f m i n i m i z i n g t h e s e l o s s e s i s t o o p t i m i z e t h e s h a p e o f t h e i n c i - d e n t p u l s e f o r a g i v e n j o i n t . S u c h o p t i m u m s h a p e s h a v e b e e n
o b t a i n e d i n p a p e r C, f o r p u l s e s w i t h a f i x e d d u r a t i o n . The e f f i c i e n c i e s o f e n e r g y t r a n s m i s s i o n a r e e v a l u a t e d f o r t h e o p t i m a l l y s h a p e d p u l s e s as w e l l as f o r t h e c o r r e s p o n d i n g r e c t a n g u l a r p u l s e s . The i m p r o v e m e n t s i n t h e e f f i c i e n c y t u r n o u t t o be g e n e r a l l y s m a l l . M o r e o v e r , i t may n o t be p o s s i b l e t o r e a l i z e s u c h p u l s e s t h r o u g h i m p a c t [ 2 1 ] .
A n o t h e r way o f i m p r o v i n g t h e e f f i c i e n c y i s t o o p t i m i z e t h e s h a p e o f t h e j o i n t f o r a f i x e d ( s a y r e c t a n g u l a r ) i n c i d e n t p u l s e . T h i s i s t h e s u b j e c t o f p a p e r • w h e r e o p t i m u m s h a p e s
( o r i m p e d a n c e d i s t r i b u t i o n s ) a r e o b t a i n e d f o r a j o i n t w i t h a f i x e d mass and l e n g t h . The e f f i c i e n c i e s f o r t h e o p t i m a l l y s h a p e d j o i n t s a r e c o m p a r e d w i t h t h o s e f o r t h e c o r r e s p o n d i n g c y l i n d r i c a l j o i n t s . S i g n i f i c a n t i m p r o v e m e n t s i n t h e e f f i c i e n c y t u r n o u t t o be p o s s i b l e t h r o u g h j o i n t o p t i m i z a t i o n . F o r a c a s e s t u d i e d i n d e t a i l i m p r o v e m e n t s o f up t o 30 p e r c e n t a r e o b t a i n e d ,
4
The o p t i m u m j o i n t s h a p e s t u r n o u t t o i n c l u d e l a r g e a n d a b r u p t c h a n g e s i n i m p e d a n c e o v e r t h e j o i n t l e n g t h . I m p e d a n c e r a t i o s o f up t o 90 a r e e n c o u n t e r e d f o r t h e c a s e m e n t i o n e d a b o v e . F o r s u c h j o i n t s h a p e s t h e t h r e e - d i m e n s i on a 1 e f f e c t s may become i m p o r t a n t a n d h e n c e t h e r e may be d o u b t s a b o u t t h e v a l i d i t y o f t h e o n e - d i mens i on a 1 m o d e l e m p l o y e d . T h e r e f o r e , e x p e r i m e n t s w e r e p e r f o r m e d on some o p t i m a l l y s h a p e d j o i n t s as w e l l as on c y l i n d r i - c a l j o i n t s . R e s u l t s o f t h e s e e x p e r i m e n t s a r e r e p o r t e d i n p a p e r E. T h e r e s u l t s s u p p o r t t h e v a l i d i t y o f t h e t h e o r e t i c a l m o d e l u s e d I n p a p e r s C a n d D. U s e f u l n e s s o f t h e o n e - d i m e n s i ona 1 m o d e l i s c l e a r l y d e m o n s t r a t e d e v e n i n c a s e s w h e r e r e l a t i v e l y l a r g e a n d a b r u p t c h a n g e s o c c u r i n t h e i m p e d a n c e o f a wave t r a n s m i t t i n g r o d .
N e x t f o l l o w s t h e d e t a i l e d s u m m a r i e s o f p a p e r s A t o E.
SUMMARY OF APPENOED PAPERS
P a p e r A: L o n g i t u d i n a l e l a s t i c i m p a c t b e t w e e n a f i n i t e c o n i c a l r o d a n d a l o n g c y l i n d r i c a l r o d i s s t u d i e d ( i l e x p e r i m e n t a l l y ,
( i i ) a n a l y t i c a l l y , by u s i n g o n e - d i m e n s i on a 1 w a v e t h e o r y t o o b t a i n a c l o s e d - f o r m s o l u t i o n , a n d ( i i i ) n u m e r i c a l l y , by u s i n g a t h r e e - d i m e n s i o n a l a x i s y m m e t r i o f i n i t e e l e m e n t m o d e l . The r e s u l t s f r o m ( i ) - ( i i i ) a r e c o m p a r e d f o r f o u r c o n i c a l r o d s . To o b t a i n i n c r e a s i n g l y t h r e e - d i m e n s i o n a l b e h a v i o u r , c o n i c a l r o d s w i t h h a l f a p e x - a n g l e s o f 5 ° , 1 0 ° , 15° a n d 2 5 ° a r e i n v e s t i g a t e d . The o n e - d i m e n s i on a 1 m o d e l a c c u r a t e l y p r e d i c t s t h e r e s p o n s e f o r t h e 5 ° - c o n e . The d i s c r e p a n c i e s b e t w e e n o n e - d i m e n s i o n a l a n a l y t i c a l r e s u l t s a n d e x p e r i m e n t a l o r f i n i t e e l e m e n t r e s u l t s i n c r e a s e f o r i n c r e a s i n g c o n e a n g l e s . T h e a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l a n d t h e f i n i t e l e m e n t
r e s u l t s i s q u i t e g o o d i n g e n e r a l . The e f f e c t o f c o n t a c t
c o n d i t i o n s b e t w e e n t h e i m p a c t i n g s u r f a c e s i s a l s o i n v e s t i g a t e d . F r i c t i o n b e t w e e n t h e I m p a c t i n g s u r f a c e s i s f o u n d t o h a v e
n e g l i g i b l e e f f e c t s on t h e s t r a i n r e s p o n s e . D e t a i l s r e g a r d i n g t h e o n e - d i m e n s i o n a l a n a l y t i c a l r e s u l t s a r e g i v e n i n [ 2 2 ] .
P a p e r B: The p u r p o s e o f t h i s b r i e f n o t e i s t o p o i n t o u t a f r e q u e n t l y o v e r l o o k e d r o l e o f t h e c o n c e p t o f i m p e d a n c e i n o n e - d i m e n s i ona 1 w a v e p r o p a g a t i o n p r o b l e m s . F o r t h e c a s e o f l o n g i t u d i n a l w a v e s i n r o d s , i t i s e m p h a s i z e d t h a t t h e i n f l u e n c e o f v a r i a t i o n s i n t h e c r o s s - s e c t i o n a l a r e a A, Y o u n g ' s m o d u l u s E, a n d d e n s i t y p can be c o m b i n e d i n a s i n g l e p a r a m e t e r c a l l e d t h e i m p e d a n c e Z = Ape = AE/c, w h e r e c = ( E / p ) ^ ^ i s t h e w a v e s p e e d . T h u s , any c o m b i n a t i o n s o f A, p a n d E
w h i c h c o r r e s p o n d t o t h e same Z s h o u l d y i e l d t h e same r e s u l t s . T h i s means t h a t i n p a p e r A t h e r e s u l t s on c o n e s a r e a l s o v a l i d f o r o t h e r c o m b i n a t i o n s o f A, p a n d E w h i c h y i e l d a q u a d r a t i c v a r i a t i o n o f t h e i m p e d a n c e (.same as t h a t o f t h e c r o s s - s e c t i on a 1 a r e a s o f t h e c o n e s ) .
P a p e r C: The p r o b l e m t r e a t e d i n t h i s p a p e r c o n c e r n s t h e o p t i m i z a t i o n o f t h e s h a p e o f an i n c i d e n t p u l s e o f g i v e n d u r a t i o n s u c h t h a t t h e e n e r g y t r a n s m i t t e d t h r o u g h a g i v e n r o d - j o i n t s y s t e m i s m a x i m i z e d . The i n t e r e s t i n t h e p r o b l e m i s m a i n l y due t o i t s a p p l i c a t i o n s i n p e r c u s s i v e d r i l l i n g . H o w e v e r , by m a k i n g use o f t h e c o n c e p t o f i m p e d a n c e d i s c u s s e d i n p a p e r B, t h e r e s u l t s c a n be d i r e c t l y i n t e r p r e t e d i n o t h e r f i e l d s l i k e e l e c t r o m a g n e t i c w a v e s i n t r a n s m i s s i o n l i n e s a n d s h a l l o w w a t e r w a v e s .
6
The o p t i m i z a t i o n p r o b l e m i s f o r m u l a t e d i n g e n e r a l t e r m s a p p l i c a b l e t o a l l s u c h f i e l d s . The o p t i m u m w a v e s h a p e s a r e o b t a i n e d f o r t w o s p e c i a l c a s e s : ( i ) a j o i n t w i t h c o n s t a n t i m p e d a n c e and ( i i ) a j o i n t w i t h c o n c e n t r a t e d m a s s , ( i ) l e a d s t o a m a t r i x e i g e n v a l u e p r o b l e m and a n o n - u n i q u e s o l u t i o n , w h e r e a s , ( i i ) l e a d s t o an e i g e n v a l u e p r o b l e m f o r an i n t e g r a l e q u a t i o n and a u n i q u e s o l u t i o n . I n b o t h c a s e s t h e e f f i c i e n c i e s o f e n e r g y t r a n s m i s s i o n f o r t h e o p t i m u m w a v e s h a p e s a r e - c o m p a r e d w i t h t h o s e f o r t h e
c o r r e s p o n d i n g r e c t a n g u l a r w a v e s ( a s t h e l a t t e r a r e c o m m o n l y e m p l o y e d i n m o d e r n p e r c u s s i v e d r i l l i n g m a c h i n e s ) [ 2 3 ] . The g a i n s i n t h e e f f i c i e n c y t h r o u g h s u c h an o p t i m i z a t i o n g e n e r a l l y t u r n o u t t o be s m a l l ( a f e w p e r c e n t ) . H o w e v e r
t h e r e s u l t s c l a r i f y how much t h e e f f i c i e n c i e s c a n be i m p r o v e d by o p t i m i z i n g t h e wave s h a p e .
P a p e r D: An a l t e r n a t i v e way o f I m p r o v i n g t h e e f f i c i e n c y o f e n e r g y t r a n s m i s s i o n f o r t h e r o d - j o i n t p r o b l e m t r e a t e d i n p a p e r C
i s t o o p t i m i z e t h e s h a p e o f t h e j o i n t r a t h e r t h a n t h e s h a p e o f t h e i n c i d e n t p u l s e . T h i s i s t h e s u b j e c t o f p a p e r D. The I n c i d e n t p u l s e i s a s s u m e d t o be o f r e c t a n g u l a r s h a p e w i t h a g i v e n d u r a t i o n . O p t i m u m s h a p e ( s ) ( i . e . i m p e d a n c e d i s t r i b u 1 1 o n i s ) J a r e d e t e r m i n e d f o r a j o i n t h a v i n g a g i v e n mass and l e n g t h . The m e t h o d e m p l o y e d i s as f o l l o w s . By d i v i d i n g t h e j o i n t l e n g t h i n t o N s e g m e n t s , t h e j o i n t i m p e d a n c e f u n c t i o n i s
*. • A • 4- M 4- 4- • A , ( 1) , C 2 ) 7 CN)
d i s c r e t i z e d i n t o N c o n s t a n t i m p e d a n c e s Z Q , Zg , Z ^ U s i n g t h e t h e o r y d e v e l o p e d i n p a p e r C, t h e e f f i c i e n c y o f
e n e r g y t r a n s m i s s i o n n i s e x p r e s s e d as a n o n - l i n e a r f u n c t i o n o f t h e N v a r i a b l e s , i . e . ,
7
f7( 1) 7( 2 J 71 IM) , f, .
The c o n s t a n t mass c o n s t r a i n t i m p l i e s a l i n e a r r e l a t i o n b e t w e e n t h e N v a r i a b l e s
Z Q +Zg + 0 = c o n s t a n t . IZI
A l s o s i n c e t h e i m p e d a n c e s m u s t be p o s i t i v e we h a v e
Z j1] > 0, Z j2] > 0 Z ^ > D. ( 3 )
Thus t h e p r o b l e m i s t o m a x i m i z e t h e f u n c t i o n n g i v e n by 11) a n d s u b j e c t e d t o t h e c o n s t r a i n t s C2J a n d [ 3 ) . T h i s n o n - l i n e a r p r o g r a m m i n g p r o b l e m w i t h l i n e a r c o n s t r a i n t s i s t h e n s o l v e d n u m e r i c a l l y u s i n g t h e r e d u c e d g r a d i e n t m e t h o d w h i c h i s a s l i g h t l y m o d i f i e d f o r m o f t h e m e t h o d o f s t e e p e s t d e s c e n t s .
The r e s u l t s show t h a t q u i t e s i g n i f i c a n t i m p r o v e m e n t s i n t h e e f f i c i e n c y c a n Oe a c h i e v e d by o p t i m i z i n g t h e j o i n t i m p e d a n c e d i s t r i b u t i o n s . F o r a c a s e s t u d i e d i n d e t a i l t h e i m p r o v e m e n t i s a b o u t 30 p e r c e n t . H o w e v e r , t h e o p t i m a l l y s h a p e d j o i n t s t u r n o u t t o i n c l u d e l a r g e and a b r u p t c h a n g e s i n t h e j o i n t i m p e d a n c e s o v e r t h e i r l e n g t h s , i m p e d a n c e r a t i o s o f a b o u t 30 t o 90 a r e e n c o u n t e r e d f o r t h e c a s e s t u d i e d i n d e t a i l .
P a p e r F: O c c u r r e n c e o f l a r g e and a b r u p t c h a n g e s o f i m p e d a n c e i n t h e o p t i m u m j o i n t s o f p a p e r 0 r a i s e s t h e q u e s t i o n o f v a l i d i t y o f t h e o n e - d i m e n s i ona 1 m o d e l e m p l o y e d . T h u s
e x p e r i m e n t a l i n v e s t i g a t i o n s w e r e n e e d e d . T h i s p a p e r p r e s e n t s r e s u l t s o f s u c h e x p e r i m e n t s .
8
T e s t s w e r e p e r f o r m e d on t h e o p t i m u m j o i n t s c o r r e s p o n d i n g t o N = "I , 3 a n d 5 i n p a p e r •. I n c i d e n t s t r e s s p u l s e s w e r e p r o d u c e d t h r o u g h l o n g i t u d i n a l I m p a c t b e t w e e n a hammer a n d a r o d , e m p l o y i n g a c o m p r e s s e d a i r g u n . U s i n g s t r a i n g a u g e s a t c o n v e n i e n t l o c a t i o n s , t h e i n c i d e n t , r e f l e c t e d a n d t r a n s m i t t e d w a v e s w e r e r e c o r d e d o n a t r a n s i e n t r e c o r d e r . T h e e x p e r i m e n t a l v a l u e s o f t h e e f f i c i e n c y c o u l d t h u s be o b t a i n e d . C o m p a r i s o n s w i t h t h e t h e o r e t i c a l v a l u e s show a d i f f e r e n c e o f l e s s t h a n
5 p e r c e n t i n a l l c a s e s . A l s o t h e c o m p l e t e f o r m s o f t h e e x p e r i m e n t a l l y o b t a i n e d i n c i d e n t , r e f l e c t e d a n d t r a n s m i t t e d w a v e s w e r e c o m p a r e d w i t h t h e c o r r e s p o n d i n g c u r v e s a c c o r d i n g
t o t h e o n e - d i m e n s i o n a l m o d e l . The a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t c l e a r l y s u p p o r t s t h e v a l i d i t y o f t h e o n e -
d i m e n s i o n a l m o d e l a n d c o n f i r m s t h e r e s u l t s o b t a i n e d i n p a p e r D.
ACKNOWLEDGEMENTS
The r e s e a r c h w o r k p r e s e n t e d i n t h i s t h e s i s h a s b e e n c a r r i e d
o u t d u r i n g t h e y e a r s 1 9 7 6 - 1 3 7 9 a t t h e D i v i s i o n o f S o l i d M e c h a n i c s , U n i v e r s i t y o f Luleå, Luleå.
I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e t o P r o f e s s o r B e n g t L u n d b e r g f o r p r o p o s i n g t h e s u b j e c t a n d f o r g u i d a n c e d u r i n g t h e c o u r s e o f t h i s w o r k .
The a s s i s t a n c e r e c e i v e d f r o m Mr. B r u n o N i l s s o n f o r t h e e x p e r i m e n t a l p a r t s i n p a p e r s A a n d E was i n v a l u a b l e a n d i s h i g h l y a p p r e c i a t e d . I a l s o t h a n k my c o l l e a g u e s a t t h e D i v i s i o n o f S o l i d M e c h a n i c s f o r t h e i r i n t e r e s t , a n d my w i f e N a r e s h f o r h e r s u p p o r t d u r i n g t h e c o u r s e o f t h i s s t u d y .
g
The f i n a n c i a l s u p p o r t g i v e n by T e c h n i c a l D e v e l o p m e n t ( S T U ) a n d p a p e r s D a n d E i s h i g h l y a p p r e c
t h e N a t i o n a l S w e d i s h B o a r d f o r S a n d v i k AB f o r w o r k i n t h e i a t e d .
REFERENCES
1 . T. P ö s o h l a n d D e r S t o s s , Handbuch der Physik, Band B, B e r l i n , ( 1 9 2 8 ) , p . 5 0 1 .
2. B. de Sa i n t - V e n a n t , J o u r n a l de M a t h e r n a t i q u e s , 2e s e r i e 1 2 , 2 3 7 - 3 7 6 , ( 1 8 6 7 ) .
3. B. de S a i n t - V e n a n t , C o m p t e s R e n d u s , 6 6 , ( 1 8 B 8 ) , p . 6 5 0 - 6 5 3 .
4. L. H. D o n n e i l , " L o n g i t u d i n a l w a v e t r a n s m i s s i o n a n d i m p a c t " , T r a n s a c t i o n s o f t h e ASME, 5>2, 1 5 3 - 1 67 , ( 1 9 3 0 ) .
5. W. G o l d s m i t h , Impact, E d w a r d A r n o l d , L o n d o n , ( 1 9 6 0 ) .
6. H. C. F i s c h e r , On Longitudinal Impact 1-V1, T h e Hague M a r t i n u s N i j h o f f , ( 1 9 6 0 ) .
7. R. D. M i n d l i n a n d G. H e r r m a n n , " A o n e - d i m e n s i o n a l t h e o r y o f c o m p r e s s i o n a l w a v e s i n an e l a s t i c r o d ", P r o c . 1 s t . U.S.
N a t . C o n g r e s s A p p l . M e c h . , C h i c a g o , 1 8 7 - 1 9 1 , ( 1 9 5 1 ) .
8. L. P o c h h a m m e r , " U e b e r F o r t p 1 a n z u n g s g e s c h w i n d i g k e i t e n k l e i n e r S c h w i n g u n g e n i n e i n e m u n b e g r e n z t e n i s o t r o p e n K r e i s z y l i n d e r " , J . f . r e i n e u. a n g e w . M a t h . C r e l l e , 8 1 , ( 1 8 7 6 ) .
9. P. K. D u t t a , " The d e t e r m i n a t i o n o f s t r e s s w a v e f o r m s p r o d u c e d by p e r c u s s i v e d r i l l p i s t o n s o f v a r i o u s g e o m e t r i c a l d e s i g n s " , I n t . J . R o c k Mech. M i n . S e i . , 5, 5 0 1 - 5 1 8 , ( 1 9 6 8 ) .
1 0 . T. H a y a s h i , Y. F u z i m a r a a n d K. M o r i s a w a , " I m p a c t s t r e s s w a v e s i n c o m p o s i t e s t r u c t u r e s " , P r o c e e d i n g s o f t h e F i r s t I n t e r n a t i o n a l C o n f e r e n c e on S t r u c t u r a l M e c h a n i c s i n R e a c t o r T e c h n o l o g y , B e r l i n , S e p t . 2 0 - 2 4 , ( 1 9 7 1 ) .
1 o
K. T a n a k a a n d T. K u r o k a w a , " S t r e s s w a v e p r o p a g a t i o n i n a
b a r o f v a r i a b l e c r o s s - s e c t i o n " , B u l l e t i n o f t h e JSME J_6 ( 9 3 ) , 4 8 5 - 4 9 1 , ( 1 9 7 3 ) .
Z. W a s i u t y n s k i a n d A. B r a n d t , " The p r e s e n t s t a t e o f k n o w l e d g e i n t h e f i e l d s o f o p t i m u m d e s i g n o f s t r u c t u r e s " , A p p l i e d M e c h a n i c s R e v i e w s , j J 3 , 3 4 4 - 3 5 0 , ( 1 9 6 3 ) .
F. I . N i o r d s o n a n d P. P e d e r s e n , " A r e v i e w o f o p t i m a l
s t r u c t u r a l d e s i g n " , P r o c . 1 3 t h . I n t . C o n g r . T h . A p p l . M e c h . , Moscow, 2 6 4 - 2 7 8 , ( 1 9 7 3 ) .
F. I . N i o r d s o n , " A m e t h o d f o r s o l v i n g i n v e r s e e i g e n v a l u e p r o b l e m s " , Recent Progress in Applied Mechanics, The Folke Odquist Volume, W i l e y , New y o r k , ( 1 9 6 8 ) .
W. P r ä g e r a n d J . E. T a y l o r , " P r o b l e m s o f o p t i m a l s t r u c t u r a l d e s i g n " , ASME W i n t e r A n n u a l M e e t i n g , P a p e r 67-WA/APM-29, P i t t s b u r g h , P a . , N o v . ( 1 9 6 7 ) .
J . E. T a y l o r , " M i n i m u m mass b a r f o r a x i a l v i b r a t i o n a t s p e c i f i n a t u r a l f r e q u e n c y " , AIAA J . 5 ( 1 0 ) , 1 9 1 1 - 1 9 1 3 , ( 1 9 6 7 ) .
L. 3. I c e r m a n , " O p t i m a l s t r u c t u r a l d e s i g n f o r g i v e n d y n a m i c c o m p l i a n c e " , M. S. t h e s i s , U n i v . C a l i f o r n i a , San D i e g o , ( 1 9 6 8 )
R. M. B r a c h , " M i n i m u m d y n a m i c r e s p o n s e f o r a c l a s s o f s i m p l y s u p p o r t e d beam s h a p e s " , I n t . 3. Mech. S e i . 1 0 , 4 2 9 - 4 3 9 , ( 1 9 6 7 )
E. E i s n e r , " C o m p l e t e s o l u t i o n s o f t h e W e b s t e r h o r n e q u a t i o n "
The J o u r n a l o f A c o u s t i c a l S o c i e t y o f A m e r i c a , 4j2_ ( 4 ) , 1 1 2 6 - 1 1 4 6 , ( 1 9 6 7 ) .
B. L u n d b e r g , " Some b a s i c p r o b l e m s i n p e r c u s s i v e r o o k
d e s t r u c t i o n " , D o c t o r a l t h e s i s a t C h a l m e r s U n i v . o f T e c h n o l o g y , G ö t e b e r g , ( 1 9 7 1 ) .
11
B. L u n d b e r g a n d M. L e s s e r , " Gn imp a c t o r s y n t h e s i s " , J . Soun V i b . 56 ( 1 h 5 - 1 4 , [ 1 9 7 8 ) .
R. G u p t a , " E l a s t i c i m p a c t b e t w e e n a f i n i t e c o n i c a l r o d and a l o n g c y l i n d r i c a l o n e : A t h e o r e t i c a l s t u d y " , U n i v e r s i t y o f L u l e å , T e c h n i c a l R e p o r t 1 9 7 9 : 0 3 T.
R. G u p t a , " T r a n s m i s s i o n o f a r e c t a n g u l a r e l a s t i c wave t h r o u g h a d r i l l r o d j o i n t " , U n i v e r s i t y o f L u l e å , T e c h n i c a l R e p o r t 1 9 7 9 : 0 2 T.
Journal of Sound and Vibration (1978) 60(4), 555-563
ELASTIC I M P A C T B E T W E E N A F I N I T E C O N I C A L R O D A N D A L O N G C Y L I N D R I C A L R O D
R . B . G U P T A
Department of Mechanical Engineering, University of Luleå, Luleå, Sweden
A N D
L . N I L S S O N
Department of Structural Mechanics, Chalmers University of Technology, Gothenburg, Sweden
(Received 13 February 1978, and in revised form 27 May 1978)
Journal of Sound and Vibration (1978) 60(4), 555-563
ELASTIC I M P A C T B E T W E E N A F I N I T E C O N I C A L R O D A N D A L O N G C Y L I N D R I C A L R O D
R . B . G U P T A
Department of Mechanical Engineering, University of Luleå, Luleå, Sweden
A N D
L . N I L S S O N
Department of Structural Mechanics, Chalmers University of Technology, Gothenburg, Sweden
(Received 13 February 1978, and in revised form 21 May 1978)
Elastic impact between a truncated finite conical r o d and a long cylindrical r o d is studied (i) experimentally, ( i i ) analytically, by using one-dimensional wave theory t o o b t a i n a closed-form solution, and ( i i i ) numerically, by using a three-dimensional axisymmetric finite element model. The results are compared f o r cones o f different lengths but w i t h the same end diameters. The agreement between the results f r o m studies (i) and ( i i i ) is very good i n general. As expected, the deviation o f the results o f study (Li) f r o m those o f studies (i) and (iii) becomes increasingly apparent as the slenderness o f the cones decreases.
1. I N T R O D U C T I O N
T h e s t u d y o f w a v e p r o p a g a t i o n i n c o n i c a l rods is o f interest since this g e o m e t r i c a l f o r m is o f t e n m e t i n t e c h n o l o g y . A considerable n u m b e r o f results have been p u b l i s h e d f r o m t h e o r e t i - c a l as w e l l as e x p e r i m e n t a l i n v e s t i g a t i o n s o f elastic waves i n c y l i n d r i c a l r o d s w i t h c i r c u l a r cross-section. T h e s tudy o f waves i n c o n i c a l r o d s , h o w e v e r , has l a r g e l y been r e s t r i c t e d t o o n e - d i m e n s i o n a l t h e o r e t i c a l investigations. T h i s is a consequence o f the d i f f i c u l t i e s i n v o l v e d i n t h e a n a l y t i c a l t r e a t m e n t o f the general t h r e e - d i m e n s i o n a l case.
L a n d o n a n d Q u i n n e y [1] o b t a i n e d a t r a v e l l i n g wave s o l u t i o n f o r a n i n f i n i t e cone, nearly fifty years ago. T h e y also e m p l o y e d the H o p k i n s o n b a r m e t h o d f o r o b s e r v a t i o n o f the pulse.
K e n n e r a n d G o l d s m i t h [2] have also c a r r i e d o u t e x p e r i m e n t a l a n d o n e - d i m e n s i o n a l a n a l y t i c a l i n v e s t i g a t i o n s . N o paper has been f o u n d c o n c e r n i n g t h r e e - d i m e n s i o n a l a x i s y m m e t r i c finite element s o l u t i o n o f the p r o b l e m o f i m p a c t between c o n i c a l rods a n d c y l i n d r i c a l rods.
G o u d r e a u [ 3 ] , h o w e v e r , used the finite element m e t h o d ( F E M ) i n a s t u d y o f stress wave p r o p a g a t i o n i n s h o r t c y l i n d r i c a l rods w h i c h i m p a c t o n a r i g i d surface. H u g h e s et al. [4] give a finite element f o r m u l a t i o n o f general i m p a c t p r o b l e m s . A m o n g o t h e r p r o b l e m s , one- d i m e n s i o n a l i m p a c t o f rods was s t u d i e d . Recently R a m a m u r t i a n d R a m a n a m u r t i [5] have p r o p o s e d a finite element m e t h o d f o r s t u d y i n g i m p a c t o n s h o r t l e n g t h bars.
T h e present i n v e s t i g a t i o n consists o f ( i ) e x p e r i m e n t a l , ( i i ) o n e - d i m e n s i o n a l a n a l y t i c a l , a n d ( i i i ) t h r e e - d i m e n s i o n a l n u m e r i c a l ( F E M ) studies o f the p r o b l e m o f i m p a c t b e t w e e n a c y l i n d r i - cal r o d a n d a finite t r u n c a t e d c o n i c a l r o d . Results o f the three p a r t s are c o m p a r e d . A x i a l s t r a i n at a l o c a t i o n close t o t h e i m p a c t e n d o f the c y l i n d r i c a l r o d is used as a basis f o r the c o m p a r i s o n . T o o b t a i n increasingly t h r e e - d i m e n s i o n a l b e h a v i o u r , c o n i c a l r o d s w i t h h a l f apex-angles o f 5 ° , 1 0 ° , 15° a n d 2 5 ° were investigated.
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F o r increased c o n i c i t y o f the r o d , a n increased need f o r e m p l o y i n g a t h r e e - d i m e n s i o n a l m o d e l is d e m o n s t r a t e d . A x i s y m m e t r i c finite element analysis results show excellent agreement w i t h those o f the e x p e r i m e n t . A s expected, o n e - d i m e n s i o n a l analysis leads t o accurate results o n l y f o r slender cones.
2. E X P E R I M E N T
F i g u r e 1 shows a schematic d i a g r a m o f the e x p e r i m e n t a l set-up e m p l o y e d f o r m e a s u r i n g the a x i a l surface s t r a i n e at a p o s i t i o n P, 40 m m a w a y f r o m t h e i m p a c t e n d o f the c y l i n d r i c a l r o d . T h e c y l i n d r i c a l r o d was d r o p p e d v e r t i c a l l y d o w n t o i m p a c t against a cone w h i c h was resting u p o n a s o f t a n d s p o n g y m a t e r i a l A x i a l s t r a i n £ was measured b y u s i n g a p a i r o f s t r a i n gauges ( G , G ' ) m o u n t e d d i a m e t r i c a l l y o p p o s i t e o n the r o d . T h e gauges ( G , G ' ) were connected t o o p p o s i t e branches o f the W h e a t s t o n e b r i d g e W B , such t h a t c o n t r i b u t i o n f r o m b e n d i n g was cancelled (the s u m o f t h e t w o strains was r e c o r d e d ) . T w o m o r e pairs o f s t r a i n gauges ( G , , G | ) a n d ( G „ , Gn) were e m p l o y e d at p o s i t i o n Q (see F i g u r e 1) i n o r d e r t o m o n i t o r the m a x i m u m b e n d i n g s t r a i n eb. ( G | , G | ) f o r m e d n e i g h b o u r i n g branches o f the W h e a t s t o n e b r i d g e W B I , i n o r d e r t o measure the b e n d i n g s t r a i n et f r o m ( G „ G j ) . S i m i l a r l y b r i d g e W B I I measures b e n d i n g s t r a i n s „ f r o m ( G „ , G'u). T h e m a x i m u m b e n d i n g s t r a i n at Q is t h e n sb = (sj + eu)L'2. T h e r a t i o o f eb a n d e p r o v i d e s a measure o f the q u a l i t y o f the i m p a c t . G o o d i m p a c t was achieved b y r e q u i r i n g | ee/e J 1.
I n order t o o b t a i n i n c r e a s i n g l y t h r e e - d i m e n s i o n a l b e h a v i o u r , f o u r cones w i t h h a l f apex- angles 0 o f 5 ° , 10°, 15°, 2 5 ° were chosen. T h e t w o end diameters d= 8 m m (same as t h e c y l i n d r i c a l r o d d i a m e t e r ) a n d D = 32 m m were e q u a l f o r a l l o f the cones. T h e cones a n d the
Figure 1. Schematic diagram of the experimental set-up for measuring the strains e and ab.
I M P A C T B E T W E E N C O N E A N D C Y L I N D E R 557
c y l i n d r i c a l r o d were m a d e o f steel w i t h Y o u n g ' s m o d u l u s 208 G P a , Poisson's r a t i o 0'3 a n d w a v e speed (c = \ / E / p ) 5080 m/s. L i n e a r l y elastic b e h a v i o u r a n d s m a l l d e f o r m a t i o n s were g u a r a n t e e d t h r o u g h s m a l l i m p a c t velocities. T h e d r o p h e i g h t h was 51 m m w h i c h c o r r e s p o n d s t o a n i m p a c t v e l o c i t y V = yjlgh o f 1 -00 m / s . T h e i m p a c t i n g faces o f a l l the r o d s were s l i g h t l y r o u n d e d near the edges i n o r d e r t o m a k e a d j u s t m e n t s less c r i t i c a l . T h e c y l i n d r i c a l r o d used w a s 2 m l o n g a n d was g u i d e d b y 3 n y l o n bearings p o s i t i o n e d at respectively 150, 750 a n d 1350 m m f r o m the i m p a c t end.
3. T H E O R Y 3 . 1 . O N E - D I M E N S I O N A L M O D E L
A t t h e m o m e n t o f i m p a c t , t = 0, the i m p a c t s i t u a t i o n is as s h o w n i n F i g u r e 2. T h e c o n i c a l i m p a c t o r i m p a c t s w i t h a v e l o c i t y V against t h e s t a t i o n a r y s e m i - i n f i n i t e c y l i n d r i c a l r o d . T h i s i m p a c t s i t u a t i o n is o p p o s i t e t o the e x p e r i m e n t a l s i t u a t i o n described p r e v i o u s l y ( w h e r e the
o — 0 t
d-
Figure 2. The impact situation at the moment of contact / = 0.
cones are k e p t s t a t i o n a r y ) , b u t o n l y the r e l a t i v e i m p a c t v e l o c i t y V influences the results. T h e c o - o r d i n a t e x is chosen w i t h o r i g i n 0 at the i m a g i n e d apex o f the cone. T h e o n e - d i m e n s i o n a l w a v e e q u a t i o n g o v e r n i n g the m o t i o n o f the c o n e - r o d system is
A(x)d2 u/dt2 = c2 S/dx[A(x)du/dx], (1)
w h e r e u = u(x, t) is the displacement a l o n g the x - a x i s , t is t i m e a n d c is the elastic wave speed.
T h e area f u n c t i o n A(x) represents the v a r i a t i o n o f cross-sectional area o f t h e r o d s a l o n g the x - a x i s ,
(
Al, — c c < x < x A [ ( x / x , )2, x , < x < x2 A(x)--A x i a l s t r a i n e(x, t ) a n d p a r t i c l e v e l o c i t y v(x, t ) are related t o the a x i a l d i s p l a c e m e n t uix, t) b y t h e e q u a t i o n s
e(x, t) = du/dx, v(x,t) = du/dr. (2)
A t a s e c t i o n x0, the s t r a i n e ( x0, t) c a n be d e t e r m i n e d f r o m d ' A l e m b e r t ' s s o l u t i o n o f e q u a t i o n s (1) a n d (2) a l o n g w i t h the i n i t i a l c o n d i t i o n s
e ( x , 0 ) = 0
K ( X , 0 )
x < x2, fO, x < x ,
\ - V , x , < x < x2
a n d t h e b o u n d a r y c o n d i t i o n s at t h e t r a c t i o n f r e e e n d x = x2 e ( x2, / ) = 0, ' > 0 .
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C l o s e d f o r m s o l u t i o n s f o r 0 < t < 6(x2 - x , ) / c are p l o t t e d together w i t h t h e n u m e r i c a l a n d e x p e r i m e n t a l results i n F i g u r e s 5 a n d 6 o f section 4. T h e d i s c o n t i n u i t i e s i n the o n e - d i m e n s i o n a l s o l u t i o n are due t o the a b r u p t a r r i v a l s o f r e f l e c t i o n s f r o m the free e n d o f the cone.
3.2. T H R E E - D I M E N S I O N A L ( A X I S Y M M E T R I C ) F I N I T E E L E M E N T M O D E L
C l o s e d f o r m s o l u t i o n s are i n general n o t a v a i l a b l e f o r t h r e e - d i m e n s i o n a l e l a s t o - d y n a m i c p r o b l e m s . T h e r e f o r e , t h e present i m p a c t p r o b l e m is solved n u m e r i c a l l y b y u s i n g t h e finite element m e t h o d ( F E M ) .
A s t a n d a r d d i s p l a c e m e n t f o r m u l a t i o n o f the finite element m e t h o d is used. O n l y a b r i e f o u t l i n e o f t h e d e r i v a t i o n o f the finite element e q u a t i o n s o f m o t i o n is g i v e n here. M o r e details can be f o u n d i n t e x t b o o k s o n the subject (e.g., t h a t o f Z i e n k i e w i c z [ 6 ] ) .
T h u s , t h e p i s t o n - r o d system is m o d e l l e d b y a n assembly o f f o u r - n o d e q u a d r i l a t e r a l r i n g elements. W i t h i n each element, t h e displacements are i n t e r p o l a t e d f r o m t h e i r values a t t h e n o d a l p o i n t s o f the element. B i l i n e a r p o l y n o m i a l s i n the n a t u r a l c o - o r d i n a t e s o f the element are used f o r t h i s i n t e r p o l a t i o n .
F u l l c o n t i n u i t y at c o m m o n n o d a l p o i n t s o f n e i g h b o u r i n g elements requires t h a t the associated element n o d a l displacements are e q u a l : i.e., t h e y are i d e n t i f i e d w i t h i d e n t i c a l c o m p o n e n t s i n the g l o b a l n o d a l displacement v e c t o r u „ ( r ) . T o a l l o w f o r a f r i c t i o n l e s s i n t e r - face, t h i s c o n t i n u i t y r e q u i r e m e n t can be relaxed f o r the t a n g e n t i a l c o m p o n e n t s o f t h e n o d a l displacements. I n the present a p p l i c a t i o n , c o n t a c t between the p i s t o n a n d r o d is m a i n t a i n e d f r o m the t i m e o f i m p a c t . T h e r e f o r e , i t is n a t u r a l t o consider the p i s t o n - r o d system as one s t r u c t u r a l u n i t , a n d thus the n o d a l c o n t i n u i t y r e q u i r e m e n t s m u s t h o l d also f o r t h e c o n t a c t i n g nodes.
A f t e r e s t a b l i s h i n g k i n e m a t i c a l a n d c o n s t i t u t i v e r e l a t i o n s , a p p l i c a t i o n o f t h e p r i n c i p l e o f v i r t u a l w o r k yields the discrete equations o f m o t i o n o f t h e f i n i t e element system:
[M]d2Urt/dt2 + [S]uN = 0. (3)
Since a l l b o u n d a r i e s are f r e e f r o m t r a c t i o n s the r e s u l t a n t s t r u c t u r a l n o d a l l o a d v e c t o r is a zero v e c t o r . T h e s t r u c t u r a l consistent mass m a t r i x [M] a n d the s t r u c t u r a l stiffness m a t r i x [S] consist o f c o n t r i b u t i o n s f r o m element masses a n d stiffnesses. F o r the s o l u t i o n o f the e q u a t i o n s o f m o t i o n (3), i n i t i a l c o n d i t i o n s are needed. T h e y are ( w i t h k = 1, 2, . . . , TV)
«*(0) = 0, ( 4 ) f— V i n the a x i a l d i r e c t i o n o f p i s t o n n o d a l p o i n t s )
* j o otherwise
T h r o u g h t h e i m p a c t a steep stress wave is created. S m a l l element sizes are r e q u i r e d t o a v o i d s m o o t h i n g effects o n the stress f r o n t . F u r t h e r m o r e , the d i s c r e t i z a t i o n i n t i m e m u s t be m a d e w i t h c o n s i d e r a t i o n t o t h e h i g h f r e q u e n c y c o n t e n t o f the excited modes. T h i s calls f o r s m a l l t i m e steps i n the t i m e i n t e g r a t i o n p r o c e d u r e .
Based o n the p r e v i o u s requirements, a v e l o c i t y f o r m u l a t e d c e n t r a l d i f f e r e n c e i n t e g r a t i o n o f e q u a t i o n (3) is c h o s e n :
Bu%+"2/dt = 3 u ; - "2/ a r - At [M]-' [S]u"N, (5)
u "N + 1 = u ; + Atdutfi,2/dt. (6)
T h e subscripts i n d i c a t e discrete times. I f the c e n t r a l d i f f e r e n c e scheme is used i n c o n j u n c t i o n w i t h a d i a g o n a l ( l u m p e d ) mass m a t r i x , the i n v e r s i o n o f [M] is t r i v i a l . F u r t h e r m o r e , the p r o d u c t [ S ] u " c a n be o b t a i n e d as a n assembly o f element c o n t r i b u t i o n s , i n w h i c h case n o s t r u c t u r a l stiffness m a t r i x has t o be established. W i t h i m p l e m e n t a t i o n o f these p r o c e d u r e s .
I M P A C T B E T W E E N C O N E A N D C Y L I N D E R 559 the d e m a n d f o r c o m p u t e r storage is m i n i m i z e d . I n t h e present study, t h e mass l u m p i n g scheme p r o p o s e d b y H i n t o n et al. [7] is used. T h u s , the « t h d i a g o n a l t e r m is o b t a i n e d f r o m
Mn = Mn„ » 7 ( 2 M„„), (7)
jv
w h e r e M„„ is the n t h d i a g o n a l t e r m o f the consistent mass m a t r i x , a n d W is t w i c e t h e t o t a l w e i g h t o f t h e system.
T h e c e n t r a l d i f f e r e n c e scheme is o n l y c o n d i t i o n a l l y s t a b l e: i.e., the a c t u a l t i m e step A t m u s t be less t h a n a c r i t i c a l value AtCT. T h i s c r i t i c a l t i m e step c a n be estimated as
At„ = yAx"""/cp, (8)
where Ax™1" is the smallest distance between t w o a d j a c e n t n o d a l p o i n t s , c„ is the speed o f t h e d i l a t a t i o n wave, a n d y is a p o s i t i v e f a c t o r less t h a n u n i t y ( c f . t h e paper b y B e l y t s c h k o et al.
[8]). I n the present a p p l i c a t i o n t h e c r i t i c a l t i m e step does n o t y i e l d a severe r e s t r i c t i o n , since the h i g h f r e q u e n c y c o n t e n t o f the response i n itself d e m a n d s s m a l l t i m e steps. U s u a l l y , a c o n s t a n t t i m e step c o r r e s p o n d i n g t o y = 0-8 is chosen ( c f . T a b l e 1).
T A B L E 1
Data for element meshes and time steps
Cone N o . o f
half-apex elements in
angle, 0 N o . o f piston N o . o f rod radial Time-step Contact
(degrees) elements, N2 elements, TV] direction, N3 (ja) c o n d i t i o n
5 36 18 1 0-238 Point contact
10 IS 84 1 0-238 Point contact
15 18 88 1 0 1 6 Point contact
25 18 100 3 0 0 9 1 element
frictionless
25 18 100 3 0-09 1 element
f u l l f r i c t i o n
25 18 100 3 0 0 9 2 element
f r i c t i o n less
25 18 100 3 0-09 2 element
f u l l f r i c t i o n
25 18 86 1 0 0 9 point contact
E q u a t i o n (6) yields a step-by-step s o l u t i o n f o r t h e n o d a l displacements as f u n c t i o n s o f t i m e . W i t h these displacements, the strains are o b t a i n e d t h r o u g h the k i n e m a t i c a l r e l a t i o n s . Since t h e displacements a n d strains are s o l u t i o n s o f t h e discrete e q u a t i o n s o f m o t i o n (3) a n d n o t o f the c o n t i n u u m p r o b l e m , they w i l l c o n t a i n s p u r i o u s o s c i l l a t i o n s a n d d i s p e r s i o n effects.
T h e frequencies o f the s p u r i o u s o s c i l l a t i o n s are inversely p r o p o r t i o n a l t o t h e element sizes.
T h u s , i f the element sizes are increased these frequencies decrease. I f a u n i f o r m element mesh is used, the s p u r i o u s o s c i l l a t i o n f r e q u e n c y corresponds t o t h e highest f r e q u e n c y o f the system.
I t has been c u s t o m a r y t o compensate f o r the spurious o s c i l l a t i o n b y e m p l o y i n g some f o r m o f d a m p i n g . A n ideal d a m p i n g s h o u l d o n l y a f f e c t the s p u r i o u s o s c i l l a t i o n s . I n r e a l i t y , n o such i d e a l d a m p i n g exists, a n d t h e r e f o r e also l o w e r frequencies are a f f e c t e d . T h e r e f o r e , n o a t t e m p t is m a d e i n t h i s s t u d y t o exclude the s p u r i o u s o s c i l l a t i o n s . T h e d i s p e r s i o n effect, o n t h e o t h e r h a n d , w i l l m a n i f e s t itself as a d i f f e r e n c e i n phase velocities o f d i f f e r e n t modes o f the
AB
560 R . B . G U P T A A N D L . N I L S S O N
response. T h u s , w i t h increasing m o d e f r e q u e n c y the associated phase v e l o c i t y w i l l decrease.
T h e d i s p e r s i o n effect is a n i n h e r e n t p r o p e r t y o f the f i n i t e elements, a n d i t c a n o n l y be reduced b y t h e use o f h i g h f r e q u e n c y d a m p i n g o r f i l t e r i n g techniques. A g a i n , these techniques also a f f e c t l o w e r modes o f the response, a n d are n o t used i n the present s t u d y .
F i g u r e 3 shows a l a y o u t f o r the finite element mesh o f the p i s t o n - r o d system. A l s o d e f i n e d i n t h e figure are the n u m b e r o f elements i n the a x i a l d i r e c t i o n A/, ( c y l i n d e r ) a n d N2 (cone) a n d the n u m b e r o f elements i n the r a d i a l d i r e c t i o n N3. T a b l e 1 shows the e l e m e n t meshes a n d t i m e steps f o r the v a r i o u s finite element m o d e l s .
Figure 3. Elements for the finite element model.
( a ) ( b ) ( c ) Figure 4. Various contact conditions, (a) Point contact; (b) one element contact; (c) two element contact.
I t s h o u l d be emphasized t h a t no convergence study is made i n t h e s p a t i a l o r t e m p o r a l d o m a i n s . T h e s t a b i l i t y r e s t r i c t i o n o f s m a l l t i m e steps ensures d i m i n i s h i n g t r u n c a t i o n errors i n the t i m e i n t e g r a t i o n .
I n o r d e r t o m i n i m i z e the c o m p u t e r e f f o r t , linear c o n d i t i o n s between the p i s t o n a n d the r o d have been assumed. T h e r e f o r e , the a c t u a l c o n t a c t areas have t o be p r e d e t e r m i n e d . W i t h one r a d i a l element the t w o possibilities o f p o i n t c o n t a c t o r f u l l c o n t a c t exist. W i t h three r a d i a l elements also the possibilities o f one a n d t w o element c o n t a c t m u s t be a d d e d . T h e i m p a c t faces o f the pistons were s l i g h t l y r o u n d e d , a n d the a c t u a l c o n t a c t areas i n the e x p e r i m e n t a l s t u d y are believed t o be closer t o p o i n t c o n t a c t t h a n t o f u l l c o n t a c t . T h e p o s s i b i l i t y o f f u l l c o n t a c t has t h e r e f o r e been e x c l u d e d i n the n u m e r i c a l s t u d y . T h u s p o i n t c o n t a c t has been assumed f o r the m o d e l s w i t h one element i n the r a d i a l d i r e c t i o n . F o r t h e m o d e l s c o n s i s t i n g o f three elements i n the r a d i a l d i r e c t i o n the p o s s i b i l i t y o f a f r i c t i o n l e s s c o n t a c t was also i n v e s t i g a t e d . T o m a i n t a i n the l i n e a r i t y o f the p r o b l e m o n l y the t w o extreme possibilities o f f u l l f r i c t i o n ( n o s l i d i n g ) o r n o f r i c t i o n ( f r e e s l i d i n g ) exist. T h e f u l l f r i c t i o n case is m o d e l l e d t h r o u g h f u l l n o d a l c o n t i n u i t y , w h i l e t h e c o n t i n u i t y r e q u i r e m e n t is relaxed f o r t a n g e n t i a l displacements i n the case o f f r i c t i o n l e s s c o n t a c t . T a b l e 1 shows the d i f f e r e n t c o n t a c t c o n d i t i o n s w h i c h have been used. T h e c o r r e s p o n d i n g c o n t a c t meshes, i n the case o f three r a d i a l elements, are s h o w n i n F i g u r e 4.
I M P A C T B E T W E E N C O N E A N D C Y L I N D E R 561
4. R E S U L T S A N D D I S C U S S I O N
T h e a x i a l s t r a i n s i n the r o d 40 m m f r o m the c o n t a c t surface ( F i g u r e 3) was chosen f o r the c o m p a r i s o n o f the 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 results. A l l results are g i v e n i n n o n - d i m e n - sional f o r m s . T h u s , the n o r m a l s t r a i n £ has been n o r m a l i z e d against V/c w h i c h c o r r e s p o n d s t o t w i c e the m a x i m u m s t r a i n o b t a i n e d f o r apex angle 0 ° . T h e t i m e t has been n o r m a l i z e d against the d o u b l e t r a n s i t t i m e 2L/c t h r o u g h the a c t u a l p i s t o n o f length L. F o r t h i s p u r p o s e , t h e speed o f p r o p a g a t i o n c is i n accordance w i t h o n e - d i m e n s i o n a l t h e o r y (c = \ / E / p ) .
D u e t o p r a c t i c a l d i f f i c u l t i e s , the t i m e f o r i n i t i a l c o n t a c t c o u l d n o t be recorded i n the e x p e r i - m e n t s . T h e r e f o r e , this t i m e was estimated i n such a w a y t h a t the t i m e f o r m a x i m u m s t r a i n c o r r e s p o n d s t o the p r e d i c t i o n o f the o n e - d i m e n s i o n a l t h e o r y .
1
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l-D closed form solution
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3 - D F E M solution
Figure 5, (c) 9 = 15°.
Non-dimensional strain vs. non-dimensional time, (a) Cone half apex angle 6=5"; (b) 6= 10:
F i n i t e element results a n d e x p e r i m e n t a l results are i n g o o d agreement f o r pistons w i t h h a l f apex angles 5 ° , 1 0 ° , a n d 1 5 ° (see F i g u r e s 5(a), 5(b) a n d 5(c)). T h e h i g h f r e q u e n c y responses i n the finite element s o l u t i o n s are p a r t l y due t o s p u r i o u s o s c i l l a t i o n s . H o w e v e r , c o n t r i b u t i o n s t o these o s c i l l a t i o n s also o r i g i n a t e f r o m r a d i a l i n e r t i a a n d Poisson effects. T h e scatter i n t h e e x p e r i m e n t a l results was less s i g n i f i c a n t t h a n these effects. F o r the p i s t o n w i t h h a l f apex angle 5 ° , the o n e - d i m e n s i o n a l m o d e l accurately predicts the s t r a i n response (see F i g u r e 5(a)).
A s expected, the discrepancies between o n e - d i m e n s i o n a l results a n d e x p e r i m e n t a l as w e l l as finite e l e m e n t results increase w i t h the apex angle.
I n o r d e r t o o b t a i n s a t i s f a c t o r y results f o r the 2 5 ° cone, m o r e t h a n one element is r e q u i r e d i n the r a d i a l d i r e c t i o n (see T a b l e 1). F u r t h e r m o r e , i t is f o u n d t h a t a p p r o p r i a t e c o n t a c t c o n - d i t i o n s i n the finite element models are essential. A s can be seen f r o m Figures 6(a) t o 6 ( d ) , the area o f c o n t a c t is the m o s t i m p o r t a n t f a c t o r . T h u s the element d i v i s i o n s c o r r e s p o n d i n g t o a one element c o n t a c t c o n d i t i o n give results i n best agreement w i t h the e x p e r i m e n t a l results (see F i g u r e s 6(a) a n d 6 ( b ) ) . A n increase i n c o n t a c t areas gives rise t o discrepancy b e t w e e n
A3
f i n i t e element results a n d e x p e r i m e n t a l results (see F i g u r e s 6(c) a n d 6 ( d ) ) . T h e a m o u n t o f f r i c t i o n i n t h e c o n t a c t zone is f o u n d t o have a n e g l i g i b l e effect o n the a c t u a l s t r a i n (see F i g u r e s 6(a) a n d 6 ( b ) , o r 6(c) a n d 6 ( d ) ) .
A C K N O W L E D G M E N T
T h e a u t h o r s w i s h to t h a n k M r B r u n o N i l s s o n f o r his assistance i n t h e e x p e r i m e n t a l i n v e s t i - g a t i o n a n d Professor Bengt L u n d b e r g f o r suggesting the p r o b l e m a n d f o r several h e l p f u l discussions.
R E F E R E N C E S
1 . J . W . L A N D O N and H . Q U I N N E Y 1 9 2 3 Proceedings of the Royal Society London, A103, 6 2 2 - 6 4 3 . Experiments w i t h the H o p k i n s o n pressure bar.
2 . V . H . K E N N E R and W . G O L D S M I T H 1 9 6 8 Experimental Mechanics 8, 4 4 2 - 4 4 9 . Elastic waves i n truncated cones.
3. G . L . G O U D R E A U 1 9 7 0 Ph.D. Thesis University of California, Berkley. Evaluation o f discrete methods f o r the linear dynamic response o f elastic and viscoelastic solids.
I M P A C T B E T W E E N C O N E A N D C Y L I N D E R 563 4 . T . J . R . H U G H E S , R . L . T A Y L O R , J . L . S A C K M A N , A . C U R N I E R and W . K A N O K N U K U L C H A I 1 9 7 6
Computer Methods in Applied Mechanics and Engineering 8 , 2 4 9 - 2 7 6 . A finite element method f o r a class o f contact-impact problems.
5. V . R A M A M U R T I and P . V . R A M A N A M U R T I 1 9 7 7 Journal of Sound and Vibration 53, 5 2 9 - 5 4 3 . Impact o n short length bars.
6. O. C. Z I E N K I E W I C Z 1 9 7 1 The Finite Element Method in Engineering Science. L o n d o n : M c G r a w - H i l l .
7 . E. H I N T O N , T . R O C K and O. C . Z I E N K I E W I C Z 1 9 7 6 Earthquake Engineering and Structural Dynamics 4, 2 4 5 - 2 4 9 . A note o n mass l u m p i n g and related processes i n the finite element method.
8. T . BELYTSCHKO, N . H O L M E S and R . M U L L E N 1 9 7 5 The American Society of Mechanical Engineers 14, 1 - 2 1 . Explicit integration—stability, solution properties, cost.
PROPAGATION OF E L A S T I C WAVES IN RODS WITH V A R I A B L E CROSS-SECTION
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i s t h e [ v a r i a b l e ) wave s p e e d . Hence A, p, a n d E c a n o n l y i n f l u e n c e t h e r e s u l t s i n t h e c o m b i n a t i o n d i c t a t e d by e q u a t i o n s ( 3 ) a n d [ 4 ) .
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a n d m a k i n g u s e o f t h e n o t a t i o n s u ( y , t ] = u [ x ( y ) , t ) , E ( y ) E ( x [ y ) ) , A ( y ) = A ( x ( y ) ) e t c . , e q u a t i o n ( 1 ) i s r e d u c e d t o t h e f o r m
w h i c h c l e a r l y s h o w s t h a t t h e i n f l u e n c e o f A, p , a n d E i s c o n t a i n e d i n a s i n g l e p a r a m e t e r Z = Äpc = Ä É / c .
x d x ' y y C x J = J
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