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Wave propagation in elastic bars coupled by viscoelstic element Kapoor, Arun Kumar 1969

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WAVE PROPAGATION IN ELASTIC BARS COUPLED BY VISCOELASTIC ELEMENT by ARUN KUMAR KAPOOR B. Tech. Indian I n s t i t u t e of Technology, Kanpur, I n d i a , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT 0 THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department o f Mechanical E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969 In presenting this thesis in p a r t i a l fuK'iment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thes,is for financial gain shall not be allowed without my written permission. ( Arun Kumar Kapoor ) Department of Mechanical Engineering The University of B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 29 , 1969 ABSTRACT The s p l i t Hopkinson p r e s s u r e bar has been used to study the a c o u s t i c i s o l a t i o n t h a t can be achieved by i n s e r t i n g a compliant element i n t o the sound path i n an e l a s t i c system, and t o study the dynamic response of the m a t e r i a l c o m p r i s i n g the compliant element. The specimens v/ere i n s e r t e d between two s t e e l t r a n s d u c e r b a r s . The i n c i d e n t s t r e s s p u l s e , of about 100 microsecond d u r a t i o n , was produced by s t r i k i n g the f r e e end of one of the t r a n s d u c e r bars by a round-headed s t r i k e r bar. The i n c i d e n t p u l s e i n the f i r s t bar and'the t r a n s m i t t e d p u l s e i n the second bar were sensed by s t r a i n gages and d i s p l a y e d on an o s c i l l o s c o p e . The comparison of computed F o u r i e r transforms ( w i t h i n the a c o u s t i c frequency range) of both the i n c i d e n t and t r a n s m i t t e d p u l s e s showed t h a t i n g e n e r a l a g r e a t e r r e d u c t i o n i n t r a n s m i s s i o n of v i b r a t i o n a c r o s s a specimen i s a c h i e v e d : i ) by i n c r e a s i n g the l e n g t h of the specimen; i i ) by u s i n g a m a t e r i a l w i t h a h i g h e r a t t e n u a t i o n c o n s t a n t or h i g h e r v i s c o u s damping; i i i ) by i n c r e a s i n g the impedance mismatch between the spec-imen and the s t e e l t r a n s d u c e r b a r s . A l s o , i t was found t h a t i s o l a t i o n i s g r e a t e r a t h i g h f r e q u e n c i e s than a t low f r e q u e n c i e s . CORRIGENDA Page Line Abstract 13 List of Figures 12 Nomenclature 14 1 6 up 2 9 5 1 5 4,6,7,13 5 1.1 7 9 up 8 12 15 3 38 . 10 up 45 3 up 57 1 64 20 Change audible not acoustic Sine not since Inverse not Ini_verse specimens_ not specimen audible not acoustic (3.1.10) not (3.1.8) c* not c or c (icot) not (icoT) [12] not [6] [13] not [6] - ^ M t + f ) not h(t+£ ) o o lateral not letral specimens_ not specimen Time not strain Volterra-not Voltera TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 2 STATEMENT OF PROBLEM . 2 3 REVIEW 3 3.1 Review of T h e o r e t i c a l Work 3 3.1.1 Wave Propagation i n E l a s t i c and V i s c o e l a s t i c Bars 3 3.1.2 Waveform Produced by Mechanical Impact 5 3.2 Review of Experimental Work 7 4 Theory 9 4.1 P r o p a g a t i o n of L o n g i t u d i n a l S t r e s s P u l s e i n Coupled Bars 9 4.2 D u r a t i o n o f S t r e s s Pulse . 26 4.3 Momentum A n a l y s i s 29 5 E x p e r i m e n t a t i o n and Ins t r u m e n t a t i o n 34 5.1 E x p e r i m e n t a l Setup 34 5.1.1 Mechanical System 34 5.1.2 Transducer and Recording System. . . 34 5.2 D e t a i l s o f Experimental Apparatus 36 5.2.1 T r i g g e r i n g C i r c u i t 36 5.2.2 S t r a i n Gage L o c a t i o n s . . . . . . . 36 5.2.3 S i z e of S t r a i n Gages 38 5.2.4 Diameter of Bars 38 5.2.5 S e l e c t i o n o f S t r i k e r . . . . . . . . 38 5.2.6 S e l e c t i o n o f Specimens 44 Chapter Page 6 R e s u l t s and D i s c u s s i o n . 4 6 6 . 1 Performance of the System and i t s L i m i t a t i o n s 4 6 6 1 . 1 D u r a t i o n of Pulse 4 6 6 . 1 . 2 No D i s t o r t i o n of P ulse 4 6 6 . 1 . 3 E f f e c t of Adhesive . . 5 0 6.2 Checks on Observations and R e s u l t s . . . . 5 0 7 C o n c l u s i o n s and Remarks 5 9 8 Suggestions f o r F u r t h e r Work 6 1 8 . 1 Improvements and Changes i n P r e s e n t E x p e r i m e n t a l Setup 6 1 8.2 Suggestions f o r F u r t h e r A n a l y s i s 6 3 B i b l i o g r a p h y 6 4 Appendix A T e s t Procedure 6 5 Appendix B C a l c u l a t i o n s 6 6 Appendix C Two-Triggering Pulse E l e c t r o n i c C i r c u i t 7 0 Appendix D A d d i t i o n a l F i g u r e s 7 1 LIST OF TABLES T a b l e ' Page 4.1. A S t r e s s - T i m e - D i s t a n c e R e l a t i o n s h i p f o r Three-Bar System 15 5.2. A R e s u l t s o f S t r i k e r Bar S t u d i e s . . . 42 5.2.B M a t e r i a l P r o p e r t i e s 45 LIST OF FIGURES F i g u r e Page 4.1.1 Two-Bar System 10 4.1.2 Three-Bar System 13 4.1.3 o-x-t P l o t : Three-Bar System . . . . 16 4.1.4 I n c i d e n t R e f l e c t e d and T r a n s m i t t e d Waves i n Three-Bar System 18 4.1.5 P r o p a g a t i o n of Rectangular Pulse i n Three-Bar System 20 4.1.6 o-x-t P l o t f o r I n c i d e n t Rectangular Pulse i n Three-Bar System 21 4.1.7 V i s c o e l a s t i c Bar C o u p l i n g Two E l a s t i c Bars 22 4.2.1 F o u r i e r Transform f o r a H a l f Since Wave . . . . 27 5.1.1 Schematic Diagram of Experimental Setup . . . . 35 5.2.1 E r r o r i n Measuring Dynamic S t r a i n w i t h S t r a i n Gages 37 5.2.2 S t r i k i n g Arrangements 40 6.1.1 F o u r i e r Transform of Observed Pulse 47 6.1.2 Setup to check c o n d i t i o n of No D i s t o r t i o n of P u l s e - 48 6.1.3 Record Showing No D i s t o r t i o n of P ulse 48 6.1.4 Setup to Check E f f e c t of Adhesive . 49 6.1.5 Record f o r P a i n t as an Adhesive . . . . . . . . 49 6.1.6 Record f o r Grease as an Adhesive 49 6.1.7 Record without an Adhesive at J u n c t i o n 49 6.1.8 F o u r i e r Transform of P u l s e s Recorded f o r P a i n t . . . . . . . . 51 F i g u r e Page 6.2.1 Stress-Time R e l a t i o n s h i p Record . . 52 i ! 6.2.2 F o u r i e r Transforms and Reduction C o e f f i c i e n t s . . ,54 6.2.3 Average Reduction C o e f f i c i e n t . . 55 6.2.4 Stress-Time R e l a t i o n s h i p Record 57 8.1.1 Suggested Mechanical System . . . . . . 62 8.1.2 H y p o t h e t i c a l Stress-Time R e l a t i o n s h i p 62 C l C i r c u i t f o r Two T r i g g e r i n g Pulses 69 D . l Average Reduction C o e f f i c i e n t f o r 0.25 i n long 7% A n t i m o n i a l Lead 72 D.2 Average Reduction C o e f f i c i e n t f o r 0.50 i n long 7% A n t i m o n i a l Lead -7 3 D.3 Average Reduction C o e f f i c i e n t f o r 0.75 i n long 7% A n t i m o n i a l Lead -. 74 D.4 Average Reduction C o e f f i c i e n t f o r 0.25 i n l o n g 3% A n t i m o n i a l Lead 75 D.5 Average Reduction C o e f f i c i e n t f o r 0.50 i n long 3% A n t i m o n i a l Lead 76 D.6 Average Reduction C o e f f i c i e n t f o r 0.75 i n long 3% A n t i m o n i a l Lead 77 D.7 Average Reduction C o e f f i c i e n t f o r 0.25 i n long Pure Lead 78 D.8 Average Reduction C o e f f i c i e n t f o r 0.50 i n long Pure Lead 79 D.9 Average Reduction C o e f f i c i e n t f o r 0.75 i n long Pure Lead 80 D.10 Average Reduction C o e f f i c i e n t f o r 0.25 i n long Nylon 81 D . l l Average Reduction C o e f f i c i e n t f o r 0.50 i n long Nylon 82 D.12 Average Reduction C o e f f i c i e n t f o r 0.75 i n long Nylon 83 F i g u r e Page D . 1 3 T r a n s m i s s i o n C o e f f i c i e n t For 7% A n t i m o n i a l Lead 86 D . 1 4 T r a n s m i s s i o n C o e f f i c i e n t For 3% A n t i m o n i a l Lead 87 D . 1 5 T r a n s m i s s i o n C o e f f i c i e n t For Pure Lead . . . . . 88 D . 1 6 T r a n s m i s s i o n C o e f f i c i e n t For Nylon . 89 D . 1 7 E x p e r i m e n t a l Setup 90 NOMENCLATURE A (to.) Amplitude i n the Cosine F o u r i e r Transform B(u>) Amplitude i n the Sine F o u r i e r Transform E Young's Modulus E = E , + i E ' 1 , Complex Modulus E 1 Real P a r t of Complex Modulus E 1 1 Imaginary P a r t of Complex Modulus En (to) Energy of a wave over i t s p e r i o d T 0 G (O J ) Modulus i n Complex F o u r i e r Transform [ = ( A 2 ( u ) ) + B 2 ( a ) ) ) 1 / 2 ] I (to) I n t e n s i t y of a Wave of Frequency to L(to) Reduction C o e f f i c i e n t L (w) Average Reduction C o e f f i c i e n t avg 3 F ( u ( t ) ) = U ( O J ) , F o u r i e r Transform of u(t) wi t h r e s p e c t t o t F ^ ( U ( C J ) ) = u ( t ) , I n i v e r s e F o u r i e r Transform o f U ( o o)with r e s p e c t to to T Time D u r a t i o n of Pulse T Q Time P e r i o d of a Wave of Frequency to _ -c E l a s t i c Wave Propagation V e l o c i t y c Phase V e l o c i t y i n L i n e a r V i s c o e l a s t i c M a t e r i a l x 3 f ^ - | ) f ' ( t — ) D e r i v a t i v e with Respect t o Argument (= ) 3 ( t - f ) • .f D e r i v a t i v e w i t h r e s p e c t to Time 1 Length of the F i n i t e Bar i n Three-Bar System p Frequency of H a l f Sine Wave t Time vi P a r t i c l e Displacement u P a r t i c l e V e l o c i t y x D i s t a n c e from O r i g i n z Impedance z D Impedance of S e m i - I n f i n i t e Bar i n Three-Bar System a T(w) T r a n s m i s s i o n C o e f f i c i e n t a t Frequency OJ e S t r a i n <f>, xp Phase D i f f e r e n c e X 1 = ( z Q - z ) / ( z 0 + z) X 2 =2z 0/(z 0+z) v P o i s s o n ' s R a t i o p D e n s i t y a S t r e s s a F ( a ( t ) ) , F o u r i e r Transform of a(t). T Time taken by a S t r e s s Wave to T r a v e l the Length 1 of F i n i t e Bar OJ C i r c u l a r Frequency S u p e r s c r i p t s i , r , t r e p r e s e n t I n c i d e n t , R e f l e c t e d and Tr a n s m i t t e d P u l s e or Wave. S u b s c r i p t s 1, 2, 3 r e f e r t o bars 1, 2 and 3. ACKNOWLEDGEMENT The author wishes to express h i s deep g r a t i t u d e and a p p r e c i a t i o n f o r the i n v a l u a b l e a d v i c e and guidance g i v e n him throughout a l l stages o f i n v e s t i g a t i o n by Dr. H. Ramsey. S i n c e r e thanks are a l s o extended t o Dr. T.E. Siddon f o r h i s comments i n the f i n a l stages of the work. S p e c i a l thanks are due t o the e n t i r e t e c h n i c a l s t a f f o f Mechanical E n g i n e e r i n g Department, f o r t h e i r p r a c t i c a l a d v i c e and h e l p . Support f o r t h i s r e s e a r c h was p r o v i d e d by the N a t i o n a l Research C o u n c i l of Canada, through grant No. NRC67-1685. 1. INTRODUCTION The problem of i s o l a t i o n of b u i l d i n g s and s t r u c t u r e s from a c o u s t i c n o i s e and v i b r a t i o n i s q u i t e o l d . Lead-asbestos pads have been used [8] f o r the l a s t f o r t y - f i v e y e ars i n the f o u n d a t i o n s of b u i l d i n g s to prevent t r a n s m i s s i o n of v i b r a t i o n s from the ground i n t o the b u i l d i n g s . S i m i l a r i l y l e a d a l s o f i n d s a p p l i c a t i o n i n many other areas to c o n t r o l sound and v i b r a t i o n s [ 9 ] . The phenomenon of wave propaga t i o n i n s o l i d bars i s w e l l known. Ex p e r i m e n t a l i n v e s t i g a t i o n s were under way by the b e g i n -n i n g of the 2Cth c e n t u r y . The " s p l i t Hopkinson p r e s s u r e bar" method was i n t r o d u c e d i n mid 20th century to study the behaviour of s m a l l specimens under dynamic l o a d i n g . S e v e r a l methods have been employed i n the p a s t u s i n g e x p l o s i v e charge and mechanical d e v i c e s to produce a p u l s e f o r dynamic compression t e s t i n g of s o l i d s . So f a r s c i e n t i s t s have c o n c e n t r a t e d t h e i r a t t e n t i o n towards dynamic compression t e s t i n g of s m a l l specimen i n order to study the dynamic behaviour of m a t e r i a l s and to determine dynamic s t r e s s - s t r a i n curves, e t c . Setups s i m i l a r t o s p l i t Hopkinson p r e s s u r e bar-method were used. The p r e s e n t work i s aimed at the study of the frequency dependence of m a t e r i a l s , and t h e i r v i b r a t i o n i s o l a t i o n c h a r a c t e r i s t i c s . 2. STATEMENT OF PROBLEM The problem was to design and c o n s t r u c t equipment ade-quate f o r the study of the frequency response of a m a t e r i a l and t o analyse the behaviour of the specimens of d i f f e r e n t l e n g t h s w i t h a view t o r e d u c i n g the t r a n s m i s s i o n of a c o u s t i c v i b r a t i o n s a c r o s s them. The above problem was d i v i d e d i n t o the f o l l o w i n g sub-problems : i ) Design and c o n s t r u c t a system t o produce s t r e s s p u l s e s c o v e r i n g the a c o u s t i c frequency band, with s m a l l s t r e s s amplitudes; i i ) Set up i n s t r u m e n t a t i o n t o r e c o r d the o b s e r v a t i o n s of experiments conducted; i i i ) A nalyse the re c o r d s t o i n v e s t i g a t e the e f f e c t s of spec-imen l e n g t h and m a t e r i a l p r o p e r t i e s on the specimen's c a p a c i t y t o reduce the amplitude of v i b r a t i o n t r a n s -m i t t e d a c r o s s i t . 3. REVIEW 3.1 Review of T h e o r e t i c a l Work 3.1.1 Wave Propagation i n E l a s t i c and V i s c o e l a s t i c Bars In 1807 Thomas Young gave laws governing the pr o p a g a t i o n of e l a s t i c waves i n b a r s . The s o l u t i o n of the governing d i f f e r e n t i a l e q u a t i o n 92u 32u p — ^ = E — 5 - (3 .1 .1 ) 3t^ 3x which i s u = f (t+|) + g ( t - £ ) , (3 .1 .2 ) where c = / ^ " (3 .1 .3 ) X X d e s c r i b e s the two waves f (t+—) and g ( t - — ) p r o p a g a t i n g w i t h v e l -o c i t y c i n the ne g a t i v e and p o s i t i v e x d i r e c t i o n s r e s p e c t i v e l y . L a t e r H. Kolsky [1] analysed one-dimensional wave pr o p a g a t i o n i n v i s c o e l a s t i c s o l i d s by i n t r o d u c i n g the concept of the complex e l a s t i c " m o d u l u s f o r d e s c r i b i n g s i n u s o i d a l wave prop-a g a t i o n . The complex modulus E i s d e f i n e d as the r a t i o of s i n u -s o i d a l l y v a r y i n g s t r e s s a = a0 exp(iwt) (3 .1 .4) 4 t o the s i n u s o i d a l l y v a r y i n g s t r a i n e = e o e x p ( i o j t - i 6 ) . ( 3 . 1 . 5 ) In ( 3 . 1 . 4 ) and ( 3 . 1 . 5 ) r e s p e c t i v e l y , o 0 and e 0 are r e a l , 6 i s the phase l a g between the a p p l i e d s t r e s s a and the r e s u l -t a n t s t r a i n e. The complex modulus i s then g i v e n by E = E'+iE' 1 = - = 2 i e X p ( i 6 ) = E* exp(i<5) ( 3 . 1 . 6 ) £ £ o where E* = [ E , 2 + E ' - 2 ] 1 / 2 ( 3 . 1 . 7 ) and tan 6 = E ' 1 / E 1 . ( 3 . 1 . 8 ) Hence f o r s i n u s o i d a l l y v a r y i n g displacement u = U ( x ) e i w t i t f o l l o w s from ( 3 . 1 . 6 ) t h a t 0 = E f i r • • ( 3 . 1 . 9 ) R e p l a c i n g E i n the equ a t i o n o f motion ( 3 . 1 . 1 ) by E, y i e l d s , f o r s i n u s o i d a l l y v a r y i n g displacement, _ 32u 32u E j = p — - . (.3.1.10) 3x dt* 5 The s o l u t i o n of (3.1.8; f o r a p r o g r e s s i v e s i n u s o i d a l wave of frequency oo/2ir, whose displacement a t the o r i g i n i s g i v e n by u G e x p ( i w t ) , i s u = u 0 exp (-ax+ia) (t-*) ) (3.1.11) c where c = ( E * / p ) 1 / 2 sec | (3.1.12). a = (u/c) tan 6/2 . (3.1.13) The p r o p a g a t i o n of a p u l s e has been t r e a t e d by F o u r i e r s y n t h e s i s and thus the displacement a t the o r i g i n (x=0.0) i s expressed as the F o u r i e r i n t e g r a l u(0,t) = J _ o o A(io)exp(iojT)da) . (3.1.14) T h e r e f o r e , , 0 0 I x u(x,t ) = J A (to) exp (-ax+iu (t-—) ) dw (3.1.15) where A(a>) i s a complex f u n c t i o n of the frequency o>. 3.1.2 Wave form Produced by Mechanical Impact: V a r i o u s authors have analysed the problem of f i n d i n g the l e n g t h or d u r a t i o n of a s t r e s s p u l s e produced i n a long bar by the impact between a s t a t i o n a r y long bar and moving o b j e c t s of d i f f e r e n t geometries. Timoshenko [2] has shown t h a t the l e n g t h of the s t r e s s p u l s e produced by the l o n g i t u d i n a l impact between two bars i s 21^, where 1^ i s the l e n g t h of the s h o r t e r bar. Hertz [3] analysed the problem of a b a l l of r a d i u s r and mass m, s t r i k i n g a s e m i - i n f i n i t e bar wi t h v e l o c i t y v Q . He d e r i v e d the d i f f e r e n t i a l e q uation 3/2 d2a k da ' k + + _ a V ^ = 0 (3.1.16) d t 2 pc 0fi dt m wit h the i n i t i a l c o n d i t i o n s a=0 and v=v D f o r t=0 , where a i s the i n d e n t a t i o n of the impinging b a l l and n i s the area of c r o s s s e c t i o n of the bar. The constant k i n (3.1.16) i s g i v e n by k = 4 - — — r 1 / 2 (3.1.17) (1-v 2) where E, p and v r e f e r to the m a t e r i a l of the bar. The s o l u t i o n of (3.1.16) was found to be i n c l o s e agree-ment wi t h the experimental r e s u l t s [3]. W.A. Prowse [4] a p p l i e d the Hertz theory to analyse the impact of a round-headed bar and found the e x p r e s s i o n ? 1+6+62 26+1 B = =1— + 2 l o g [ ] - 4/ 3 tan 1 — — (3.1.18) /3~ ( 1 - 6 ) 2 / 3 where and a i s the i n d e n t a t i o n at time t , i s the maximum i n d e n t a -t i o n , and v Q i s the v e l o c i t y of the impinging bar. ( . 3.2 Review of Exp e r i m e n t a l Work In 1914 Bertram Hopkinson [5] proposed an experimental t e c h n i q u e , now known as the Hopkinson p r e s s u r e bar method, f o r measuring the p r e s s u r e produced by the impact of a p r o j e c t i l e o r e x p l o s i v e charge on the end of a long bar. The apparatus employed a s h o r t bar, c a l l e d the time p i e c e , stuck by means of grease t o the end of a s t e e l bar, s e v e r a l f e e t l o n g , suspended h o r i z o n t a l l y by means of s t r i n g s . The p r e s s u r e t o be measured was a p p l i e d a t the end of the l o n g s t e e l bar, o p p o s i t e to the end to which the time p i e c e was a t t a c h e d . The time p i e c e flew o f f when the p r e s s u r e wave, r e f l e c t e d as t e n s i o n wave from the end of the time p i e c e , reached the j u n c t i o n of the time p i e c e and the long s t e e l bar. The momenta of time p i e c e s of v a r i o u s lengths were reco r d e d t o c a l c u l a t e the p r e s s u r e p u l s e . In 1949, Kolsky [6] i n t r o d u c e d the " s p l i t Hopkinson . p r e s s u r e bar" apparatus. T h i s employed a d i s c specimen sandwiched between two long s t e e l bars suspended h o r i z o n t a l l y by s t r i n g s . An e x p l o s i v e charge a t the end of the s t e e l a n v i l bar caused a s h o r t d u r a t i o n (~1 micro-sec.) s t r e s s p u l s e i n the a n v i l bar. The p u l s e s u f f e r e d d i s p e r s i o n d u r i n g p r o p a g a t i o n through bar due t o the geometric and mechanical p r o p e r t i e s of the bar. The displacement-time r e l a t i o n s h i p was measured by a p a r a l l e l p l a t e condenser a t the f r e e end of the second bar. The 8 s t r e s s - t i m e and s t r a i n - t i m e r e l a t i o n s h i p s were d e r i v e d by n u m e r i c a l a n a l y s i s of the displacement-time r e l a t i o n s h i p s o b t a i n e d i n the absence and presence of the specimen. Davies and Hunter (1962), [ 6 ] , s t u d i e d the mechanical b e h a v i o r of s o l i d s under dynamic compression t e s t i n g . T h e i r method d i f f e r e d from Kolsky's mainly i n specimen geometry. For K o l s k y ' s specimen, s i n c e a/h=10 (a i s the r a d i u s and h i s the t h i c k n e s s of the specimen), s t r e s s e s i n the t a n g e n t i a l and r a d i a l d i r e c t i o n s arose due to f r i c t i o n a l f o r c e s between the specimen-bar i n t e r f a c e s . A c r i t e r i o n to i n d i c a t e when f r i c t i o n a l f o r c e s can be n e g l e c t e d , was d e r i v e d by H i l l i n 1950, [6]. T h i s c r i t e r i o n r e q u i r e d a/h<<25 i n the experiment conducted by Davies and Hunter, and t h i s c o n d i t i o n was s a t i s f i e d . The c i r c u l a r faces of bars and specimen i n the Davies and Hunter experiments were f i n i s h e d t o o p t i c a l f l a t n e s s . The specimens were h e l d i n p o s i t i o n by "wrung j o i n t s . " In r e c e n t years the use of s t r a i n gages f o r measurements of s t r a i n p u l s e s has been e s t a b l i s h e d . Experiments conducted by R.M. Davies, D.H. Edwards and D.E. Thomas i n 1950 [7] l e d to the c o n c l u s i o n t h a t s t a t i c and dynamic gage f a c t o r s are p r a c -t i c a l l y the same. 4. THEORY In o r d e r t o p r o v i d e some t h e o r e t i c a l b a s i s t o v e r i f y i q u a l i t a t i v e l y the experimental i n v e s t i g a t i o n s , the f o l l o w i n g a n a l y s i s has been developed. 4.1 P r o p a g a t i o n o f a L o n g i t u d i n a l S t r e s s  P u l s e i n Coupled Bars F i r s t of a l l , we c o n s i d e r the case of two s e m i - i n f i n i t e bars o f the same diameter but of d i f f e r e n t m a t e r i a l s coupled t o g e t h e r (Figure 4.1.1). From (3.1.1) and (3.1.2) waves p r o p a g a t i n g i n bars 1 and 2 may be d e s c r i b e d by U l = f l ( t + l 1 ) + 9 l C t - c J 1 ) (4.1.1) u 2 = f 2 ( t + £ ) + g 2 ( t - | ) . (4.1.2) 2 2 T h e r e f o r e p a r t i c l e v e l o c i t y i s gi v e n by u x = f 1 l ( t + | ) + q±'(t-|^) (4.1.3) u 2 = f 2 , ( t + l } + g 2 , ( t ~ l ) * (4.1.4) 2 2 Since the s t r e s s i n e l a s t i c bar i s r e l a t e d t o s t r a i n by Hooke's law, the s t r e s s and o 2 are gi v e n by E, E i o1 = — f 1 ' ( t + | ) - — g ; L ' ( t - | ) (4.1.5) 1 c-^  1 I I fc Barl 1-2 Bar2 Junct ion F i g u r e 4.1.1 Two-Bar System o 11 ct2 = % V ( t + c V - - 9 2 , ( t - i > • ( 4 - 1 - 6 ) 2 1 c 2 2 An incident stress wave given by • i 1 - 0 ( t - | ) i s considered to t r a v e l i n the po s i t i v e x d i r e c t i o n i n bar 1 and to ar r i v e at the junction x=0.0 at time t=0. Therefore, i n (4.1.5) f '(t+- ) = 0 for t<0 (4.1.7) 1 C l and 9 i , ( t - c y = - ^  ° ( t - ^ • ( 4 - i - 8 ) As there i s no wave t r a v e l l i n g i n negative x d i r e c t i o n i n bar 2, f,'(t+*-) =0 (4.1.9) 2 C 2 Continuity of stress and p a r t i c l e v e l o c i t y requires, at x=0.0, °1 = °2 U l = U2 (4.1.10) The boundary conditions (4.1.10) for t>0 applied to (4.1.3), (4.1.4), (4.1.5), (4.1.6) y i e l d the set of equations 12 f 1 ' ( t ) + g 1 ' ( t ) = f 2 ' ( t ) + g 2 ' ( t ) } (4.1.11) E, E, E „ E „ - ^ ' ( t ) - — g ^ (t) = - i - f 2 * ( t ) - - ^ - g 2 ' (t) . S o l v i n g (4.1.11) to g e t h e r with (4.1.8) and (4.1.9) ( f o r x=0.0), f o r g^ 1 and f 2 1 and then s u b s t i t u t i n g these ex-p r e s s i o n s i n (4.1.5) and (4.1.6) y i e l d s o^(x,t) and (^(X/t) g i v e n by the e x p r e s s i o n s , z „ - 2 . a,(x,t) = a ( t - £ )+ -= a(t+ x-) , (4.1.12) C l z2+Z± C l where the r e f l e c t e d wave i n bar 1 i s z„-z, a 1 r ( x , t ) = — a(t+~-) , (4.1.13) z 2 + z l 1 and t r a n s m i t t e d wave i n bar 2. t 2 2 2 x a 2 (x,t) = a 2 ( x , t ) = — - °"(t-f-) (4.1.14) Z 2 + 2 1 2 The impedances z^ and z 2 of bar 1 and bar 2 r e s p e c t i v e l y are g i v e n by Z l = p l ° l = E l / / C l a n d Z2 = P 2 C 2 = E 2 / / c 2 * (4.1.15) The above a n a l y s i s shows t h a t phenomena of r e f l e c t i o n and 0 ->- X E 0 P„ C e 1-2 E P C 2-3 Junctions Co ft C, F i g u r e 4.1.2 Three-Bar System t r a n s m i s s i o n t a k e p l a c e a t t h e j u n c t i o n , w h e n t h e i n c i d e n t w a v e r e a c h e s t h e j u n c t i o n b e t w e e n t w o b a r s . I n a n o t h e r e x a m p l e , a n e l a s t i c b a r o f l e n g t h 1 i s s a n -d w i c h e d b e t w e e n t w o s e m i - i n f i n i t e b a r s ( F i g u r e 4.1.2). T h e s e m i - i n f i n i t e b a r s a r e o f t h e s a m e m a t e r i a l , w i t h p h y s i c a l p r o -p e r t i e s E Q , p 0 a n d c 0 . T h e f i n i t e b a r i s o f d i f f e r e n t m a t e r i a l w i t h p h y s i c a l p r o p e r t i e s E , p a n d c . W e n o w d e f i n e T = 1 / c A 1 = (z 0-z)/(z c+z) . (4.1.16) = 2 z 0 / ( z 0 + z) = A 2 (2-A 2) = ( l - A ^ = 2z/(z Q+z) . A s t r e s s w a v e i n t h e f o r m o f a H e a v i s i d e s t e p f u n c t i o n , g i v e n b y a 1 i ( x , t ) = a ( t - ~ ) = h ( t - £ ) (4.1.17) ± c l c i i s a s s u m e d t o t r a v e l i n b a r 1. B e g i n n i n g a t t i m e t=0 w h e n i t a r r i v e s a t t h e j u n c t i o n b e t w e e n b a r 1 a n d b a r 2, i t i s p a r t l y r e f l e c t e d a n d p a r t l y t r a n s m i t t e d t o b a r 2. T h e w a v e t r a n s m i t t e d t o b a r 2, w h e n i t a r r i v e s a t t h e j u n c t i o n 2-3, i s p a r t l y r e -f l e c t e d a n d p a r t l y t r a n s m i t t e d t o b a r 3. T h u s s t r e s s w a v e s t r a v e l b a c k a n d f o r t h i n b a r 2 a n d g i v e r i s e t o s u c c e s s i v e r e -f l e c t i o n a n d t r a n s m i s s i o n a t j u n c t i o n s 1-2 a n d 2-3. F r o m (4.1.12), (4.1.14) a n d (4.1.16), t h e a n a l y s i s h a s T h e n a n d 4 Bar 1 Bar 2 Bar 3 # A f t - * T -( 3 - A , y £?-A,)A* Aft-st •+*/<:) •1 ^ f 2 ">• J ) A -uA^A ft +i)T - A / O TABLE 4.1.A STRESS-TIME-DISTANCE RELATIONSHIP FOR SUCCESSIVE REFLECTION AND TRANSMISSION FOR AN ELASTIC BAR SANDWICHED BETWEEN TWO SEMI-INFINITE ELASTIC BARS 16 17 been c a r r i e d out to develop the s t r e s s - t i m e - d i s t a n c e r e l a t i o n -s h i p s and r e s u l t s f o r s u c c e s s i v e r e f l e c t i o n and t r a n s m i s s i o n are shown i n the Table (4.1.A) and F i g u r e 4.1.3. The s t r e s s e s i n bars 1 and 3, a ^ ( x , t ) and a^(x,t) r e -s p e c t i v e l y can be o b t a i n e d by s u p e r p o s i t i o n of the s t r e s s e s l i s t e d i n Table ( 4 . 1 .A ) . Thus a 1 ( x r t ) = h ( t - | - ) - X 1 h ( t + | - ) + ( l - A 1 2 ) A 1[h(t-2 T+|-) (4.1.18) + A 2 h ( t - 4 x + — ) + + A-, 2 ( n~ 1 ), ~ ^x Nj_ . 1 c 0 1 h(t-2nx+—)+....] and, a 3 ( x , t ) = (1-A X 2) [h(t-x-|-) + A 1 2 h ( t - 3 x - X - ) + A 1 4 h ( t - 5 T - £ - ) + . . .+ A 2 ( n _ 1 ) h ( t - ( 2 n - l ) T - ~ ) + ] . (4.1.19) 1 C A Then a^(x,t) can be expressed as ' a L ( x , t ) = c ^ 1 ( x , t ) + a x r (x,t) (4T1.20) where (4.1.20) a/u.t) = h ( t - J - ) i s the i n c i d e n t wave t r a v e l l i n g i n the p o s i t i v e x d i r e c t i o n , and o ^ r ( x , t ) i s the r e f l e c t e d wave t r a v e l l i n g i n the n e g a t i v e x d i r e c t i o n i n bar 1 a f t e r time t=0. The s t r e s s c r ^ ^ t ) i n bar 3 i s a s s o c i a t e d w i t h a t r a n s m i t t e d wave a 3 ^ ( x , t ) t r a v e l l i n g i n F i g u r e 4.1.4 I n c i d e n t R e f l e c t e d and T r a n s m i t t e d Waves i n T h r e e - B a r S y s t e m 19 • i r p o s i t i v e x d i r e c t i o n i n bar 3 when t>0. The waves , i r t and ./• CTi a n c ^ °3 a r e p l o t t e d i n F i g u r e 4.1.4, which shows the form of the i n c i d e n t r e f l e c t e d and t r a n s m i t t e d waves. Now.we i n v e s t i g a t e the problem of pr o p a g a t i o n of an i n c i d e n t r e c t a n g u l a r p u l s e of d u r a t i o n T (Figure 4.1.5a) propa-g a t i n g i n the p o s i t i v e x d i r e c t i o n i n bar 1 i n a t h r e e - b a r system (Figure 4.1.2). The r e c t a n g u l a r p u l s e can be expressed as o / ( x , t ) = h ( t - X - ) - h ( t - T - x - ) 1 Cft C« r t Thus (x,t) and ( x , t ) , the r e f l e c t e d and t r a n s m i t t e d waves r e s p e c t i v e l y , can be o b t a i n e d by s u p e r p o s i t i o n of the r e f l e c t e d and t r a n s m i t t e d waves f o r h ( t - — ) and h ( t - T - x / c Q ) u s i n g (4.1.18) o and (4.1.19). The r e s u l t s f o r a p a r t i c u l a r case (T=4T) have been der-i v e d s y s t e m a t i c a l l y i n F i g u r e 4.1.5 and F i g u r e 4.1.6 by the s u p e r p o s i t i o n of waves shown i n F i g u r e 4.1.4 and F i g u r e 4.1.3. The f o l l o w i n g a n a l y s i s has been c a r r i e d out f o r a f i n i t e - v i - s c o l e a s t i c bar sandwiched between two s e m i - i n f i n i t e e l a s t i c bars of the same m a t e r i a l (Figure 4.1.7). The i n c i d e n t , t r a n s -m i t t e d and r e f l e c t e d displacement p u l s e s are giv e n as u 1 1 ( x , t ) = f 3 _ ( t _ | — ) a n d u 1 r ( x , t ) = g 1 ( t + x — ) u 3 ( x , t ) = u 3 t ( x , t ) = f 3 ( t - X - ) (4.1.21) u (x,t) = f (t-p-)+g (t+*-) x x o 1 C Q 20 F i g u r e 4.1.5 P r o p a g a t i o n o f R e c t a n g u l a r P u l s e i n T h r e e - B a r S y s t e m 21 2 2 ( b •) For t co F i g u r e 4.1.7 V i s c o e l a s t i c Bar C o u p l i n g Two E l a s t i c B a rs 23 Fourier transforms are defined as, | F - 1 [ F ( u ) ) ] = f (t) = - i - [ °° F ( c o ) e " i w t du (4.1.-22) /- 00 F [ f ( t ) ] = F(u) = — — J f ( t ) e l w t dt (4.1.23) (-iujPtu) e " i u , t du> . (4.1.24) v „ -°° Therefore, F ( f ( t ) ) = -iuF (u) and F ( f ' ( t ) ) = -o) 2F(u)) The equation for sinusoidal wave propagation i n bar 2 can be written as 32u„ , 32u~ ^7 5^ = 0 , 0 4 X 4 1 (4.1.26) 3x 2 c 3t where -2 E 1 +iE 1' c = for v i s c o e l a s t i c materials. For the spe c i a l case when bar 2 i s e l a s t i c ^ i s replaced by c = / E // p • Applying the Fourier trans-formation to (4.1.26) y i e l d s 2 U 2 ' 1 (u>)+^j U 2 (u) = 0 . (4.1.27) c The solution of (4.1.27) may be written 24 U 2(O J) = P(OJ) cos ~ + Q(oj)sin ~ (4.1.28) The inverse Fourier transformation y i e l d s , u 2 (x,t) r m /2TT J rr> / \ „„„ I D X , . . O J X , - i o j t , [ P ( O J ) C O S — + Q(w)sm — ] e doj At x=0, (4.1.29) u 2 ( 0 , t ) = u 2 ( 0 , t ) /277 P (OJ) e l w * " doj — oo /2TT y r" <-I O J ) P (OJ) e do) (4.1.30) (4.1.31) u 2'(0,t) = • 2TT OJ ^, \ - i o j t , — Q (OJ) e doj (4.1.32) The requirement of continuity of stress and p a r t i c l e v e l o c i t y at the junctions (x=0 and x=l) y i e l d s the following equations: — f ' ( t ) + — g ' (t) = - i -c 0 c 0 /2TT — E Q (OJ) e doj (4.1.34) f 1 , . ( t ) + g 1 ' (t) /2TT ( - i o j ) P ( o j ) e ~ i w t doj (4.1.34) u 2 ( l , t ) = f 3 ' ( t - 5 i ) (4.1.35) f3'(t-i-.= - i /2ir ' O J E R . . . O J I — [ - P (OJ) s i n — Q ( u ) ) c o s ^ ] e " i w t d o j . (4.1.36) 25 Combining and s i m p l i f i n g (4.1.35), (4.1.36) and (4.1.29) y i e l d s E 0 0)1 , ~ / \ • col - — ( - i o o ) [P(u)cos — + Q(o))s.in — ] U)E r T - . / \ • ^1 , ~ / \ (1)1 n — [-P(u)sin — + Q(w)cos — J or i c E i c E [cos ^ s i n ^ ] P(u) + [ s i n ~ . +' cos ^ i ] Q ( u ) = 0 . c c E 0 c c c E Q c (4.1.37) From (4.1.21), i t can be shown t h a t i ° x . t f,'(t-£ ) = a , w ( x , t ) , (4.1.38) c ° hence i n c o n j u n c t i o n w i t h the F o u r i e r t r a n s f o r m of (4.1.35), we o b t a i n the eq u a t i o n io)[P(o))cos — + Q(co) s i n — ] = — o * (co) . (4.1.39) c c E 0 S i m i l a r l y , by (4.1.33) and (4.1.34), i t can be shown t h a t E ° F - i io) P(u)+ cof- Q(co) = 2 a ± (oo) . (4.1.40) c„ c P ( t o ) and Q ( i o ) can be found from (4.1.37) and (4.1.40) t r and hence (x,t) and (x,t) can be computed. Since the v a l u e s f o r c ( c o ) are not a v a i l a b l e r e a d i l y , (4.1.37), (4.1.39) 26 and (4.1.40) can be s o l v e d to g i v e the f o l l o w i n g e q u a t i o n : r 0 , 1 . ! a) 1 , ~ . , p c s u l , , pc ,2 . OJ 1 , pc , 1 s m — + 2 i (—- ) cos — + ( — — ) s i n — = 2 i { — — — ) — . ; C P o C 0 C P O C O C P O C O - t (4.1.41) When a^ 1(x,t) and a . j t ( X / t ) are known, c(o))can be found by solv-i n g (4.1.41). Then, s i n c e E = E'+iE'' = p [ c ( u ) ) 2 the complex modulus of the m a t e r i a l of bar 2 can be found, 4.2 D u r a t i o n o f S t r e s s Pulse The f o l l o w i n g a n a l y s i s has been c a r r i e d out to show the e f f e c t of the d u r a t i o n of a p u l s e on i t s frequency spectrum. The frequency spectrum of a p u l s e a(t) i s expressed by i t s F o u r i e r t r a n s f o r m . a(t) can be w r i t t e n as : t ) = - i n o ( t )  — J [ A ( O J ) C O S ojt + B ( O J ) s i n ojt] doj TT 0 1 TT o f (4.2.1) G(OJ) s i n (ojt+<f>) d u where G(w) = [ A 2 ( O J ) + B 2 ( O J ) ] 1 / 2 (4.2.2) i s the modulus i n the complex F o u r i e r t r a n s f o r m , and <j> = t a n - 1 lA(w) / B(u) ] (4.2.3) i s the argument i n the complex F o u r i e r t r a n s f o r m 0 T 4 a F i g u r e 4.2.1 F o u r i e r T r a n s f o r m f o r a H a l f S i n e Wave 28 Also, A(oi) = j °° a (t) cos tot dt (4.2.4) t — CO and B(u) = / o(t)- s i n tot dt . (4.2.5) For the p a r t i c u l a r case of a stress pulse of the form of a half sine wave given by a(t) = s i n pt for 0<? t£ T = 0 for Q>s t> T where P = T T / T , A(to), B(to) and G(to) have been calculated as follows: From (4.2.4) A(to) = J s i n pt cos tot dt [1+cos —5- J (4.2.6) 2 2 1 2 D - C O and from (4.2.5) / T B(to) = J s i n p t s i n tot d t p . toT for w^ p . (4.2.7) -to "2 2 s i n 2 29 T h e r e f o r e , a c c o r d i n g to (4.2.2) „ . » 2p TTO) G(w) = 2 r cos — . (4.2.8) G(u) has been p l o t t e d a g a i n s t frequency i n F i g u r e 4.2.1. F i g u r e 4.2.1 shows t h a t the amplitude (the modulus i n the complex F o u r i e r Transform), f o r u>2.5p, i s l e s s than 20% of the maximum val u e a t u=0. Hence t o cover a l a r g e r frequency band a p u l s e of s h o r t e r d u r a t i o n would be r e q u i r e d . 4.3 Momentum A n a l y s i s By the law of c o n s e r v a t i o n o f l i n e a r momentum, the momen-tum of the i n c i d e n t p u l s e should be equal to the a l g e b r a i c sum of the momenta of the r e f l e c t e d and the t r a n s m i t t e d p u l s e s . The momentum of the i n c i d e n t p u l s e i s and the t o t a l momentum of the r e f l e c t e d and t r a n s m i t t e d p u l s e s i s (4.3.1) M P 0 c I • t 0 d t + J u-. p 0 c o d t o (4.3.2) i t f o l l o w s t h a t (4.3.3) 30 The i n c i d e n t and t r a n s m i t t e d p u l s e s are t r a v e l l i n g i n the p o s i -t i v e x d i r e c t i o n , and the r e f l e c t e d p u l s e i n the n e g a t i v e x d i r e c t i o n . Then U 1 X = g l ( t _ ~ )r a n d u x = 91 ( t _3" ) = CT11(x,t) ° 0 E Q Ul" = fl(t+c0)' a n d Ul" =fl'(t+cV = — a 1 r ( x , t ) (4.3.4) u 2 = g 3 ( t - | ), and u 3 = g 3 ' ( t - X - ) = a 3 t ( x , t ) . ° ° E Q In view of (4.3.4), (4.3.3) may be w r i t t e n , /° . / °° f °° o , x ( x , t ) d t = -J a , r ( x , t ) d t + J a 7 t ( x , t ) d t , (4.3.5) which shows t h a t the area under the s t r e s s - t i m e curve of the i n c i d e n t p u l s e i s equal to the cor r e s p o n d i n g area of r e f l e c t e d p u l s e s u b s t r a c t e d from the area of the t r a n s m i t t e d p u l s e . Examination of (4.2.4), (4.2.5) and (4.2.2) f o r OJ=0 shows t h a t G(0) = J a ( t ) d t — 00 (4.3.6) 31 and hence by (4.3.5) j G 1 ( 0 ) = G r ( 0 ) + G t ( 0 ) . (4.3.7) For an e l a s t i c t h r e e - b a r system, i t was proved a n a l y t i c a l l y f o r the case of the p r o p a g a t i o n of the r e c t a n g u l a r p u l s e of d u r a t i o n T=4x and u n i t h e i g h t , d i s c u s s e d i n s e c t i o n 4.1, t h a t /°° • r T a 1 1 ( t ) d t = J a 1 1 ( t ) d t = 4-f J o 1 r ( t ) d t = 0 a ^ t ( t ) d t = 4T . Hence (4.3.5) i s s a t i s f i e d . A s i m i l a r check was made f o r the case when bar 2 i s v i s c o e l a s t i c . In the l i m i t as OJ approaches zero (4.1.37) becomes v 11 p ( u ) + Q C u ) = Q C ° c or (4.3.8) i c E 0 Q (0 ) ) = P(u) . c c E S u b s t i t u t i o n i n (4.1.40) y i e l d s , when w approaches zero, i u E 0 i c E G 32 or iojE P ( O J ) = o-r(u) . (4.3.9) By (4.1.39), iojE 0 . P ( O J ) = a 3 r(oj) (4.3.10) and t h e r e f o r e , [ a , 1 ( t ) d t = J a , t ( t ) d f . (4.3.11) S o l v i n g (4.1.33) and (4.1.34) f o r the r e f l e c t e d p u l s e y i e l d s E ° E - r - i o j — - P ( O J ) + OJ Q ( u ) = 2a, ( O J ) or (4.3.12) _ E 0 E c E Q o"-, (OJ) = [ - i o j — + OJ — i ] P ( O J ) . ± c 0 - - c 0 P ( O J ) and Q ( O J ) are f i n i t e f o r OJ=0 , p r o v i d e d the displacement u 2 ( 0 , t ) and the s t r a i n u 2 ' ( 0 , t ) are i n t e g r a b l e over the i n t e r v a l -OO<T<°O . T h e r e f o r e as — r to + 0, o 1 (OJ)-> 0 , hence (4.3.13) | a , r ( t ) d t = 0 . Thus {4.3.5) o r (4.3.7) are s a t i s f i e d from the r e s u l t s (4.3.11) and (4.3.13). 5. EXPERIMENTATION AND INSTRUMENTATION The p r e s e n t s e c t i o n d e a l s w i t h the d e s i g n aspects of the e x p e r i m e n t a l setup, and c e r t a i n experiments c a r r i e d out t o a r r i v e a t the f i n a l d e s i g n . The s p l i t Hopkinson p r e s s u r e bar apparatus i n t r o d u c e d by Kolsky [6] was the b a s i s f o r the design of the experimental setup. Changes were i n t r o d u c e d to take account of the p r e s e n t requirements. 5.1 E x p e r i m e n t a l Setup A schematic diagram of the setup i s shown i n F i g u r e 5.1.1. The o v e r a l l d e s i g n c o n s i s t s of two main systems d e s c r i b e d as f o l l o w s . 5.1.1 Mechanical System: Two 0.25 i n . diameter s t e e l bars of lengths 4.0 f t . and 5.0 f t . were suspended by s t r i n g s and the specimen to be t e s t e d was bonded between the two bars with p a i n t . The i n c i d e n t p u l s e was i n t r o d u c e d a t f r e e end of the 4.0 f t . long bar by the impact of a round-headed s t e e l bar (1.5 i n . long and 0.25 i n . diameter) h e l d l o o s e l y i n a bronze bushing at the end of a pendulum. 5.1.2 Transducer and Recording System: Budd s t r a i n gages (type C6-121 Budd M e t a f i l m s t r a i n gages, red c o l o u r ) were employed f o r the purpose of s e n s i n g the i n c i d e n t , r e f l e c t e d and t r a n s m i t t e d p u l s e s t r a v e l l i n g i n bar 1 and bar 3. The gage on bar 1 was mounted 18 i n . away 35 6 0" 3 0" H Active. Specimen Gag 1 Compensating Gage - T r i g g e r i n g C i r cu i t S t r iker 1_, 0-25" T ti/jfu nr. 1-5" F i g u r e 5 . 1 . 1 S c h e m a t i c D i a g r a m o f E x p e r i m e n t a l A p p a r a t u s 36 from the f r e e end and on bar 3, 30 i n . away from the f r e e end. Two compensating gages were mounted on two d i f f e r e n t bars of s t e e l f o r temperature compensation. The output from each of the h a l f b r i d g e s , c o n s i s t i n g of one a c t i v e and one compensating gtge, was f e d i n t o separate B r i d g e A m p l i f i e r Meters (BAM). The a m p l i f i e d output from the BAM was observed and recorded on an o s c i l l o s c o p e . These s t r a i n - t i m e r e l a t i o n s h i p s were photographed (Figure 6.2.1) to o b t a i n a f i n a l r e c o r d . 5.2 D e t a i l s o f Experimental Apparatus 5.2.1 T r i g g e r i n g C i r c u i t : By p r oper adjustment of the l e v e l s w i t c h , the o s c i l l o -scope can be t r i g g e r e d when an e l e c t r i c a l p u l s e bf more than 2 v o l t s amplitude i s a p p l i e d to the e x t e r n a l t r i g g e r i n g t e r m i n a l s of the scope. The t r i g g e r i n g c i r c u i t has been shown i n F i g u r e 5.1.1. A 12V p u l s e , produced by the c o n t a c t of the s t r i k e r bar and bar 1, was employed t o t r i g g e r the scope. 5.2.2 S t r a i n Gage L o c a t i o n s : The s t r a i n gages were mounted on the bars more than h a l f the l e n g t h o f the i n c i d e n t p u l s e away from the ends, i ) to a v o i d any i n t e r f e r e n c e of the i n c i d e n t and t r a n s -m i t t e d p u l s e s with the r e f l e c t e d p u l s e s from the ends. i i ) to a v o i d any l o c a l i z e d t h r e e - d i m e n s i o n a l e f f e c t s a t the ends. 37 F i g u r e 5.2.1 E r r o r i n M e a s u r i n g D y n a m i c S t r a i n w i t h S t r a i n G ages 38 5.2.3 S i z e of S t r a i n Gages: Some e r r o r [10] r e s u l t s when a s t r a i n gage i s used t o measure a s t r a i n p u l s e , due to the f a c t t h a t a s t r a i n gage measures an average s t r a i n over i t s l e n g t h A l , r a t h e r than s t r a i n a t a p o i n t . T h i s e f f e c t i s i l l u s t r a t e d i n F i g u r e 5.2.1. A gage l e n g t h of 0.25 i n . was chosen. Since the wave l e n g t h of h i g h e s t frequency (20,000 cps) of i n t e r e s t i s 10 i n . , the r a t i o o f gage l e n g t h to wave l e n g t h (1/40) was q u i t e low. T h e r e f o r e the e r r o r i n measuring instantaneous s t r a i n a t a p o i n t was n e g l i g i b l e . 5.2.4 Diameter of Bars: Kolsky [11] has shown t h a t the e r r o r i n the determin-a t i o n of s t r a i n u s i n g one-dimensional wave pr o p a g a t i o n theory 2 2 i s p r o p o r t i o n a l t o a /A (where a i s the diameter of the bar and-A i s the wave l e n g t h of s t r a i n wave). Since the s h o r t e s t wave-l e n g t h of i n t e r e s t i n the s t e e l bar i s 10 i n . ( f o r 20,000 c p s ) , the e r r o r i n v o l v e d i s l e s s than 1% f o r 0.25 i n . diameter s t e e l bar, and hence three dimensional e f f e c t s , such as l e t r a l i n e r t i a , are n e g l i g i b l e i n the p r e s e n t circumstances. 5.2.5 S e l e c t i o n of S t r i k e r : The h i g h e s t frequency of i n t e r e s t was chosen as 20,000 cps. From F i g u r e 4.2.1 which shows the modulus of the F o u r i e r t r a n s -form of a h a l f s i n e wave, we have concluded t h a t the amplitude f o r w>2.5p i s l e s s than 20% of the maximum amplitude a t zero frequency. T h e r e f o r e i n order t h a t the p u l s e c o n t a i n s i g n i f i c a n t components up to a frequency of 20,000 cps, i t i s necessary t h a t , 2 . 5 P > 2 T T X 2 0 , 0 0 0 where p=Tr/T and the p u l s e d u r a t i o n i s T, so i • T <(2.5/2x20,000) sec. Thus a p u l s e d u r a t i o n of l e s s than 62.5 micro-sec i s r e q u i r e d . Davies and Hunter [6] have mentioned t h a t i n g e n e r a l mechanical systems are used f o r experiments i n which l o a d i s a p p l i e d i n times of o r d e r of m i l l i s e c o n d s or g r e a t e r , w h i l e f o r p u l s e s of microsecond d u r a t i o n , e x p l o s i v e s are used. However, i n the case of l o a d i n g by e x p l o s i v e charges, the i n t e n s i t y of s t r e s s i s 3 very h i g h (>10 p s i ) and there i s l i t t l e c o n t r o l on the form of the p u l s e . T h e r e f o r e a pendulum s t r i k e r was chosen t o produce a s t r e s s p u l s e , f o r there i s g r e a t e r c o n t r o l on r e p r o d u c i b i l i t y of the p u l s e form and the s t r e s s e s i n v o l v e d are of the order of 100 p s i . A number of t h e o r i e s have been p o s t u l a t e d and experiments conducted r e g a r d i n g the d u r a t i o n of a p u l s e produced by the impact between the bodies of d i f f e r e n t geometries, and these have been reviewed i n Chapter 3. I t was not f e a s i b l e t o achieve the i d e a l c o n d i t i o n s of the t h e o r i e s i n p r e s e n t work. However, d i f f e r e n t g e o m e t r i c a l c o n f i g u r a t i o n s , of the s t r i k e r were i n -v e s t i g a t e d e x p e r i m e n t a l l y , and the time d u r a t i o n of the p u l s e s r e c o r d e d have been t a b u l a t e d i n Table 5.2.A. [ b ] 1!) (j \ i \ \ T f I o F i g u r e 5.2.2 S t r i k i n g Arrangements • 41 C o n f i g e r a t i o n 1:-A f l a t - h e a d e d bar of l e n g t h 1 was r i g i d l y mounted a t the!end of the pendulum arm (Figure 5.2.2a). C o n f i g u r a t i o n 2:-T h i s c o n f i g u r a t i o n was s i m i l a r t o 1 except the s t r i k i n g head of the bar was rounded (Figure 5.2.2b). C o n f i g u r a t i o n 3:-A b a l l b e a r i n g suspended by a heavy wire was used to s t r i k e the f r e e end of the bar 1 (Figure 5.2.2c). C o n f i g u r a t i o n 4:-A b a l l - b e a r i n g of diameter d was shot by a s p r i n g loaded shot gun, to achieve a f r e e impact between the b a l l and the bar. (Figure 5.2.2d). C o n f i g u r a t i o n 5:-T h i s arrangement was used to achieve impact between two f r e e b a r s . The s t r i k e r bar r e s t e d i n a s e m i - c i r c u l a r s l o t i n the pendulum (Figure 5.2.2e). C o n f i g u r a t i o n 6:-T h i s arrangement was s i m i l a r t o 5 except t h a t the s l o t was removed and the s t r i k e r bar was f r e e to s l i d e i n a smooth bronze bushing (Figure 5.2.2f). TABLE 5.2.A RESULTS OF STRIKER BAR STUDIES s. No. c o n f i g -u r a t i o n •1 i n . d • i n . S t r i k e r Geometry M a t e r i a l Time p Sec Remarks 1 1 3 1/2 1 o S t e e l > 300 S.-No.l through 5 gave r e s u l t s 2 1 3 3/8 -do- -do- 150 w i t h poor r e p r o d u c i b i l i t y 3 1 3 1/4 -do- -do- 100 P u l s e d u r a t i o n was i n c r e a s e d 4 1 3 3/16 -do- -do- 125 by i n c r e a s i n g impact v e l o c i t y . 5 1 3 1/4 -do- Alumimimum 125 6 1 1.5 1/4 -do- S t e e l 125 7 1 1.5 3/16 -do- -do- 125 8 2 3 1/4 1 > o -do- 125 9 2 1.5 1/4 -do- -do- > 100 Imperfect impacts. 10 3 - 9/16 o o B a l l B e a r i n g 80-150 Higher impact v e l o c i t y gave a 11 3 3/4 -do- -do- 100-200 s m a l l e r p u l s e d u r a t i o n . 12 4 — 9/16 -do- -do- - Nothing c o u l d be re c o r d e d due t o r a t t l i n g o f the b a l l . 13 5 3 3/8 1 1 o S t e e l 150-200 14 5 3 1/4 -do- -do- -do-15 5 3 3/16 -do- S t e e l -do-16 5 1.5 3/8 -do- -do- -do-17 5 1.5 1/4 -do- -do- , -do-18 5 1.5 3/8 1 ) o -do- -do-19 5 1.5 1/4 -do- -do- -do-to Table 5.2.A (continued) s. c o n f i g - 1 d S t r i k e r Time No. u r a t i o n i n . i n . • Geometry M a t e r i a l p Sec Remarks' 20 6 1.5 1/4 -do- Wood 130 1 I i 21 6 1.5 1/4 i 1 O S t e e l 80-110 i Poor r e p r o d u c i b i l i t y . 22 6 1.5 3/8 -do- -do- -do- -do-23 6 1.5 1/4 i — ) o -do- - 100 Good r e s u l t s . 24 6 1 1/4 -do- -do- ~ 100 co As Table 5.2.A shows, i n the twenty-four d i f f e r e n t arrangements t r i e d , i t was not p o s s i b l e to produce a p u l s e of s m a l l e r d u r a t i o n than 62 y sec. The b e s t r e s u l t s were o b t a i n e d f o r c o n f i g u r a t i o n 6 with a 1.5 i n . long round-headed s t e e l bar of 0.25 i n . diameter s l i d i n g i n the bronze bushing. 5.2.6 S e l e c t i o n of Specimens: I t was.shown i n Chapter 4 t h a t the g r e a t e r the d i f f e r -ence i n the impedance (z=E/c) i n the two contiguous m a t e r i a l s , the l e s s the energy t r a n s m i t t e d . Hence the f o l l o w i n g were the c r i t e r i a i n s e l e c t i o n of a specimen: i ) The r a t i o (A^) of the d i f f e r e n c e and sum of the impe-dances of s t e e l and the m a t e r i a l of the specimen should be l a r g e . i i ) The m a t e r i a l should be s t r o n g enough to support enough l o a d so t h a t i t may be used i n b u i l d i n g s and s t r u c t u r e s , i i i ) Some m a t e r i a l s e x h i b i t the p r o p e r t y of i n t e r n a l f r i c -t i o n , which, i n many i n s t a n c e s , can be t r e a t e d by . l i n e a r v i s c o e l a s t i c i t y . The energy of a s t r e s s wave i s d i s s i p a t e d w h ile p r o p a g a t i n g through such m a t e r i a l . i v ) The l e n g t h of the specimen has been chosen (a/h<<25) to e l i m i n a t e the e f f e c t s of r a d i a l and t a n g e n t i a l f r i c t i o n ~ [6] but s h o r t enough to be used e c o n o m i c a l l y i n a p p l i -c a t i o n s . Table 5.2.B shows approximate valu e s of d i f f e r e n t r e l e -vant m a t e r i a l p r o p e r t i e s 45 TABLE 5.2.B MATERIAL PROPERTIES E X 1 0 - 6 c x i o 4 Z M a t e r i a l p s i i n / s e c . p s i - s e c . / i n . Aluminum 10 6 170 .94 S t e e l 30 20 150 1 Copper 15.6 4 390 .56 Lead 2.35 4.34 54 . 5 •: ' Nylon .18-.45 5.36 3.3-8.3 .07 The m a t e r i a l s t e s t e d were pure l e a d , 3% a n t i m o n i a l l e a d , 7% a n t i m o n i a l l e a d and n y l o n . Nylon was chosen as a t y p i c a l h i g h polymer. A l l specimen were 0.25 i n . i n diameter, and 0.25 i n . , 0.50 i n . and 0.75 i n . i n l e n g t h . For these propor-t i o n s c r i t e r i o n IV, l i s t e d was s a t i s f i e d . 6. RESULTS AND DISCUSSION i The f i r s t p a r t of t h i s chapter d e a l s w i t h the perform-ances of the system, and shows the ext e n t t o which the r e q u i r e -ments are s a t i s f i e d and what l i m i t a t i o n s were encountered. The second p a r t d e a l s with the checks on the o b s e r v a t i o n s and r e s u l t s . 6.1 Performance of the System and i t s L i m i t a t i o n s 6.1.1 D u r a t i o n of P u l s e : The system f i n a l l y s e l e c t e d to produce the s t r e s s p u l s e gave a p u l s e of about 100 microseconds d u r a t i o n as a g a i n s t the r e q u i r e d 62 microsecond or l e s s . However, s i n c e the p u l s e pro-duced was not a h a l f s i n e wave, the F o u r i e r Transform of the p u l s e covered f r e q u e n c i e s , f r o m zero to 20,000 cps as shown i n F i g u r e 6.1.1. 6.1.2 No D i s t o r t i o n of P u l s e : As shown i n F i g u r e 5.1.1, s t r a i n gages on bars 1 and 3 were mounted 30.0 i n . away from the specimen. To check t h a t t h e r e was no d i s t o r t i o n of a p u l s e as i t propagates i n the s t e e l t r a n s d u c e r bars 1 and 3, experiments were c a r r i e d out on a continuous 8 f t . s t e e l bar (Figure 6.1.2). No d i s t o r t i o n was found (Figure 6.1.3) f o r the pr o p a g a t i o n of a p u l s e over a d i s t a n c e of 120 i n . I t was p o s s i b l e to observe the wave over t h i s d i s t a n c e by t a k i n g account of the r e f l e c t e d p u l s e . There-f o r e i t can be concluded t h a t the wave forms rec o r d e d by the s t r a i n gages on the t r a n s d u c e r bars 1 and 3 g i v e an ac c u r a t e i n d i c a t i o n of the wave a t the j u n c t i o n s w i t h the specimen, bar 2. 4 7 F i g u r e 6.1.1 F o u r i e r T r a n s f o r m of Observed P u l s e o S c o p e F i g u r e 6.1.2 Setup t o check C o n d i t i o n of No D i s t o r t i o n Pulse F i g u r e 6.1.3 Record Showing No D i s t o r t i o n of Pulse 49 IZD F i g u r e 6.1.4 Setup to Check E f f e c t of Adhesive Records to Show the E f f e c t of Adhesives F i g u r e 6.1.5 P a i n t F i g u r e 6.1.6 Grease F i g u r e 6.1.7 No Adhesive 50 6.1.3 E f f e c t of Adhesive: To h o l d the specimen i n - p o s i t i o n , p a i n t was used as an adhesive. To make sure t h a t the p o i n t does not cause any r e f l e c t i o n o f the p u l s e , an. experiment was conducted by j o i n -i n g w i t h p a i n t the two s t e e l bars of 0.25 i n . diameter (Figure 6.1.4) and r e c o r d i n g the p u l s e i n each bar. The r e c o r d (Figure 6.1.5) shows t h a t there i s no r e f l e c t i o n of the p u l s e a t the j o i n t . F i g u r e 6.1.6 and F i g u r e 6.1.7 were recorded f o r s i m i l a r experiments conducted by u s i n g grease a t the j o i n t and by keeping the bar i n c o n t a c t j u s t by a p p l y i n g s l i g h t p r e s s u r e (no adhesive used). Both F i g u r e 6.1.6 and F i g u r e 6.1.7 show r e f l e c t i o n o f the p u l s e at the j o i n t as compared t o no r e f l e c -t i o n i n case of F i g u r e 6.1.5. The F o u r i e r transforms f o r the i n c i d e n t and t r a n s m i t t e d p u l s e s have been shown i n F i g u r e 6.1.8 f o r the case of two bars i n F i g u r e 6.1.4 j o i n e d t o g e t h e r with p a i n t . T h i s e s t a b l i s h e s t h a t the presence of p a i n t causes no r e f l e c t i o n o r d i s t o r t i o n a t the j u c t i o n . S ince the BAM had a constant g a i n up to 15,000 cps, and s i n c e the modulus of the F o u r i e r t r a n s f o r m of the i n c i d e n t p u l s e a t 15,000 cps i s l e s s than 10% of the maximum amplitude at zero frequency (Figure 6.1.1) the r e s u l t s were p l o t t e d up to 15,000 cps o n l y . 6.2 Checks on Observations and R e s u l t s A t y p i c a l s t r e s s - t i m e r e l a t i o n s h i p r e c o r d i s shown i n F i g u r e 6.2.1. The f o l l o w i n g checks and o b s e r v a t i o n s were made: F i g u r e . 6 . 1 . 8 F o u r i e r T r a n s f o r m o f P u l s e s Recorded f o r P a i n t 52 F i g u r e 6.2.1 S t r e s s - T i m e R e l a t i o n s h i p 53 i ) The c a l c u l a t e d v e l o c i t y of sound i n s t e e l i s about 2 0 x l 0 4 i n / s e c (c = / E / p , E = 3 0 x l 0 6 p s i p = .282 l b / i n 3 ) ; t h e r e f o r e the t r a n s m i t t e d and r e f l e c t e d p u l s e s should commence 300 microsecond a f t e r the commencement of the i n c i d e n t p u l s e f o r a p a i r of s t r a i n gages mounted 30.0 i n . away from the j u n c t i o n s . T h i s checks w i t h . F i g u r e 6.2.1. i i ) In the presence of a specimen, the t r a n s m i t t e d p u l s e appeared at t e n u a t e d (Figure 6.2.1) as compared t o the i n c i d e n t p u l s e . i i i ) D i s p e r s i o n was observed i n the t r a n s m i t t e d and r e -f l e c t e d p u l s e s (Figure 6.2.1) with r e s p e c t t o the i n c i d e n t p u l s e . i v ) The t r a n s m i t t e d p u l s e was always observed t o be com-p r e s s i v e f o r a compressive i n c i d e n t p u l s e . v) The r e f l e c t e d p u l s e was t e n s i l e i n i t i a l l y and changed to compression. A l l o b s e r v a t i o n s , i i through v, check w i t h the simple theory developed i n Chapter 4 f o r an e l a s t i c specimen, bar 2. The i n f i n i t e d u r a t i o n of the t r a n s m i t t e d and r e f l e c t e d p u l s e s (found t h e o r e t i c a l l y ) c o u l d not be observed p r a c t i c a l l y due to sm a l l amplitudes a s s o c i a t e d w i t h the t r a i l i n g p a r t of the p u l s e s . The r e f l e c t e d p u l s e , being l o n g e r than 36.0 i n . , c o u l d not be i s o l a t e d from i t s own r e f l e c t i o n ( f i g u r e 6.2.1) from the f r e e end of the bar 1, which was 18 i n . away (Figure 5.1.1) from the gage. F i g u r e 6.2.2 shows G 1 (to) , G t ( t u ) and L(O J) p l o t t e d a g a i n s t frequency; G 1 ( O J ) and G ^ ( O J) are the moduli of the oo in K c p s F i g u r e 6.2.2 F o u r i e r T r a n s f o r m s and R e d u c t i o n C o e f f i c i e n t l O O C H L a v ? 0 80.0 -60.0 ~ 40.0-20.0,-Frequency in Kcps CO —r> -20.0 •* 2-0 4-0 6-0 8-0 10«0 1 2 - 0 Ave rage Reduction Coer r . _ . S t anda rd Dev ia t ion F i g u r e 6.2.3 Average R e d u c t i o n C o e f f i c i e n t f o r 0.75 i n Lead en F o u r i e r transforms o f the i n c i d e n t and the t r a n s m i t t e d p u l s e s and L ( t o ) , the r e d u c t i o n c o e f f i c i e n t (= [G 1 ( c o)-G t (co) JxlOO/G"*" (OJ) ) i : the p e r c e n t r e d u c t i o n i n amplitude. The f i n a l r e s u l t s p r e -sented, F i g u r e 6.2.3, show L a V g ( ( J J ) (r.m'. s. average) wi t h standard d e v i a t i o n l i n e s bounding i t to gi v e an i n d i c a t i o n of the s c a t t e r i n the r e s u l t s . Most of the r e s u l t s show t h a t the r e d u c t i o n c o e f f i c i e n t i s not zero a t zero frequency. T h i s o b s e r v a t i o n c o u l d not be accounted f o r by the simple t h e o r i e s developed i n Chapter 4 f o r the case of e l a s t i c and l i n e a r v i s c o e l a s t i c specimens, s i n c e these t h e o r i e s p r e d i c t t h a t the r e d u c t i o n c o e f f i c i e n t a t zero frequency should be zero. T h i s e f f e c t may be due to non-l i n e a r behaviour of the m a t e r i a l . Checks on the c o n s e r v a t i o n of momentum c o u l d not be made a c c u r a t e l y as the complete i n f o r m a t i o n about the r e f l e c t e d p u l s e was not a v a i l a b l e . However, approximate checks made on some photographs s a t i s f i e d (4.3.5). Some cases have shown a p e c u l i a r behaviour o f a neg a t i v e r e d u c t i o n c o e f f i c i e n t f o r a range of s m a l l f r e q u e n c i e s . Such behaviour i s not i n accord with the l i n e a r t h e o r i e s developed •~ i n Chapter 4. One such case i s shown i n F i g u r e 6.2.4. The area and hence the momentum of the t r a n s m i t t e d p u l s e are; g r e a t e r than the area and the momentum of the i n c i d e n t p u l s e . A l s o " s i n c e the area of the compressive p a r t of the r e f l e c t e d p u l s e i s g r e a t e r ( i n a b s o l u t e value) than the area of the t e n s i l e -p a r t o f the r e f l e c t e d p u l s e , the r e f l e c t e d p u l s e has p o s i t i v e momentum. The amplitude of the F o u r i e r t r a n s f o r m a t zero F i g u r e 6.2.4 S t r e s s S t r a i n R e l a t i o n s h i p F i g u r e 6.2.4 S t r e s s S t r a i n R e l a t i o n s h i p 5 8 frequency i s p r o p o r t i o n a l to the momentum of the p u l s e . Hence a n e g a t i v e r e d u c t i o n c o e f f i c i e n t i s r e l a t e d t o the d i s t r i b u t i o n of momentum d i s c u s s e d above. The energy c a r r i e d i n the i n c i d e n t p u l s e i s p r o p o r t i o n a l t o J oo ^ l a / ^ t ) | ] 2 d t x x< 0 By c a l c u l a t i n g the cor r e s p o n d i n g i n t e g r a l f o r the t r a n s m i t t e d p u l s e i n bar 3, i t was found t h a t there was a r e d u c t i o n i n the energy t r a n s m i t t e d , even f o r t h i s case of a ne g a t i v e r e -d u c t i o n c o e f f i c i e n t . The r e s u l t s can be c o n s i d e r e d u s e f u l o n l y up t o 1 2 , 0 0 0 cps as the d e v i a t i o n f o r o>>12,000 cps are r a t h e r h i g h . The hi g h e r frequency components i n the F o u r i e r t r a n s f o r m a r i s e due to sudden changes i n sl o p e s which c o u l d not be d e f i n e d a c c u r a t e l y w i t h the photographs taken i n the pr e s e n t work. T h i s may be the reason f o r l a r g e r d e v i a t i o n s f o r OJ > 1 2 , 0 0 0 cps. 7. CONCLUSION AND REMARKS • I t has been found t h a t the approach adopted i n the'-p r e s e n t work i s a very good method f o r s t u d y i n g the frequency response of d i f f e r e n t m a t e r i a l s . The impact of the s t r i k e r bar produces a p u l s e which i s of s u i t a b l y s m a l l amplitude (~250 p s i ) and c o n t a i n s f r e q u e n c i e s up to 15,000 cps. E x p l o s -i v e charges, on the o t h e r hand, produce h i g h amplitude p u l s e s and the waveform i s more d i f f i c u l t to reproduce. I t i s very d i f f i c u l t t o produce a s t r e s s p u l s e w i t h the d u r a t i o n of microsecond range by mechanical d e v i c e s . A s t r e s s p u l s e ( w i t h i n the e l a s t i c l i m i t ) p r o p a g a t i n g through a s t e e l bar does not d i s t o r t . The use of p a i n t to h o l d the specimen between the two s t e e l t r a n s d u c e r bars i s an ex-c e l l e n t technique which does not d e s t r o y the s u r f a c e s i n c o n t a c t . The f o l l o w i n g p o i n t s , e s t a b l i s h e d t h e o r e t i c a l l y f o r the cases of e l a s t i c specimens, bar 2, were v e r i f i e d e x p e r i m e n t a l l y f o r the specimens used i n the p r e s e n t r e s e a r c h : i ) The t r a n s m i t t e d p u l s e has the same s i g n as the i n c i d e n t ' p u l s e ; i . e . i f the i n c i d e n t p u l s e was compressive, the trans-"~ ~ m i t t e d p u l s e was a l s o found to be compressive. i i ) The t r a n s m i t t e d p u l s e was found to be a t t e n u a t e d and" d i s p e r s e d as compared to the i n c i d e n t p u l s e . i i i ) I t was a l s o v e r i f i e d t h a t the r e f l e c t e d p u l s e had both compressive and t e n s i l e p a r t s . i v ) Energy t r a n s m i t t e d to bar 3 was l e s s than the energy i n the i n c i d e n t p u l s e i n bar 1. . As opposed t o the r e s u l t s of simple l i n e a r t h e o r y , the momentum of the t r a n s m i t t e d p u l s e was not found t o be equal to the momentum of the i n c i d e n t p u l s e . T h i s r e s u l t may be a t t r i b u t e d t o n o n l i n e a r v i s c o e l a s t i c behaviour o f the m a t e r i a l of the specimen. R e s u l t s show t h a t r e d u c t i o n c o e f f i c i e n t i s a f u n c t i o n of frequency, and a l s o of t h i c k n e s s and m a t e r i a l of the spec-imen. In g e n e r a l the r e d u c t i o n c o e f f i c i e n t i n c r e a s e d w i t h the f o l l o w i n g f a c t o r s : i ) An i n c r e a s e i n t h i c k n e s s o f the specimen. T h i s can be observed by comparison o f the L a V g (w) curves f o r the specimens of d i f f e r e n t l e n g t h s of the same m a t e r i a l . A p u l s e p r o p a g a t i n g i n a lon g e r v i s c o e l a s t i c m a t e r i a l i s atte n u a t e d more than i n a s h o r t e r one. i i ) An i n c r e a s e i n the r a t i o o f d i f f e r e n c e and sum of the impedances of the coupled m a t e r i a l s . T h i s can be observed by comparing L a vg(to) curves f o r the specimens of the same l e n g t h but of d i f f e r e n t m a t e r i a l s . i i i ) An i n c r e a s e i n the v i s c o u s damping of the m a t e r i a l . For pure l e a d , 3% and 7% a u t i m o n i a l l e a d , the val u e o f q u a n t i t y i s almost the same, but a comparison of L a vg(co) curves f o r 0.75 i n . long specimens of the three m a t e r i a l s d i s t i n c t l y shows an i n c r e a s e d r e d u c t i o n c o e f f i c i e n t f o r pure l e a d , which has the h i g h e s t v i s c o u s damping c o e f f i c i e n t of a l l the t h r e e . i v ) An i n c r e a s e i n frequency. Every L a v g (OJ ) curve shows an ' i n c r e a s e i n the r e d u c t i o n c o e f f i c i e n t with i n c r e a s i n g f r e -quency except a few which show a decrease f o r h i g h e r frequency. 8. SUGGESTIONS FOR FURTHER WORK The f o l i o v / i n g suggestions can be d i v i d e d i n t o two groups. 8.1 Improvements and Changes i n the Present Experimental Setup I. For h i g h e r s e n s i t i v i t y , h i g h e r v o l t a g e output i s r e -q u i r e d ( i t i s d e s i r a b l e to achieve a s e n s i t i v i t y of a t l e a s t 10 psi/mv and p r e f e r a b l y 1 psi/mv or l e s s ) . The s e n s i t i v i t y can be i n c r e a s e d i n f o l l o w i n g ways: (a) Using a f o u r gage b r i d g e ; (b) Employing an a d d i t i o n a l d.c. source i n s e r i e s with the BAM, depending on the power r e s t r i c t i o n of the gages. (c) Using a BAM w i t h a h i g h e r a m p l i f i c a t i o n f a c t o r or a separate a m p l i f i e r t o a m p l i f y output from the BAM. I I . A band-pass f i l t e r (80 cps to 40,000 cps) may be used to e l i m i n a t e any 60 Hz pick-up and h i g h frequency n o i s e . I I I . To get a b e t t e r d e s c r i p t i o n of the p u l s e , the time s c a l e of the scope should be e n l a r g e d . T h i s can be accomplished by u s i n g the t r i g g e r i n g c i r c u i t shown i n Appendix C, F i g u r e C l and u s i n g the mechanical system shown i n F i g u r e 8.1.1. T h i s w i l l g i v e a s t r e s s - t i m e r e l a t i o n s h i p h y p o t h e t i c a l l y c o n s t r u c t e d i n F i g u r e 8.1.2. IV. I t would be e a s i e r to mount a speciment on a 0.5 i n . diameter bar. V. Laminated specimens can be t r i e d as they are b e i n g F i g u r e 8.1.2 H y p o t h e t i c a l S t r e s s - T i m e R e l a t i o n s h i p used i n the foundations of b u i l d i n g s . 8.2 Suggestions f o r F u r t h e r A n a l y s i s i F u r t h e r a n a l y s i s has been suggested which c o u l d be c a r r i e d out u s i n g the data o b t a i n e d from the setup suggested. I. I t may be shown f o r each s t r e s s - t i m e r e l a t i o n s h i p t h a t : . CO CO CO j a 1 1 ( t ) d t =-j a 1 r ( t ) d t +/ a 2 t ( t ) d t . I I . Depending on the r e s u l t s o b t a i n e d f o r each specimen, v i s c o e l a s t i c model may be a p p r o p r i a t e to i n t e r p r e t the r e s u l t s I I I . E = E(co) and c = c(w) can be computed n u m e r i c a l l y from (4.1.41) by an i t e r a t i v e technique. B I BL IOGRAPHY 1. K o l s k y , H. " I n t e r n a t i o n a l Symposium on S t r e s s Wave Prop-a g a t i o n i n M a t e r i a l s , " Norman Davids ( e d i t o r ) , I n t e r s c i e n c e P u b l i s h e r s , Inc., New York. 2. Timoshenko, S. "Theory of E l a s t i c i t y , " (1st E d i t i o n ) , McGraw H i l l , (1934), pp. 387-389. 3. H e r t z , Ei "Uber Die Beruhrung F e r t e r E l a s t i s c h e r Korper," J . Reine Angew. Math., V o l . 92 (1881), p. 156. R e p r i n t e d i n E n g l i s h i n Hertz M i s c e l l a n e o u s Papers, London, MacMillan. 4. Prowse, W.A* "The Development of Pressure Waves During the L o n g i t u d i n a l Impact of Bars," P h i l o s o p h i c a l Magazine, V o l . 22 (1936), pp. 209-39. 5. Hopkinson, Bi "A Method of Measuring P r e s s u r e Produced i n the Detonation of High E x p l o s i v e s or by Impact of B u l l e t s , " P h i l o s o p h i c a l T r a n s a c t i o n s of Royal S o c i e t y o f London, S e r i e s A, V o l . 213, p. 411. 6. Davies, E.D.H. and Hunter, S.C. "The Dynamic Compression T e s t i n g of S o l i d s by Method of the S p l i t Hopkinson P r e s -sure Bar," J o u r n a l of Mechanics and P h y s i c s of S o l i d s (1963), Vol.11, pp. 155-179. 7. V o l t e r a ^ E . and Zachmanoglou, E.C. "Use of R e s i s t e n c e S t r a i n Gages and P i z o e l e c t r i c Gages," from "Dynamics of V i b r a t i o n , " C h a r l e s E. M e r r i l l Books, Inc., (1965), p. 565. 8. "Current P r a c t i s e s i n I s o l a t i n g B u i l d i n g s from V i b r a t i o n w i t h Lead Asbastos Pads," AlA, No. 39-D, Lead I n d u s t r i e s A s s o c i a t i o n , N.Y. (1969) . 9. "Lead to C o n t r o l Sound and V i b r a t i o n , " 9 A c o u s t i c a l T r e a t -ment, Sound and V i b r a t i o n , (1969), Lead I n d u s t r i e s Assoc-i a t i o n , Inc., N.Y. 10. T a y l o r , D.A.W. "Time and Amplitude E r r o r s i n Measurement of Dynamic S t r a i n P u l s e s by R e s i s t a n c e S t r a i n Gages," . I n t e r n a t i o n a l J o u r n a l of Mechanical Sciences (1964). 11. K o l s k y , H. and Douch, L.S. "Experimental S t u d i e s i n P l a s t i c Wave Pr o p a g a t i o n , " J o u r n a l of Mechanics and P h y s i c s of S o l i d s , (1962), V o l . 10, pp. 105-123. * From the book l i s t e d i n [7]. 12. Kolsky, H.** Proceedings of Physics Society (1949), B 62, p. 676 13. H i l l , R** "The Mathematical Theory of Plasticity," (1950), Clarendon Press, Oxford. * ** From the paper listed in [6]. « APPENDIX A T e s t Procedure;-The specimen was glued with p a i n t between two s t e e l bars suspended by s t r i n g s . The pendulum was r e l e a s e d from a c e r t a i n angle and the s t r i k e r bar was made to s t r i k e the f r e e end of bar 1. The impact of s t r i k e r bar with bar 1 t r i g g e r e d the scope. The s t r e s s - t i m e r e l a t i o n s h i p s produced by the o s c i l l o s c o p e beam were photographed. For each specimen, about ten photographs were taken with a 35 mm A s h a i Pentax 2 camera on a Plus X f i l m . These were e n l a r g e d t o g i v e 1 i n . on 2 the p r i n t f o r 1 cm. on the scope. The time a x i s on the p r i n t was 0 . 1 m s e c / i n . The p u l s e s were d i g i t i z e d by r e a d i n g p o i n t s a t 10 m i c r o - s e c . i n t e r v a l s . The r e s u l t s o b t a i n e d from such photographs were very e r r a t i c s i n c e the q u a l i t y o f the photographs was very poor. To improve the q u a l i t y , the s e n s i t i v i t y of the BAM was i n c r e a s e d by a d d i t i o n a l d.c. v o l t a g e . A photograph thus o b t a i n e d has been shown i n F i g u r e 6 . 2 . 1 . APPENDIX B C a l c u l a t i o n s : -The i n c i d e n t and the t r a n s m i t t e d p u l s e s , d e s c r i b e d i n Appendix A and F i g u r e 6.2.1, were normalized by d i v i d i n g the o r d i n a t e s o f the p u l s e s by maximum s t r e s s i n the i n c i d e n t p u l s e . Thus A (oi ) , B(u>) and G(w) were d e f i n e d as f o l l o w s f o r computing the r e s u l t s : I 00 A(o)) = J c o s u t dt l — OO Q max B(u) = / ^4^- s i n ut dt y — co n a max G(a>) = [A 2(co) + B 2 ( O J ) ] 1 / 2 A(co) and B( I D) were computed by numer i c a l i n t e g e r a t i o n . . To check the n u m e r i c a l method adopted numerical and a n a l y t i c a l r e s u l t s were computed f o r a h a l f s i n e wave of d u r a t i o n of 100 micro-second on which the p o i n t s were d e s c r i b e d a t 10 microsecond i n t e r v a l s . The maximum e r r o r was l e s s than 1%, up to 20,000 cps, The r e d u c t i o n c o e f f i c i e n t L(O J) was c a l c u l a t e d a f t e r f i n d i n g G' 1 (CD) and G t(co) f o r the i n c i d e n t and the t r a n s m i t t e d p u l s e s r e s p e c t i v e l y , where L(u>) i s given by ( U) = G 1 (co)-G t (co) , x 1 0 Q _ G1(co) The r e d u c t i o n c o e f f i c i e n t L(w)was p l o t t e d f o r e i g h t to ten s t r e s s - t i m e r e l a t i o n s h i p records f o r each specimen. A 67 l a r g e r s c a t t e r (more than 4%) of p o i n t s was observed i n c e r t a i n cases p a r t i c u l a r l y a t f r e q u e n c i e s lower than 4,000 cps and h i g h e r than 10,000 cps. The h i g h frequency components a r i s e from sudden changes i n slope w h i l e low frequency components are due to s m a l l g r a d i e n t s i n the p u l s e . T h e r e f o r e the i n a c c u r -acy and e r r o r i n the r e s u l t s a t low and h i g h f r e q u e n c i e s are p o s s i b l y r e l a t e d to the f o l l o w i n g p o i n t s . i ) S ince the p o i n t s on the recorded p u l s e s c o u l d not be read a t s m a l l e r i n t e r v a l s than 10 microsecond on the h o r i z o n t a l time axes, these measurements are w i t h i n an accuracy of ±5 micro-second . Such an e r r o r was g e n e r a l l y encountered e i t h e r a t the l e a d i n g or the t r a i l i n g end of the p u l s e , where the s l o p e s are q u i t e s t e e p . Thus t h i s may g i v e r i s e to e r r o r a t h i g h e r f r e -quencies . i i ) The p u l s e s c o u l d o n l y be read w i t h an accuracy of ±5 p s i . T h i s r e p r e s e n t s an a b s o l u t e e r r o r , and hence the p e r c e n t e r r o r i n G 1 (u) , G t ( c o ) , and L(to) due to t h i s a b s o l u t e e r r o r , i s g r e a t e s t a t hi g h f r e q u e n c i e s where G 1 ( C J ) and G t ( u ) are s m a l l . i i i ) The t r a n s m i t t e d p u l s e , as found t h e o r e t i c a l l y , should extend t o i n f i n i t e time. However, due to the very s m a l l ampli-tude a s s o c i a t e d w i t h the t r a i l i n g p a r t of the t r a n s m i t t e d p u l s e , i t i s i m p r a c t i c a l t o read the p u l s e a f t e r a c e r t a i n f i n i t e time. The omitted t r a i l i n g p a r t of the t r a n s m i t t e d p u l s e may have a f i n i t e area which c o n t r i b u t e s to the low frequency com-ponants of the F o u r i e r t r a n s f o r m . T h e r e f o r e , f o u r c o n s i s t e n t L ( t o ) p l o t s (out of 8-10 p l o t s ) 68 were chosen t o o b t a i n the curves f o r L (OJ) gi v e n by avg ^ 1 L (OJ) avg I4 L 2 ( O J ) / 4 j = l 3 L (OJ) was p l o t t e d i n the f i n a l r e s u l t s bounded by the avg . * J s t a n d a r d d e v i a t i o n l i n e s d e s c r i b e d as .1 L (OJ) ± avg 2 { L (OJ) j = l avg - L j (OJ) r/4 A t v o i c a l L (") bounded by the standard d e v i a t i o n 2 ^ avg l i n e s i s shown i n F i g u r e 6.2.3. lid pA/vvHS Input —> -f£ 2N1381 2 N 1304 Z2K WvW-rwv\A—1 4-.7 K +6V 1—; Output 2% K www—{" 2-7K -12 V Monostable Vibrator Invertor OV -12 V Invertor F i g u r e C.,1 C i r c u i t f o r Two T r i g g e r i n g P u l s e s APPENDIX C Two t r i g g e r i n g p u l s e - e l e c t r o n i c c i r c u i t : \ The F i g u r e C l shows the c i r c u i t t o o b t a i n two t r i g g e r -i n g p u l s e s from one e l e c t r i c a l p u l s e produced by the c o n t a c t of the s t r i k e r bar and a n v i l bar. The i n t e r v a l T between the two p u l s e s i s gi v e n by T = .7RC. For . the c i r c u i t shown^the sweep time i s 500 microsecond and about 100 microsecond have been p r o v i d e d f o r r e s e t time, making T = 600 microsecond. Three or more t r i g g e r i n g p u l s e s can be generated by adding one or more monostable v i b r a t o r s t o the c i r c u i t . E l i m i n a t i o n o f i n v e r t o r I would y i e l d a d e l a y c i r c u i t g i v i n g a d e l a y time T = .7RC. APPENDIX D A d d i t i o n a l F i g u r e s : i ) Reduction C o e f f i c i e n t : The f o l l o w i n g f i g u r e s show the average r e d u c t i o n c o e f f i -c i e n t (L (co) ) f o r twelve specimens which were t e s t e d . Frequency in Kcps CO 2-0 4-0 6-0 8-0 10-0 12-0 Ave rage Reduction Coef f , S t anda rd Dev ia t ion F i g u r e D . l Average Reduction C o e f f i c i e n t f o r 0.25 i n l o n q .7% A n t i m o n i a l Lead loo-o-T 80.0 -j 60.0 -40.0-20.0 -Frequency in Kcps CO —r>--20.0 -* 2-0 4-0 6-0 8-0 10-0 12-0 A v e r a g e Reduction Goeff , _ . S t anda rd Dev ia t ion F i g u r e D.2 Average Reduction C o e f f i c i e n t f o r 0.50 i n long 7% A n t i m o n i a l Lead Frequency in Kcps 2-0 4-0 6-0 ~ T 1 1 8-0 10«0 V 2 " 0 A v e r a g e Reduction Goeff . CO -20.0 4 S t anda rd Dev ia t ion F i g u r e D.3 Average R e d u c t i o n . C o e f f i c i e n t f o r 0 .75' i n l o n a 7% Antimonial---L'e:a'd ." 100'CV •8 0. On 60,0 -40.0-20.0,-Frequency in Kcps CO -20.0 2-0 4-0 6-0 8-0 10*0 !2-0 Ave rage Reduction Goeff , S t anda rd Dev ia t ion F i g u r e D.4 Average Reduction C o e f f i c i e n t f o r 0.25 i n long 3% A n t i m o n i a l Lead U l 80.0 n 60.0 -40.0-20.0.-F r e q u e n c y in K cps -20.0 2-0 4-0 6-0 8-0 10*0 12-0 A v e r a g e Reduc t i on C o e f f , S t a n d a r d D e v i a t i o n F i g u r e D.5 Average Reduction C o e f f i c i e n t f o r 0.50 i n long 3% A n t i m o n i a l Lead 100-0-f 80.0-60.0 ~ 40.0-20.0.--20.0 -* b-u 10*0 A v e r a g e Reduction Coef f . S t anda rd Dev ia t ion F i g u r e D.6 Average Reduction C o e f f i c i e n t f o r 0.75 i n long 3% A n t i m o n i a l Lead ' 100-0-f Lav^ 80.0,-60.0 ~ 40.0-20.0-1 6*0 Frequency in Kcps i 1 CO — T - » --20.0 -* 2-0 4-0 8-0 10-0 12-0 Ave rage Reduction Coef f . S t anda rd Dev ia t ion Figure D.7 Average Reduction C o e f f i c i e n t for 0.25 i n long Pure Lead ^ CD 80.0 60.0 ~ 40.0-20.0,-Frequency in Kcps 2-0 CO - r > -20.0 -» 4-0 6-0 8-0 10-0 12-0 Ave rage Reduction Goeff , .. S t anda rd Dev ia t ion F i g u r e D.8 Average R e d u c t i o n C o e f f i c i e n t f o r 0.50 i r l o n g Pure Lead ^ 100'0-T Lavcr 3 0 . 0 -60,0 -40.0-2 0 . 0 -Frequency in Kcps CO -20.0 2-0 4-0 6-0 8-0 10-0 » 2 * 0 Average Reduction Coeff . . Standard Deviation F i g u r e D.9 Average Reduction C o e f f i c i e n t f o r 0.75 i n long Pure Lead OD IOO'0-I 80.0 60.0 ~ 40.0-20.0.-8-0 10*0 1 2 " ° Average Reduction Goeff. Standard Deviation F i g u r e D.10 Average Reduction C o e f f i c i e n t f o r 0.25 i n long Nylon 80,0 -60.0 " 40.0-20.0.-Frequency in Kcps GO -20.0 -* 6-0 8-0 10-0 12-0 Average Reduction Goeff. Standard Deviation F o g i r e D . l l Average R e d u c t i o n C o e f f i c i e n t f o r 0.50 i n l o n g N y l o n 00 l oo - o i 8 0 . 0 i 60.0 1 4:0.0-•20.0,--20.0 H Frequency in Kcps 8-0 10-0 1 2 - 0 Average Reduction Goeff. _. Standard Deviation CO F i g u r e D.12 Average Reduction C o e f f i c i e n t f o r 0.75 i n long Nylon oo co i i ) T r a n s m i s s i o n C o e f f i c i e n t : I f a s t r e s s wave t r a v e l l i n g i n an e l a s t i c b a r i s g i v e n by u = f ( t - | ) = D (to) s i n { u ( t - | ) + 0 (9.D.1) Then s t r e s s can be e x p r e s s e d as a = - - f 1 ( t - - ) = - - D (co) cos {co ( t - - + ip} c c c c r (9.D.2) = G (co) s i n {co (t-~) +<)»•} The i n t e n s i t y o f a wave can be e x p r e s s e d as t h e average energy o v e r t h e p e r i o d o f the wave c r o s s i n g p e r u n i t a r e a p e r u n i t t i m e . The e x p r e s s i o n f o r t h e en e r g y , o v e r a p e r i o d T 0 o f t h e wave, c r o s s i n g p e r u n i t a r e a a t a d i s t a n c e x a l o n g the l e n g t h , can be w r i t t e n as T„+x/c E (co) = [ S t r a i n Energy + K i n e t i c Energy ] n x/c T 0+x/c T 0+x/c x/c x/c (9.D.3) •2 u c d t 85 c , T 0 + x / c E 2 2 / - T 0 + x / c ^ d t x / c x / c (9.D.4) I. / T o + l 4 f ' 2 ( t - f ) « x / c c (9.D.5) = £ / E /, C G 2 ( o o ) s i n 2 { o 3 ( t - X - ) + <j>} d t (9.D.6) c c T ° .2 2E G ( u ) . (9.D.7) T h e r e f o r e t h e i n t e n s i t y I (u>) c a n b e e x p r e s s e d a s c 2 (9.D.8) I ( t o ) = fg- G^(co) . T h e t r a n s m i s s i o n c o e f f i c i e n t ^ ( C D ) i s g i v e n a s t h e r a t i o o f t h e i n t e n s i t i e s o f t h e t r a n s m i t t e d w a v e t o t h e i n c i d e n t w a v e . T h u s i n p r e s e n t c o n t e x t o f a t h r e e b a r s y s t e m , » I ^ " ( t o ) r G^"(o)) ,2 , , _ , . -, 2 a T ( o j ) = —±-L = [ — — - 1 = I l - L ( w ) ] . (9.D.9) I 1 ( O J ) G 1 ( C J ) T h e f o l l o w i n g f i g u r e s ( F i g u r e s D . 1 3 t h r o u g h D.16) s h o w t h e v a r i a t i o n o f a T ( t o ) w i t h f r e q u e n c y t o , w h i c h h a v e b e e n p l o t t e d c o r r e s p o n d i n g t o L a v g ( u i ) v a l u e s s h o w n i n F i g u r e D . l t h r o u g h D . 1 2 . 2 o Specimen Length •0.25" 0.5 0" 0.7 5* o C N o 2.0 4.0 6.0 8.0 CO in Kcps 10.0 12.0 14,0 F i g u r e D.13 T r a n s m i s s i o n C o e f f i c i e n t f o r 7% A n t i m o n i a l Lead CO 00 d o Specimen Length - 0 . 2 5 " -0 .5 0 " - 0 . 7 5" CN « o 0 2.0 4.0 6.0 8.0 10.0 12.0 14,0 CO in Kcps Figure D.15 Transmission C o e f f i c i e n t For Pure Lead oo CO F i g u r e D.17 E x p e r i m e n t a l Setup V O o 

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