Dual beam complex modulus apparatus

System and method for measuring complex shear or Young's modulus of a polymeric material are described wherein first and second beams of preselected lengths and different thicknesses are disposed in parallel spaced relationship firmly held at first ends thereof and first and second spaced gripping members are attached along the beams, a specimen of polymeric material is disposed between confronting surfaces of the gripping members, a time varying force is applied to one beam, the time varying displacements of the beams are measured, and the modulus of the polymeric material is calculated from the measurements.

BACKGROUND OF THE INVENTION 
The present invention relates generally to methods and systems for 
measuring physical properties of materials, and more particularly to a 
direct measurement system for determining the complex moduli of polymeric 
materials. 
Methods for measuring complex moduli of polymeric damping materials 
generally are of two major types, namely, indirect resonant beam 
techniques and direct stiffness systems. The beam techniques are 
considered to be indirect because complex modulus values for the polymeric 
material are calculated, using appropriate formulae, from measurements of 
changes in modal damping and resonant frequency occurring in a beam of the 
polymeric material or in a beam coated with a polymer layer. Errors may 
arise because small differences between measurements on coated and 
uncoated beams may be magnified because of assumptions made in the 
mathematical treatment. 
Direct measurement systems operate by applying an appropriate time varying 
force to a specimen of the polymeric material and calculating the complex 
modulus from a measured response to that force. The measurement chain is 
often quite long and consists of many finite mass and stiffness elements 
which are major sources of error, such as unwanted resonances in the 
desired frequency range or systematic under-estimation of true specimen 
stiffness when near the system stiffness. These disadvantages are 
difficult to overcome and direct stiffness measurement systems are best 
used for low to medium range measured modulus values. 
The invention solves or substantially reduces in critical importance 
problems with prior art systems and methods as just described by providing 
system and method for measuring complex shear or Young's modulus of a 
polymeric material wherein first ends of two parallel beams of unequal 
thicknesses are attached to a base with a polymer specimen between the 
second ends, a time varying force is applied to one beam and the 
displacement responses of both beams are measured, with amplitude and 
phase angles determined by suitable displacement measuring sensors. The 
complex modulus of the polymer at the specific driving frequency and 
temperature is determined using an equation. 
It is therefore a principal object of the invention to provide improved 
system and method for measuring physical properties of materials. 
It is a further object of the invention to provide system and method for 
direct measurement of complex moduli of polymeric materials. 
It is yet another object of the invention to provide system and method for 
direct measurement of shear or Young's modulus of polymeric materials over 
wide frequency and temperature ranges. 
These and other objects of the invention will become apparent as a detailed 
description of representative embodiments proceeds. 
SUMMARY OF THE INVENTION 
In accordance with the foregoing principles and objects of the invention, 
system and method for measuring complex shear or Young's modulus of a 
polymeric material are described wherein first and second beams of 
preselected lengths and different thicknesses are disposed in parallel 
spaced relationship firmly held at first ends thereof and first and second 
spaced gripping members are attached along the beams, a specimen of 
polymeric material is disposed between confronting surfaces of the 
gripping members, a time varying force is applied to one beam, the time 
varying displacements of the beams are measured, and the modulus of the 
polymeric material is calculated from the measurements.

DETAILED DESCRIPTION 
Referring now to FIG. 1, shown therein is a sketch of system 10 
representative of the invention and useful in the practice of the method 
thereof. First and second beams 11,12 of preselected length and different 
thicknesses are disposed in preselected (adjustable) parallel spaced 
relationship, each firmly supported at one (e.g. lower) end to a 
substantially stationary base 13 through means such as flanges 14,15 and 
bolts 16. In the practical application of the method of the invention, 
beams 11,12 may ordinarily have length in the range of from about 5 to 20 
cm. Beams 11,12 have different thicknesses in order to provide a 
corresponding preselected wide difference in stiffnesses therebetween. 
Accordingly, beams 11,12 may comprise steel, aluminum, titanium, 
magnesium, or other suitable material; beam 11 will generally range in 
thickness from about 0.2 to 0.5 cm, and beam 12 will generally range in 
thickness from about 0.5 to 2.0 cm. Beams 11,12 optimally will have 
thicknesses differing by a factor of about 2 to 30, and preferably about 
5, so that a sufficiently wide range of specimen stiffnesses may be 
covered. A pair of substantially flat grip members 18,19 are attached as 
through bolts 20 at respective upper ends of beams 11,12, and are sized 
and configured to define space 22 therebetween for receiving a test 
specimen 23. It is noted that members 18,19 may, in an alternative 
structure for the invention, be disposed anywhere along the length of 
beams 11,12, which structure may provide certain versatility to 
measurement taking and specimen analysis utilizing the invention. Specimen 
23 normally is in the form of a flat sheet of material held firmly between 
confronting surfaces of members 18,19 using any suitable adhesive 24 as 
would occur to the skilled artisan guided by these teachings. Materials 
suitable for testing as specimen 23 in the practice of the invention 
include sheet materials such as silicone, urethane, acrylic, or similar 
polymer type materials. Specimen 23 thickness ordinarily ranges from about 
0.1 to 5 mm. The FIG. 1 arrangement places specimen 23 in a configuration 
for testing in shear, although arrangements comprising grip members 18,19 
of other geometries placing the specimen in compression or tension may 
also be envisioned within the scope hereof by the skilled artisan 
practicing the invention. Time varying force generating means 26 
(harmonic, non-harmonic, random or pulse) such as a piezoelectric or 
electro-dynamic shaker, magnetic force transducer, impact hammer or other 
appropriate device acts through force gauge 27 in order to selectively 
vibrate beams 11,12. The response of system 10 to force generating means 
26 is measured by suitable measuring means 29, such as an optical 
vibrometer, capacitance, non-contact or eddy current transducer, laser 
velocimeter, accelerometer or the like, attached to or placed 
appropriately close to the tip (top) ends of beams 11,12. 
Referring now to FIG. 2, shown therein is a simplified model of system 10 
for illustrating responses to force generating means 26 in a determination 
of system parameters for measuring complex moduli of specimen 23. In FIG. 
2, force generating means 26 acts with harmonic time-varying force F(t) 
upon effective masses m.sub.1,m.sub.2 supported, respectively, by 
stiffnesses k.sub.1,k.sub.2 representative of beams 11,12 and connected by 
complex stiffness k.sub.s representative of specimen 23. Parameters 
m.sub.1,k.sub.1,m.sub.2,k.sub.2 may be determined from tests on system 10 
absent specimen 23. The simple model shown in FIG. 2 is valid at 
frequencies as high as the first resonance frequency of the thinnest beam 
and as low as about zero Hz, although practical test times limit the lower 
frequency to about 1/1000 Hz. Elementary analysis applied to the FIG. 2 
model shows that the complex stiffness k.sub.s (1+i.eta..sub.s) of 
specimen 23 can be determined from measured harmonic responses X.sub.1 (t) 
and X.sub.2 (t) as follows: 
EQU k.sub.s =(k.sub.1 -m.sub.1 .omega..sup.2) R(X.sub.1 /(X.sub.1 -X.sub.2))(1) 
EQU .eta..sub.s =I(X.sub.1 /(X.sub.1 -X.sub.2))/R(X.sub.1 /(X.sub.1 
-X.sub.2))(2) 
where .eta..sub.s is the loss factor of the specimen, and .omega. is the 
driving frequency of force generating means 26 in radians per second; R 
and I are the in-phase and out-of-phase components of the system response 
and are evaluated from measured amplitudes .vertline.X.sub.1 .vertline. 
and .vertline.X.sub.2 .vertline. and phase angles .phi..sub.1 and 
.phi..sub.2 measured relative to the driving force signal 
F(t)=Fexp(i.omega.t), such that X.sub.1 =.vertline.X.sub.1 
.vertline.exp(i.phi..sub.1), R(X.sub.1)=.vertline.X.sub.1 
.vertline.cos(.phi..sub.1), I(X.sub.1)=.vertline.X.sub.1 
.vertline.sin(.phi..sub.1), etc. It then follows that, 
##EQU1## 
from which the in-phase and out-of-phase components are readily calculated 
using the theory of complex numbers, by one skilled in the art. At 
frequencies well below the first resonance of the system, m.sub.1 
.omega..sup.2 and m.sub.2 .omega..sup.2 are small compared with k.sub.1 or 
k.sub.2 and Eq (1) may be simplified accordingly. The dynamic range of the 
system may be extended by making k.sub.1 and k.sub.2 differ by one, two or 
more orders of magnitude (stiffness is proportional to the cube of beam 
thickness), and repeating the tests by exciting first one beam and then 
the other so that k.sub.1 and k.sub.2 are switched (may be done without 
disturbing the specimen). For a dynamic range of 80 db (e.g. 1/100 to 
100), values of ks may be measured within the range k.sub.1/ 100 to 
100k.sub.1. If k.sub.2= 100k.sub.1, switching excitation locations gives 
k.sub.s within a range k.sub.1/ 100 to 10000k.sub.1, and provides an 
effective dynamic range of 120 db. Measurements may be made over a wide, 
continuous, range of frequencies ranging from well below 1 Hz to 1000 Hz, 
as compared to a set of discrete frequencies characteristic of prior art 
(indirect/beam) techniques. 
Referring now to FIG. 3, shown therein is an alternative embodiment of the 
invention comprising two beams each clamped at both ends. In FIG. 3, 
various elements have functions similar to those of similarly named 
elements of the FIG. 1 embodiment. Beams 31,32 are sized and have 
different thicknesses similarly to beams 11,12 of the FIG. 1 system, but 
beams 31,32 are each firmly held at both ends 31a,b,32a,b to substantially 
stationary structure 33 in the parallel configuration suggested in FIG. 3. 
Gripping members 35,36 are disposed along beams 31,32 intermediate the 
ends thereof (preferably near the beam centers) in spaced relationship for 
supporting a specimen 37 therebetween for testing. In the FIG. 3 
arrangement, specimen 37 is held in tension or compression, although, as 
in the FIG. 1 configuration, other arrangements and grip member geometries 
may be envisioned by the skilled artisan to place specimen 37 in shear. 
Specimen 37 may be of the general form and may be held between members 
35,36 similarly to specimen 23 of FIG. 1. Time varying force generating 
means 26a selectively vibrates beams 31,32, and the response thereto of 
beams 31,32 and specimen 37 is measured by measuring means 29a disposed on 
either side of beams 31,32. Specimen 37 stiffness is then calculated from 
the measured responses using an analysis corresponding to that presented 
above. 
Referring now to FIG. 4, shown therein is another embodiment of the 
invention utilizing a single beam 41 clamped at both ends 41a,b to 
stationary structure 42. In the analysis of the system shown in FIG. 4 and 
the measurements taken therefrom in characterizing a specimen, it is seen 
that the FIG. 4 system is essentially a special case of the FIG. 3 system 
wherein one of the beams has zero thickness and stiffness. Specimen 43 may 
be held between gripping member 45 and a surface of one response measuring 
means 29b element. Time varying force generating means 26b is disposed to 
act directly into specimen 43, and the response of beam 41 and specimen 43 
is measured utilizing measuring means 29b elements. Specimen 43 stiffness 
may then be calculated as taught above considering zero stiffness of a 
second beam. 
It is noted in addition to the foregoing, and with reference again to FIG. 
1, that temperature dependence of specimen stiffness properties may be 
determined by disposing the system within a temperature controlled 
environment defined by enclosure 50, such as a furnace or refrigerator, as 
would occur to one skilled in the appropriate art guided by these 
teachings. 
The invention therefore provides system and method for direct measurement 
of complex moduli of polymeric materials. It is understood that 
modifications to the invention may be made, as might occur to one skilled 
in the field of the invention, within the scope of the appended claims. 
All embodiments contemplated hereunder which achieve the objects of the 
invention have therefore not been shown in complete detail. Other 
embodiments may be developed without departing from the spirit of the 
invention or from the scope of the appended claims.