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ABSTRACT
The Resonant Bar method f o r measuring i n t e r n a l f r i c t i o n i n s o l i d s h a s been i n v e s t i g a t e d with t h e aim of extending t h i s method t o h i g h e r frequencies i n t h e range 100 kHz t o 1 MHz. This n e c e s s i t a t e d a s t u d y o f t h e v e l o c i t y d i s p e r s i o n of t h e fundamental l o n g i t u d i n a l
mode i n rods of a p p r o p r i a t e s i z e i n t h e region where t h e wavelength is o f t h e same o r d e r of magnitude a s t h e diameter.
I t is shown t h a t t h e v e l o c i t y d i s p e r s i o n o f t h e fundamental l o n g i t u d i n a l mode i n a c y l i n d r i c a l rod obeys t h e e x a c t s o l u t i o n of t h e Pochhamrner
-
Chree equation c a l c u l a t e d by Bancroft and B r a d f i e l d , over t h e range of diameter t o wavelength r a t i o from z e r o t o 2.0.The v e l o c i t y d i s p e r s i o n o f t h e lowest l o n g i t u d i n a l mode of a rod o f square c m s s - s e c t i o n h a s proved t o be t h e same as t h a t o f a c y l i n d r i c a l r o d o f t h e same m a t e r i a l , provided t h e e q u i v a l e n t diameter of t h e square cross-sectioned r o d is taken t o be 1.155 times i t s t h i c k n e s s . The t h e o r y o f Nigro (1966, 1968) agreed o n l y p a r t i a l l y with t h e experimental v e l o c i t y d i s p e r s i o n o f a square rod (Booker 1969). A s a r e s u l t of t h i s and a r e c e n t t h e o r y by maser (19691, Nigm h a s modified h i s t h e o r v , s o t h a t a t p r e s e n t t h e r e i s p e r f e c t agreement between t h e o r y and experiment.
I n c y l i n d r i c a l r o d s many resonant responses a r e e x c i t e d t h a t are n o t due t o t h e fundamental mode ( a l s o c a l l e d t h e Young's modulus mode) when t h e diameter t o wavelenflh r a t i o becomes
a p p r e c i a b l e ( u s u a l l y about 0 . 7 ) . These resonances a r e a t t r i b u t e d t o h i g h e r o r d e r l o n g i t u d i n a l modes. I d e n t i f i c a t i o n of t h e Young's modulus mode resonances becomes impossible when many o f t h e s e h i g h e r o r d e r mode resonances occur.
On s e v e r a l occasions one of t h e e x ~ e c t e d Young's modulus mode resonances is a b s e n t , i t s p l a c e being taken by two o r more resonances extremely c l o s e t o g e t h e r i n frequency and of small v i b r a t i o n amplitude. Frequently some of t h e harmonics have a frequency above o r below t h e expected v a l u e , s o that t h e v do n o t l i e on a smooth graph of frequency v e r s u s h a m o n i c number.
It i s sho\m t h a t t h e above two phenomena occur when t h e
L I E
value of R (= -1 i s c l o s e t o one of t h e v a l u e s fl fl t o 12
t F' 1 2
'
o r R3 where, a t a value o f :
( 1 ) S l
=
QE, an end-resonance o c c u r s ;( 2 ) Sl =
nl,
t h e f i r s t complex mode becomes r e a l ;( 3 )
n
= Q,, t h e L ( O , Z ) mode c u t s t h e 9 a x i s on a n 9. v e r s u s k a graph;0
( 4 )
=
R t h e L(O , 3 1 mode c u t s t h e R a x i s . 3'
To extend t h e Sesonant Bar method t o f r e q u e n c i e s approaching 1 MHz, it has been found e s s e n t i a l t o use r o d s approximately 3? 1 i n c h e s long and t h of an inch i n diameter.
- 8
The design and c o n s t r u c t i o n o f a b a r support a b s t r a c t i n g a n e g l i g i b l e amount o f energy from such s l e n d e r r o d s proved almost impossible. Both a c e n t r a l t h r e e - p i n suppow and a c e n t r a l t h r e e - wire support have proved t o be u n s u i t a b l e . A support c o n s i s t i n g o f two p a r a l l e l h o r i z o n t a l t u n g s t e n wires of a q u a r t e r of a
thousandth of an inch diameter has however been found t o g i v e damping v a l u e s c o n s i s t e n t t o within 10%.
Graphs are drawn of damping v e r s u s frequency f o r s e v e r a l aluminium and fused s i l i c a r o d s , show in^ t h a t ( 7 ' v a r i e s hut little with frequency and t h a t on t h e whole it v a r i e s smoothly from one harmonic t o t h e n e x t .
The Resonant Bar method is i d e a l l y s u i t e d f o r t h e determination of dynamic e l a s t i c moduli. A computer Dragram has been developed and used t o c a l c u l a t e Young's modulus and P o i s s o n ' s r a t i o f o r any given rod Erom t h e resonance f r e q u e n c i e s of l o n g i t u d i n a l v i b r a t i o n , the dimensions and t h e mass.
