Chapter 6
Fourier Transform Mechanical Analysis and Phenomenological Representation of Viscoelastic Material Behavior S. N. Ganeriwala and H. A. Hartung Research Center, Philip Morris, USA, Richmond, VA 23261-6583 Fourier transform mechanical analysis (FTMA) measures material properties over broad frequency spectra by using random noise input. Modulus-frequency isotherms are obtained in just a few seconds. FTMA is superior for characterizing moisture and additive effects because i t can be carried out with minimal temperature and moisture changes. Data are obtained over a series of temperatures and then regressed with an analytical sigmoid-shaped function for modulus-temperature responses. This yields parameters with clear, distinct physical meaning which can be correlated with material variables. The groundwork needed for comprehensive studies of moisture and plasticizers in viscoelastic behavior is provided. The viscoelastic properties of polymers make them valuable for suppression of sound and vibration. A comprehensive, useful understanding of the viscoelastic damping inherent in these systems can come only from studies of mechanical properties over wide ranges of time (frequency) and temperature. If materials are moisture sensitive, the effects of water activity also should be determined. A similar rule holds for plasticizers and solvents. Techniques for measuring the dynamic mechanical properties of polymers are tabulated in figure 1 (1-9). Although each category has advantages and disadvantages, the forced vibration methods are now preferred for basic dynamic studies and tests (6). A few of the commercially available instruments for measurements of dynamic properties are listed in figure 1. Each instrument has some advantages and limitations depending upon the material, temperature and frequency range, accuracy, resolving power, and the information sought (7). Typically, these instruments measure dynamic mechanical responses to sinusoidal input. To characterize the viscoelastic properties of a material, these tests must be repeated over a range of temperatures and frequencies. This is sometimes done at a fixed frequency while the polymer specimen is heated or cooled and 0097-6156/90/0424-0092$06.00/0 © 1990 American Chemical Society
Figure
• DYNASTAT • RHEOVIBRON • DMTA • MTS • RHEOMETRICS
•DMA •RESONANT BEAM
RESONANCE METHODS
FORCED VIBRATION
DIRECT MEASUREMENT OF STRESS & STRAIN METHODS
• TORSION PENDULUM • YARZLEY OSCILLOGRAPH
FREE OSCILLATION
DO NOT INCLUDE SAMPLE INERTIA
1. Methods o f M e a s u r i n g Dynamic M e c h a n i c a l P r o p e r t i e s o f Polymers. Commercially available instruments are l i s t e d b y schemes o f measurement.
PULSE
•UNIV. OF DAYTON SYSTEM
RESONANCE METHODS
FORCED VIBRATION METHODS
WAVE PROPAGATION METHODS
HARMONIC WAVE
FREE OSCILLATION METHODS
INCLUDE SAMPLE INERTIA
METHODS OF MEASURING DYNAMIC MECHANICAL PROPERTIES OF POLYMERS
94
SOUND AND VIBRATION DAMPING WITH POLYMERS
measurements are made p e r i o d i c a l l y at different temperatures. Another method u t i l i z e s frequency v a r i a t i o n s w h i l e the temperature is held constant. I n b o t h procedures, the m a t e r i a l i s s u b j e c t e d to cyclic deformation over a period of time with uncontrolled t e m p e r a t u r e r i s e , l o s s o f v o l a t i l e s and o t h e r changes from energy d i s s i p a t e d i n the m a t e r i a l . T h e s e e f f e c t s may b e c o m p o u n d e d b y l a g s of sample from ambient temperature i f heating is carried out at constant frequency. Thus i t i s d i f f i c u l t t o o b t a i n t r u l y i s o t h e r m a l p r o p e r t i e s u s i n g most o f the c o m m e r c i a l i n s t r u m e n t s . Another problem with these instruments is determination of mechanical properties of moisture sensitive materials (10-12). Such materials have tendency to gain moisture when subjected to m e c h a n i c a l e x c i t a t i o n at constant r e l a t i v e h u m i d i t y and temperature. This makes single frequency tests impractical for isomoisture s t u d i e s over a range o f f r e q u e n c i e s . Spectral a n a l y s i s techniques to study the b e h a v i o r o f polymers s u b j e c t e d to dynamic m e c h a n i c a l loads and/or d e f o r m a t i o n is called Fourier Transform Mechanical Analysis (FTMA). FTMA m e a s u r e s t h e c o m p l e x m o d u l i o v e r a r a n g e o f f r e q u e n c i e s i n one t e s t by exciting the sample by a random s i g n a l (band l i m i t e d w h i t e n o i s e ) (13.14). FTMA o v e r c o m e s or circumvents problems inherent in other test methods because it measures dynamic m e c h a n i c a l p r o p e r t i e s over a wide range of frequency with minimal temperature and moisture changes w i t h i n the sample. Often the range o f frequency covered by an instrument is not large enough to f u l l y a n a l y z e the dynamic m e c h a n i c a l response o f a m a t e r i a l system. The p r i n c i p l e o f t i m e - t e m p e r a t u r e s u p e r p o s i t i o n i s then u t i l i z e d to o b t a i n master curves that presumably approximate a mechanical response isotherm f o r a wide range o f time (or frequency) (15-20). The u n d e r l y i n g a s s u m p t i o n i s t h a t t h e m e c h a n i c a l r e s p o n s e of a material at all temperatures is governed by the same viscoelastic mechanism. T h i s assumption has been found u n s u i t a b l e i n many c a s e s ( 1 9 . 2 0 ) . H o w e v e r , i t i s v e r y much i n u s e a n d it does p r o v i d e a l i m i t e d , s e m i - e m p i r i c a l p e r s p e c t i v e on the e f f e c t s o f time (or frequency) and temperature. It is not very suitable for treating the obvious effects of moisture, p l a s t i c i z e r s , molecular weight, etc. A l s o , i t does n o t p r o v i d e a predictive, constitutive model. With the long term objective of treating the effects of moisture and other plasticizers on the m e c h a n i c a l p r o p e r t i e s o f m a t e r i a l s , a new s c h e m e t h a t y i e l d s a c o m p l e t e c o n s t i t u t i v e m o d e l o f viscoelastic materials has been developed. The time-temperature p r i n c i p l e i s an i n t e g r a l p a r t o f t h i s m o d e l i n g w i t h a quantitative d e s c r i p t i o n o f the g l a s s t r a n s i t i o n b e h a v i o r o f p o l y m e r s . Theory Linear Viscoelasticity Theory. FTMA is based on linear v i s c o e l a s t i c i t y theory. A one dimensional form of constitutive equation for linear viscoelastic materials which are i s o t r o p i c , homogeneous, and h e r e d i t a r y ( n o n - a g i n g ) i s g i v e n b y (21): CO a(t)
=
[ G(t,t-t')7(t')dt'
(1)
6. GANERIWALA AND HARTUNG
95
Viscoelastic Material Behavior
where the k e r n e l G ( t - t ' ) i s a monotonic nonincreasing function of t i m e known as the s t r e s s r e l a x a t i o n m o d u l u s , cr(t) i s c u r r e n t s t r e s s , and -y(t') i s the s t r a i n r a t e h i s t o r y . For the case of s i n u s o i d a l s t r a i n h i s t o r y E q u a t i o n l ^ c a n be transformed to y i e l d an e x p r e s s i o n f o r the complex modulus, G ( j w ) : G*(jw)
= G'(o>) + j G " ( w )
(2)
w h e r e j - J^T, w i s t h e f r e q u e n c y , a n d G'(w) and G''(w) are the storage modulus and l o s s modulus, r e s p e c t i v e l y . G' i s r e l a t e d to t h e amount o f e n e r g y s t o r e d and r e l e a s e d i n a c y c l i c o s c i l l a t i o n and G'' indicates the energy dissipated. In damping applications Equation 2 i s expressed: G*(jw) where ratio
-
G'< 1 + ji? )
(3)
r\ i s d e s i g n a t e d a s t h e m a t e r i a l d a m p i n g f a c t o r d e f i n e d a s the of G " over G' . When t h e s t r a i n 7 ( t ) i s s i n u s o i d a l i n t i m e w i t h a n a m p l i t u d e o f
7o'« 7(t) and s t r e s s
response
a(t)
will
c(t)
-
-
7o
s
i
n
w
t
4
( )
be
)]
(5)
where a (w) is the stress a m p l i t u d e a n d 8(u>) i s t h e p h a s e a n g l e between s t r e s s and s t r a i n . N o t e t h a t q — t a n 8. Then the storage and l o s s m o d u l i are g i v e n by (13.21) 0
°o
G ' (w)
cos6
(6)
sin5
(7)
7o G"(w)« 7o T h u s , d y n a m i c m e c h a n i c a l v i s c o e l a s t i c p r o p e r t i e s may b e m e a s u r e d in tests with s i n u s o i d a l s t r a i n input at f i x e d frequency. Such t e s t s have t o be r e p e a t e d a t different frequencies over the range of i n t e r e s t to c o m p l e t e l y c h a r a c t e r i z e the m a t e r i a l . FTMA F o r m u l a t i o n . When 7 ( t ) i s that i t s F o u r i e r transform r(w)
7(t)
an a r b i t r a r y f u n c t i o n o f exists. Then ( 2 2 ) :
jWt
2n
|
r < « ) e da>
time
such
(8)
and
r).
coupled relationships Then ( 1 3 ) :
between
a(t)
and
its
T h u s , FTMA d e t e r m i n e s complex modulus as the transfer function between i n p u t s t r a i n and output s t r e s s . A p r e r e q u i s i t e i s t h a t the F o u r i e r t r a n s f o r m o f 7 ( t ) must e x i s t . White nojse should suffice since i t contains a l l frequencies. N o t e t h a t G ( j w ) i n E q u a t i o n 10 w i l l be the complex Y o u n g ' s modulus i f a ( t ) and 7 ( t ) a r e the normal stress and normal strain, respectively; and the complex shear modulus i f they are the shear s t r e s s and shear s t r a i n . Samples and Measurements. A scheme f o r s h e a r m e a s u r e m e n t s i s s h o w n i n f i g u r e 2. Two i d e n t i c a l p o l y m e r s a m p l e s o f l e n g t h L, thickness h, and w i d t h W are b o n d e d t o two r i g i d m e t a l m o u n t s . The i n n e r mount i s a t t a c h e d t o a n impedance head (a combination force and acceleration sensor), which in turn i s attached to a shaker. An a c c e l e r o m e t e r i s a t t a c h e d to the top c e n t e r o f the o u t e r mount. If f ( t ) i s the t o t a l shear f o r c e e x e r t e d on the polymer sample t h e n the s h e a r s t r e s s cr(t) i s - gj?-
a(t) and
(ID
thus
where
F(u>) i s t h e F o u r i e r t r a n s f o r m o f f ( t ) . The force f(t) c a n be d e t e r m i n e d b y "mass c a n c e l l a t i o n " a s diagrammed i n F i g u r e 3. The e q u a t i o n o f m o t i o n for the impedance h e a d - i n n e r mount a s s e m b l y c a n be e x p r e s s e d : f(t)
= f (t) x
-
(m + m.) x
x.(t)
(13)
where f-j-(t) is the force m e a s u r e d b y t h e i m p e d a n c e h e a d , m^ t h e e f f e c t i v e mass o f t h e impedance h e a d , m. the mass of the inner mount, and x. t h e a c c e l e r a t i o n o f t h e i n n e r mount m e a s u r e d b y t h e impedance head. A simple electrical circuit c a n be devised to multiply t h e x \ ( t ) s i g n a l b y (m^. + m . ) a n d s u b t r a c t t h e r e s u l t f r o m t h e f ( t ) s i g n a l , t h e r e b y p r o d u c i n g f t t ) as i t s o u t p u t . The s h e a r s t r a i n 7 ( t ) is T
7
(t)
x (t) = - i
-
x (t) 0
(14)
5
where x.(t) - x ( t ) i s found by twice i n t e g r a t i n g the a c c e l e r a t i o n difference V . (t) - x ( t ) . This c a l c u l a t i o n i s e a s i l y performed by a spectrum analyzer since i n t e g r a t i o n i n the frequency domain i s e q u i v a l e n t to d i v i d i n g the F o u r i e r transform o f the o r i g i n a l signal b y jo?. Thus 0
0
X.(w)
He*)
=
5
X (w) 0
X.(w)
^—-
2
o) h
X (CD) 0
(15)
6. GANERIWALA AND HARTUNG where their
97
Viscoelastic Material Behavior
the upper case letters lower case e q u i v a l e n t s .
represent
the
Fourier transforms
of
Wave Effects. The f o r e g o i n g t r e a t m e n t o f s t r e s s a n d s t r a i n presume no s t a n d i n g o r t r a v e l i n g waves i n t h e s a m p l e , i . e . the inertia of the sample is n e g l i g i b l e compared to v i s c o e l a s t i c forces. This c o n d i t i o n i s met when t h e length of the shear wave propagating through the sample i s much g r e a t e r t h a n t h e c r i t i c a l d i m e n s i o n o f the sample. In shear samples thickness, h, is the critical dimension. The s h e a r w a v e l e n g t h A i s g i v e n b y ( F i t z g e r a l d , E . R . T h e J o h n s H o p k i n s U n i v e r s i t y , p e r s o n a l c o m m u n i c a t i o n , 1989) ,
2n [" G ~ | ° '
2nc
~
=
=~ U
J
5
r
2(1 + t a n
I [
( 1
+
t a n
2
5 )
i/
2
8) 2 +
1
,j J
16
< >
3 where w is the frequency ( r a d / s e c ) , p t h e mass d e n s i t y ( k g / m ) , and C i s t h e s p e e d o f p r o p a g a t i o n o f sound through the material. This predicts t h a t sample i n e r t i a l e f f e c t s w i l l become s i g n i f i c a n t a t l o w e r f r e q u e n c i e s as the thickness increases. Also that the inertial effects will s h o w up a t l o w e r f r e q u e n c i e s a s t h e m o d u l u s d e c l i n e s (when t e m p e r a t u r e i s i n c r e a s e d ) . Thus, d i r e c t measurement of stress and s t r a i n w i t h f o r c e d v i b r a t i o n t e s t s i s always l i m i t e d to r e l a t i v e l y low frequencies. FTMA p r o v i d e s a d i r e c t m e t h o d f o r d e t e r m i n i n g t h e f r e q u e n c y a t w h i c h i n e r t i a l e f f e c t s become noticeable. For this purpose the outer mount is detached from the "rigid" foundation and the a c c e l e r o m e t e r i s mounted on i t . The f r e e b o d y d i a g r a m o f t h e outer mount-accelerometer assembly is shown i n Figure 4. The r e l e v a n t equation of motion is f'(t)=
(m
+ m )
a
x'o(t)
0
(17)
where f ' ( t ) i s t h e f o r c e e x e r t e d o n t h e o u t e r mount b y t h e s p e c i m e n , m the mass o f t h e a c c e l e r o m e t e r , m t h e mass o f t h e o u t e r m o u n t , and x ( t ) t h e a c c e l e r a t i o n of the outer mount measured by the accelerometer. As l o n g as t h e i n e r t i a o f the polymer sample is negligible, then f ( t ) should equal f ' ( t ) . T h e n a c c o r d i n g t o E q u a t i o n s 13 a n d 17 &
0
0
f(t)
-
If the Fourier transforms evaluated, i t follows that V r v = m
(m of
a
+ m ) 'x (t) 0
both
+ m
0
(18)
0
sides
= Constant
of
this
equation
are
(19)
Thus, the r a t i o o f the f o r c e measured by impedance head to the acceleration o f o u t e r m o u n t i n e n d - f r e e scheme m u s t r e m a i n c o n s t a n t u p t o t h e f r e q u e n c y a t w h i c h w a v e e f f e c t become s i g n i f i c a n t . Bending Effects. The f o r e g o i n g t r e a t m e n t o f s t r a i n a l s o presumes sample d e f o r m a t i o n i s o n l y s i m p l e shear, i . e . t h e r e i s no bending. An approximate a s s e s s m e n t o f the e f f e c t o f b e n d i n g c a n be o b t a i n e d
SOUND AND VIBRATION DAMPING WITH POLYMERS SHEAR SPECIMEN
ACCELEROMETER OUTER MOUNT -POLYMER SPECIMEN
-INNER MOUNT
IMPEDANCE — ^ HEAD FRONT VIEW Figure
SIDE VIEW
2. Schematic diagram o f shear
test
FREE BODY DIAGRAM OF IMPEDANCE HEAD-INNER MOUNT ASSEMBLY
FREE BODY DIAGRAM OF POLYMER-OUTER MOUNTACCELEROMETER ASSEMBLY
111t)
1
11 T
m
:
3. Analysis
o f forces
T
2
2
Figure
samples.
u s i n g "mass c a n c e l l a t i o n " .
FREE BODY DIAGRAM OF ACCELEROMETEROUTER MOUNT ASSEMBLY
f'(t) 2 Figure 4. Analysis effects.
o f forces
f'(t) 2
i n outer
mount
to investigate
wave
6. GANERIWALA AND HARTUNG by 5. and
c o n s i d e r i n g the
99
Viscoelastic Material Behavior
d e f l e c t i o n o f a c a n t i l e v e r beam s h o w n
The t o t a l d e f l e c t i o n A„, i s sum o f d e f l e c t i o n d u e s h e a r A and i s g i v e n as ( 5 . 2 3 )
to
in
Figure
bending
g
A -
= A
T
g
+
A
(20)
b
3
6Ph/5AG + P h / 3 E I
(21)
where E and G are m a t e r i a l Y o u n g ' s and s h e a r m o d u l i , A and I area a n d moment o f i n e r t i a o f t h e b e a m . F g r e l a s t o m e r s E ~ 3G, and f o r a r e c t a n g u l a r beam A - b L a n d I * y o b L . Thus E q u a t i o n 21 can be expressed A A
6Ph 1 0 , h . 2 -i 5AG L + §-(—> J
, , (") 0 0
r
T
=
1
where b , L , and h are sample d i m e n s i o n s . E q u a t i o n ( 2 2 ) shows b e n d i n g i n the simple shear deformation is minimized when t h e sample length/thickness ratio is kept s u f f i c i e n t l y large. I f L / h i s not l a r g e enough the s t r a i n and c o n s e q u e n t l y stress distribution will not be uniform. Futhermore, E q u a t i o n 15, used f o r shear s t r a i n c a l c u l a t i o n , w i l l o v e r e s t i m a t e the shear s t r a i n r e s u l t i n g in lower modulus. P e r c e n t e r r o r i n s h e a r s t r a i n c a l c u l a t e d f r o m E q u a t i o n 22 i s summarized i n Table I . Table I. Percentage of E r r o r Calculated from Equation 22 Length/Thickness 2 4 8 16
This shows that significant error
Error
i n Shear
Strain
25 % 7 % 1.7% .4%
s a m p l e l e n g t h / t h i c k n e s s m u s t b e a b o v e 10 t o i n shear s t r a i n .
avoid
Experimental A s c h e m a t i c d i a g r a m o f the e x p e r i m e n t a l a p p a r a t u s i s shown i n F i g u r e 6. Test specimens w e r e made b y c o m p r e s s i o n m o l d i n g t w o e l a s t o m e r c o m p o u n d s , N e o p r e n e r u b b e r a n d NBR ( n i t r i l e r u b b e r ) , b e t w e e n sample mounts made of aluminum. Both materials were aged f o r about 8 years. O l d samples were u s e d to prove the validity of the new apparatus. Details of sample preparation are given elsewhere (13.14). The e x a c t d i m e n s i o n s o f s a m p l e mounts v a r i e d d e p e n d i n g on the s i z e o f the polymer specimen t e s t e d . To i n s u r e t h a t t h e mounts b e h a v e d as r i g i d b o d i e s , the d i m e n s i o n s were chosen to place the r e s o n a n c e f r e q u e n c i e s o f t h e i r n a t u r a l modes o f v i b r a t i o n w e l l a b o v e the r e g i o n o f i n t e r e s t . (The lowest frequency natural mode was f o u n d t o b e t h e " t u n i n g f o r k " mode o f t h e o u t e r m o u n t . ) ( 1 4 ) A s i g n a l generator feeds band l i m i t e d w h i t e n o i s e i n t o a power amplifier which drives an electro-mechanical shaker. A p i e z o e l e c t r i c impedance head i s mounted between the s h a k e r and the
100
SOUND AND VIBRATION DAMPING WITH POLYMERS
(
i L
t — Figure
k n
5. D e f l e c t i o n analysis bending effect.
» ^ of a cantilever
beam t o e s t i m a t e
"RlGID"FOUNDATION litlUKilllllKlUlllllll
-BRACKET
M
IMPEDANCE HEAD
CHARGE AMPLIFIER
fl(0
CHARGE AMPLIFIER
X (t)
CHARGE AMPLIFIER
x (0
Figure
RANDOM SIGNAL GENERATOR
AMPLIFIER
SHAKER
MASS CANCELLATION CIRCUIT
SPECTRUM ANALYZER
S
o
6. Schematic
MO
DIFFERENTIAL AMPLIFIER
d i a g r a m o f FTMA a p p a r a t u s .
6. GANERIWALA AND HARTUNG
101
Viscoelastic Material Behavior
inner mount. A piezoelectric accelerometer i s a t t a c h e d t o the b r a c k e t t o d e t e c t the s m a l l motion o f the (supposedly rigid) foundation. Signals from both transducers are fed through charge amplifiers. The a c c e l e r a t i o n s i g n a l s x . ( t ) and x ( t ) a r e t h e n f e d into a differential a m p l i f i e r to o b t a i n x . ( t ) - x ( t ) . The force and acceleration signals f-j-(t) and k \ ( t j a r e f e d i n t o a mass c a n c e l l a t i o n c i r c u i t which produces f ( t ) as an o u t p u t ( s e e E q u a t i o n 13). The f o r c e and a c c e l e r a t i o n d i f f e r e n c e s i g n a l s f ( t ) and x . ( t ) x ( t ) a r e t h e n f e d i n t o a spectrum a n a l y z e r where t h e y are Fourier transformed, and m u l t i p l i e d by the a p p r o p r i a t e c o n s t a n t s a c c o r d i n g t o E q u a t i o n s 12 and 15. G* (jw) i s t h e n computed i n t e r n a l l y . For comparison, tests were a l s o r u n w i t h t h e same equipment u s i n g a s i n g l e frequency s i n u s o i d a l input, repeated at different f r e q u e n c i e s o v e r the range o f i n t e r e s t . A l l c o m p a r a t i v e t e s t s were r u n a t 25°C. To f u r t h e r check the f i n a l r e s u l t s , t h e p r o p e r t i e s o f the e l a s t o m e r compounds were a l s o measured u s i n g a R h e o v i b r o n . To s t u d y e f f e c t s o f temperature, the e n t i r e excitation and detection assembly was placed i n a chamber w i t h controlled temperature and h u m i d i t y . The t e s t was done a t 20 different temperatures from 4°C t o 75°C. Each t e s t took 4-5 s e c . so t h e r e c o u l d be no s i g n i f i c a n t temperature r i s e due t o i n t e r n a l f r i c t i o n i n the t e s t sample. 0
0
0
Data
Reduction
The raw d a t a was s t r i n g s o f s t o r a g e modulus, G', and f r e q u e n c y , w, a t v a r i o u s temperature l e v e l s . Data r e d u c t i o n was c a r r i e d o u t w i t h linear temperature, T, Y = l o g G', and X = l o g (co/100) . The r e d u c e d s c a l e f o r X was used so t h a t intercepts a t X=0 would correspond t o 100 Hz frequency and f a l l w i t h i n t h e e x p e r i m e n t a l range. There was noise i n the modulus d a t a attributable to mechanical and e l e c t r i c a l p e r t u r b a t i o n s . These s m a l l p e r t u r b a t i o n s were removed by smoothing w i t h simple linear and second order polynomial functions. A t h i g h temperatures the i s o t h e r m s were linear and almost level. Multiple linear r e g r e s s i o n was used to determine the s e p a r a t e e f f e c t s o f temperature and f r e q u e n c y : Y = B
0
+ B
x
T
+ B
2
X
(23)
A t lower temperatures the f i x e d f r e q u e n c y t h e r e s p o n s e o f Y as a f u n c t i o n o f T was s i g m o i d a l . Good f i t s o v e r t h e e n t i r e range o f T were o b t a i n e d by r e g r e s s i o n t o the f u n c t i o n : Y = B + B
x
T + H [ l - tanh
(T-T )/S]
(24)
g
where B, B , H, S and T! a r e parameters s e l e c t e d t o f i t t h e d a t a . At e l e v a t e d v a l u e s o f T the l a s t term o f t h i s e q u a t i o n e x t r a p o l a t e s t o 0. Thus, B i s the same as i n e q u a t i o n (23) and B = B +B X. Regressions a t a s e r i e s o f X l e v e l s gave c o r r e s p o n d i n g s e r i e s o f H, S and T . Th§ r e g r e s s i o n parameters v a r i e d w i t h X. I n t h e c a s e s o f S and T t h e r e were good l i n e a r correlations. (Statistical correlation c§efficients, R> .99.) I n the case o f t h e H p a r a m e t e r the 1
x
2
0
3
102
SOUND AND VIBRATION DAMPING WITH POLYMERS
c o r r e l a t i o n w i t h X was poor. T h i s was b e c a u s e d i d n o t c o v e r s u f f i c i e n t l y low t e m p e r a t u r e s .
of
the
experiments
Results Test Data Validation. Samples o f Neoprene and NBR w i t h d i f f e r e n t t h i c k n e s s e s were u s e d t o s t u d y wave and b e n d i n g e f f e c t s . Figure 7 is a p l o t o f the magnitude and phase o f F ( w ) / x ( w ) v e r s u s f r e q u e n c y f o r the 1.59mm Neoprene specimen (L/h=16). Both r a t i o s a r e seen to be c o n s t a n t up t o about 2500 Hz. Above t h a t f r e q u e n c y the r a t i o s b e g i n t o change because the i n e r t i a o f the specimen was no longer negligible. Note that the phase a n g l e appeared more s e n s i t i v e t o wave e f f e c t s . S i m i l a r r e s u l t s were o b t a i n e d f o r the o t h e r Neoprene and NBR specimens. The f r e q u e n c i e s where the r a t i o s began t o d e v i a t e were d i r e c t l y p r o p o r t i o n a l to the length/thickness (L/h) r a t i o s o f the samples. F i g u r e 8 shows a p p a r e n t s t o r a g e modulus s p e c t r a f o r different thickness samples o f Neoprene. Sample i n e r t i a l e f f e c t s a r e o b v i o u s a t the h i g h ends of these spectra. For the t h i n n e s t sample (L/h=16), the wave e f f e c t s began about 2500 Hz, w h i c h agrees w i t h the r e s u l t s shown i n F i g u r e 7. As e x p e c t e d , the thicker samples showed the onset o f wave e f f e c t s i n p r o p o r t i o n t o sample L/h. A d e t a i l e d d i s c u s s i o n o f wave e f f e c t s may be found elsewhere (24.25). The f r e q u e n c i e s c o r r e s p o n d i n g t o the o n s e t o f wave e f f e c t s were u s e d i n e q u a t i o n 16 t o determine wave l e n g t h / t h i c k n e s s r a t i o . In all cases the critical ratios were 14 t o 16. T h i s was midway between p r e v i o u s e s t i m a t e s ; S c h r a g (26) e s t i m a t e d 20 on t h e o r e t i c a l considerations; F e r r y (1) e s t i m a t e d 10 on the b a s i s o f e x p e r i m e n t a l results. F i g u r e 8 a l s o shows t h a t the a p p a r e n t s t o r a g e modulus d e c r e a s e s d i r e c t l y with the sample l e n g t h to thickness r a t i o , L/h. At frequencies below the o n s e t o f i n e r t i a e f f e c t s the a p p a r e n t s t o r a g e m o d u l i f o r the L/h = 8 , 4, and 2 specimens were about 3%, 8%, and 18% smaller, respectively, than the t h i n n e s t sample (L/h = 16). These a r e a p p r o x i m a t e l y the same as t h e o r e t i c a l l y p r e d i c t e d i n T a b l e 1 and t h o s e r e p o r t e d i n the l i t e r a t u r e (27.28). F i g u r e 9 shows a t y p i c a l r e s u l t of directly comparing three methods f o r determining s t o r a g e modulus on the same sample. The p o i n t s were d e t e r m i n e d w i t h a commercial Rheovibron. The dashed line was results with forced vibration using s i n g l e frequency s i n u s o i d a l i n p u t s . The s o l i d l i n e shows FTMA d a t a from random n o i s e inputs. 0
Model.
The parameters
f o r the NBR
data are l i s t e d
Table II. Parameters for NBR Data used in Equation 24
B B H S T
x
100 Hz Reference 1.32 -0.00516 0.86 13.1 11.6
Frequency C o r r e l a t i o n Slope p e r decade 0.0321 0 0 2.07 9.6
i n Table I I .
GANERIWALA AND HARTUNG
£
