Method and apparatus for determining dynamic mechanical properties of materials

A method and apparatus is provided for dynamic mechanical measurements of materials in which a plurality of coplanar coils are disposed transversely with respect to a magnetic flux field. The coils are rigidly mounted to a plate and arranged so as to eliminate inductive coupling between the coils. Alternating current is passed through some of the coils so as to vibrate or oscillate the plate because of the alternating force resulting from the interaction of the electrical current and the magnetic field. Other of the coils have an alternating motional electromotive force (voltage) generated in them because of their vibration. The mechanical impedance of the plate is determined from the vector ratio of the vibration forcing current to the velocity sensing voltage. When the plate is connected to the material to be tested in vibration, measurements of the vector ratio of electrical current to motional voltage then provide determinations of dynamic mechanical properties at vibration frequencies from 0.1 to 100,000 Hz.

FIELD OF THE INVENTION 
The present invention relates to a method and apparatus for measuring the 
dynamic mechanical properties of various materials, including viscoelastic 
materials such as gels, plastics, and rubbers. 
BACKGROUND OF THE INVENTION 
In my prior U.S. Pat. No. 2,774,239 issued Dec. 18, 1956 there is disclosed 
an apparatus for determining the dynamic mechanical properties of 
viscoelastic materials. This previously patented apparatus and method has 
yielded the complex compliance and complex modulus of materials ranging 
from viscous liquids and soft gels, thermoplastics and elastomers, to hard 
solids such as quartz, lead, and stainless steel. Although this prior 
measurement method and apparatus have been successful, there are 
substantial and significant drawbacks and limitations. First the 
measurement requires two, separate alternating current electrical bridge 
balances. Further, the two coaxial coils used for excitation and for 
velocity sensing are tightly coupled electromagnetically through a soft 
iron pole piece, and a pre-balance must be made, before each bridge 
balance, in order to eliminate or minimize the (non-motional) mutual 
inductance voltage in the velocity sensing coil by means of auxiliary 
shielding and test coils. In addition, circular, coaxial coils are 
arranged to be in annular gaps between pole pieces whose reluctance 
decreases the magnetic flux density. Thus, a very large permanent magnet 
must be used to assure the necessary high value of magnetic flux density 
in the gap. This large magnet requirement and the iron pole pieces needed 
results in a large and massive apparatus which, in turn, prevents rapid 
temperature changes and slows the acquisition of information on 
temperature dependence of dynamic mechanical properties. Finally, the 
determination of the complex modulus at each frequency, because of the two 
necessary bridge balances and the prebalance to eliminate mutual 
inductance, requires 10 to 20 minutes. 
The vibration excitation and sensing element in the prior apparatus 
consists of an aluminum alloy driving tube with coaxial coils as mentioned 
above, and it and the associated mechanical and electrical equipment are 
described in detail in U.S. Pat. No. 2,774,239. 
SUMMARY OF THE INVENTION 
The present invention overcomes the problems noted hereinbefore and 
provides a compact, efficient method and apparatus for rapidly obtaining 
accurate dynamic mechanical information on the material being tested. The 
invention employs a driving and sensing plate with flat, coplanar coils. 
The coils may be printed on the surfaces of the plate if it is a 
non-conductor, or printed on an insulating layer placed on the surfaces if 
the plate is metallic. Alternatively, the coils may be wound of fine wire 
and be cemented onto or embedded and cemented within the driving or 
sensing plate. The driving plate is disposed in a plane transverse to a 
magnetic flux field formed by pairs of magnets. A material to be tested is 
connected to the driving plate so that the test material is subject to 
stress and resulting deformation when current is passed through the coils 
in the plate. 
It is known that the magnetic flux linkage or electromagnetic coupling 
between flat, coplanar coils is much less than that between two coaxial 
coils at the same separation distance, but when the flat coils are close 
together the induced EMF in one arising from alternating current in 
another will still be appreciable. This nonmotional, induced EMF can be 
reduced by arranging two coils in a non-coplanar configuration, i.e., with 
perpendicular planes, as described in Polymer Bulletin, Vol. 18, pp. 
167-174 (1987). However, this non-coplanar coil configuration is 
inherently non-rigid at high vibration frequencies, requires a complicated 
magnetic field configuration, and as a practical matter, can not be easily 
adjusted to give zero inductive and capacitive coupling between the 
driving and sensing coils. 
By contrast the coplanar, three-coil arrangement of the present invention 
is intrinsically extremely rigid. Two of the small coils are provided to 
have identical dimensions and the same number of turns, but have reversed 
winding directions. Thus, the induced EMFs in each of these two sensing 
coils from the current in the third, driving Coil are exactly 180.degree. 
out of phase, and cancel. However, since the directions of the magnetic 
fields between pairs of permanent magnets are also reversed for each of 
the two sensing coils, then the motional EMFs in each of these coils will 
add. Thus, the present invention provides a means to eliminate the 
inductive coupling between the coils in the driving plate. Further, the 
structural arrangement provided by the present invention eliminates the 
need for two separate alternating current bridge balances and the need for 
a very large permanent magnet as was essential in the prior art device 
described hereinbefore. 
An object of the present invention is to provide a small, compact apparatus 
for rapidly producing dynamic mechanical measurements of various 
materials. 
Another object of the present invention is in providing a plurality of 
coplanar coils rigidly mounted on a plate disposed transversely in a 
magnetic flux field and so arranged as to eliminate inductive coupling 
between the coils. 
Other objects and many of the attendant advantages of the present invention 
will become more apparent when considered in connection with the 
accompanying drawings

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
The method and apparatus described herein may be used for determining the 
dynamic mechanical properties of solid, gel, and liquid materials 
including, but not limited to, plastic and rubber-like solids which are 
commonly called viscoelastic. Measurements on aluminum, steel, and other 
metals, and on crystals such as quartz, rock salt, and others can be made 
with this apparatus. It is also possible to use the present invention for 
measurements of dynamic mechanical properties of bone, intervertebral 
disks, and other animal tissues. Measurements on bio-polymers thus are 
possible and can be related to the physiological functions and state of 
the bio-material measured. 
A determination of the dynamic mechanical properties of materials is an 
important prerequisite to the design of structures and devices which will 
be subject to vibration, and also is necessary in efforts to control noise 
and/or vibration of machinery, automobiles, airplanes, and trucks. Dynamic 
mechanical measurements provide the basic information needed in attempts 
to reduce the energy loss from the heating of rolling tires and in 
calculations of power loss in tires. A reduction in the tire power loss 
will substantially increase the travelled miles per gallon of fuel burned. 
The present invention provides a means to measure the dynamic mechanical 
properties of materials and to measure these properties over a continuous 
and wide range of vibration frequencies and of temperature. Measurements 
of dynamic mechanical properties while a material is simultaneously under 
steady tensile or compressive mechanical loading can be made. The 
apparatus and method disclosed herein measure dynamic mechanical 
properties while a material is simultaneously subject to a D.C. or A.C. 
electrical field, and/or allows for simultaneous measurement of dynamic 
electrical properties while the material is undergoing mechanical 
vibrations and/or steady mechanical loading. The invention provides 
improved, simple, accurate, and precise means for these measurements. The 
invention provides measurements of complex viscosity, complex fluidity, 
complex shear modulus and shear compliance and related quantities, such as 
shear sound velocity and attenuation, as well as complex tensile and 
compressive modulus and compliance, and related quantities. The materials 
on which such measurements may be made range from liquids and soft gels to 
hard or stiff solids over continuous temperature and frequency ranges from 
-100.degree. to 150.degree. C. and 0.1 to 100,000 cycles per second. 
Referring more specifically to the drawings wherein like numerals disclose 
like parts throughout several views there is shown in FIG. 1A a plan view 
of a plate 10 having a flat, spiral, rectangular coil 11 mounted thereon. 
The plate 10 may may be made of a hard, rigid nonconductive material and 
the coil may be of fine copper or aluminum wire (e.g., 0.002 to 0.005 inch 
diameter) with insulated coating. The coils utilized in the present 
invention, may have from 500 to 1000 turns. The coil 11 may be of fine 
wire cemented to the support plate 10 with a rigid adhesive such as epoxy 
cement or alternatively the coil may be printed on the plate. FIG. 1B 
shows schematically a coil 11 mounted on one side of plate 10 and a 
further coil 11A mounted on the opposite side of plate 10. 
In the event the coils 11 and 11A are printed on the plate, the plate 10 
may be a nonconductor or metallic with an insulating layer thereon. An 
anodized surface on an aluminum plate may serve as an insulating layer or 
a high resistance silicon wafer may have coils printed directly thereon. A 
plate formed of a semiconductor may be doped to provide internal 
conducting paths in the patterns of flat, rectangular coils. 
FIG. 2 discloses schematically a drive plate 12 having a pair of velocity 
sensing coils 13 and 14 disposed between the outer surfaces of plate 12 as 
shown in FIG. 2A. The sensing coils 13 and 14 which are connected in 
series have identical dimensions and the same number of turns, from 500 to 
1000 turns, but have reversed winding directions. The larger, centrally 
located driving coil is shown at 15. The directions of the transverse 
magnetic fields across coils 15, 13, 14 are denoted in FIG. 2A by B.sub.C, 
B.sub.A, and B.sub.B respectively. The drive plate 12 has a reduced end 
portion 16. FIGS. 3 and 3A disclose in elevation and in cross section the 
assembled electromagnetic shear transducer having a base plate 17 and top 
plate 18. The base and top plates are made of a hard, rigid nonmagnetic 
material such as laminated phenolic, aluminum alloys or stainless steel. 
There are further provided pairs of end plates 19 and 20 disposed at each 
end of a set of magnets and these are provided with central slots 21 and 
22 respectively to receive the drive plate 12. 
The baseplate 17 has secured thereto a pair of blocks 23 of vibration 
isolation and insulating material which may be made of rubberized cork 
gasket material, silicone rubber or laminations of cork and rubber. In 
addition there may be plates of nonmetallic material or metal surrounded 
by rubber or cork gasket material. Fixed to the upper surfaces of the 
blocks 23 are slightly larger blocks 24 which are formed of hard rigid 
non-magnetic material similar to the material of the baseplate 17. Secured 
to the upper surfaces of the blocks 24 are blocks 25 which are identical 
to blocks 23. The combination of the base plate 17 together with the 
blocks 23, 24 and 25 provide a vibration isolation and insulating 
supporting structure for the magnets and holders. There is an identical 
supporting structure affixed to the top plate 18 in the form of blocks 26, 
27 and 28 corresponding respectively to blocks 24, 25 and 26. 
Secured to the blocks 25 and 28 are magnet holders' bottoms 29 and tops 30 
respectively. The magnet holders are made of solid or laminated soft iron 
or mild steel with high magnetic permeability and low magnetic reluctance. 
The magnets, of high strength, rare earth composition, are affixed to the 
inner surfaces of slots in the magnet holders and have the polarity as 
shown in FIGS. 3 and 3A. There are provided magnet holder spacers 31 which 
are disposed between the magnet holders 29, 30. The spacers are made of 
hard, rigid nonmagnetic material such as aluminum, stainless steel or 
plastic. The magnet sets including the end plates, magnet holders, magnets 
and spacers are held together by means of machine screws not shown in FIG. 
3, 3A. There may also be provided a housing (not shown) of cover plates to 
enclose the sides and back. 
The drive plate 12 is suspended within the air gap between the magnets and 
within the slots 21 and 22 in end plates 19 and 20 respectively. Fine, 
non-magnetic (e.g., stainless steel or phosphor bronze) wires 33 are 
attached to the driving plate 12 and affixed to adjusting screws 34 have 
grooved heads in which the support wires 33 are fitted and turned into the 
adjusting machine screws 34. The wires 33 permit longitudinal movement of 
the driving plate 12 in a plane transverse to the field of magnetic flux 
within the air gap but prevent movement of the driving plate in other 
directions. 
As shown in FIG. 3A, for dynamic mechanical measurements in shear a pair of 
samples 50, in the form of plates or disks, are pressed against the 
portion of the driving plate extending beyond the support wires 33 by 
sample clamps 51 which are fixed against motion in the direction of 
vibration. 
When dynamic measurements of a sample material in extension are to be made 
a driving plate 40 as shown in FIG. 4 has the reduced end portion 41 
provided with a clamp 42 to retain end portions of a pair of sample 
materials 43. The opposite ends of the samples are held by a clamp 44 and 
secured to a fixed support. When the driving coils in the plate are 
energized the magnetic flux within the air gap causes the plate to vibrate 
or oscillate. By means of the voltage produced in the sensing coil by the 
vibration the extensional compliance or modulus of the sample material can 
be determined from the complex/vector ratio of the driving coil current to 
the sensing coil voltage as described hereinafter. 
In FIG. 5 there is shown sample material 45 which is retained between clamp 
46 attached to a driving plate and a fixed structure at the opposite end. 
With this arrangement the flexural compliance or modulus of the sample 
material can be measured. 
In FIG. 6 there is shown a driving plate 47 which is made of sample 
material and has a reduced end portion 48 of sample material. The outer 
end of portion 48 is retained by a clamp so that the extensional 
compliance or modulus of the sample material may be determined. 
FIGS. 9A and 9B illustrate curves of complex shear and extensional 
compliance obtained from the automated flexure measurement systems, 
according to the present invention, for a polyurethane rubber and a tire 
tread stock rubber. 
FIG. 10 discloses the circuit for measurement of the ratio of the current 
in the force coil 15 to the motional electromotive force in the velocity 
sensing coils A and B corresponding to the coils 13 and 14 shown in FIG. 
2. Coils A and B are oppositely wound so as to eliminate the mutual 
inductance between the coils. The coil corresponds to the coil 15 of FIG. 
2. Coils A and B are the velocity sensing coils and coil is the current, 
driving coil although, if desired, the functions of the coils may be 
interchanged. Mutual inductance, M.sub.12, between coils A and B and coil 
is eliminated so that E.sub.2 represents only the motional EMF across the 
coils A and B and E.sub.1 the voltage across R which is I.sub.1 R where 
I.sub.1 is the current through coil and the resistance, R. 
The relevant physical properties of a material subject to vibration consist 
of a complex compliance or a complex modulus defined as the (vector) ratio 
of alternating strain to stress or alternating stress to strain, 
respectively. Thus for vibration in shear with an applied alternating, 
sinusoidal stress of frequency, f, 
EQU s(t)=s.sub.o sin 2.pi.ft (1) 
the resulting strain will be partly in phase and partly 90.degree. out of 
phase with the stress because of the elastoviscous (viscoelastic) nature 
of all real materials. 
EQU a(t)=a'.sub.o sin 2.pi.ft-a".sub.o cos 2.pi.ft (2) 
and this yields the definition of a complex shear compliance 
EQU J*=J'-iJ" (3) 
where J'=a'.sub.o /S.sub.0 ; J"=a".sub.o /S.sub.0. A complex shear modulus 
can also be used to describe the material response to a time-dependent 
stress where 
EQU G*=1/J*=G'+iG" (4) 
Materials vibrated in extension or compression can similarly be 
characterized by a complex extensional compliance, D*=D'-iD", or a complex 
extensional modulus, E*=E'+iE". Flexural or torsional vibrations can 
likewise be related to relevant complex moduli and compliances. In every 
case a proper characterization or evaluation of the vibration response of 
a material requires a knowledge of its dynamic mechanical behavior, as 
described by these moduli or compliances, over a range of temperatures, 
vibration frequencies, and as modified by steady loads, loading history, 
microstructures, together with other ambient conditions. 
Excitation/vibration of the driving plate is caused by an alternating 
current, I.sub.1 =I.sub.0 sin .omega.t, through the current (driving) coil 
15 (FIGS. 2, 2A) while it is suspended between the opposite poles of the 
permanent magnets as shown in FIGS. 3 and 3A. Since the coil "wires" (or 
printed conducting paths) are perpendicular to the magnetic field, the 
driving force according to Ampere's law is, 
EQU F.sub.1 =B.sub.1 l.sub.1 I.sub.0 sin .omega.t (5) 
where B.sub.1 (B.sub.C, FIGS. 2A, 3) is the perpendicular magnetic flux 
density in the air gap between magnets, l.sub.1 is the length of the 
driving coil turns that are in the magnetic field, and .omega.=2.pi.f, 
where f is the frequency of oscillation (vibration). 
As a result of this alternating force the plate will oscillate with the 
same frequency, but out of phase, with velocity, 
EQU v=v.sub.0 sin (.omega.t-.delta.) (6) 
and such that an alternating, open circuit, motional EMF, E.sub.2 will be 
generated in the velocity (sensing) coils 13, 14 (FIGS. 2, 2A) which are 
connected in series, according to Faraday's law, 
EQU E.sub.2 =B.sub.2 l.sub.2 v.sub.0 sin (.omega.-.delta.) (7) 
where B.sub.2 (B.sub.A, B.sub.B, FIGS. 2A, 3) is the (perpendicular) 
magnetic flux density in the gap at the location of the velocity coil, 
l.sub.2, is the length of velocity coil 13, 14 turns that are in the 
magnetic field, and .delta. is the phase shift between the driving current 
and the velocity. 
The mechanical impedance (Z.sub.Mp) of the suspended driving plate by 
definition is the vector ratio of force/velocity so that, 
##EQU1## 
where I.sub.1, E.sub.2, are the rms vector values of driving current and 
motional EMF, and K.sup.2 =B.sub.1 l.sub.1 B.sub.2 l.sub.2. The suspended 
driving plate mechanical impedance will result from air resistance, 
R.sub.a, elastance of the suspension wires, S.sub.w, and the inertial 
mass, M.sub.p, of the driving plate. 
##EQU2## 
When samples (50 FIG. 3A) of impedance, Z.sub.Ms, are clamped 51 against 
the driving plate 12, the measured mechanical impedance is increased to, 
EQU Z.sub.M '=Z.sub.Mp +Z.sub.Ms (10) 
and the measured vector ratio of driving current to motional EMF will 
change to I.sub.1 '/E.sub.1 '. Thus, in this system, it is only necessary 
to find the vector (or complex) ratio of current to voltage with and 
without a sample at each frequency of measurement to obtain the sample 
impedance, Z.sub.Ms. 
##EQU3## 
Once the mechanical impedance of the sample pair is known, the complex 
compliance or modulus can be calculated from the sample dimensions. For 
shear vibration of two disk-shaped samples, for example, 
##EQU4## 
where A is the cross sectional area of both samples, h is the thickness of 
each, i=.sqroot.-1, and Y.sub.Ms *=1/Z.sub.Ms * is the sample complex 
(vector) mechanical admittance. 
In order to get reliable results at high frequencies, it is necessary that 
the driving plate inertial mass be small so that the sample impedance is 
comparable to the driving plate impedance. If not, the driving plate 
impedance (.about.2.pi.fM.sub.p) becomes so large compared to the sample 
impedance that the difference between the current/voltage ratios with and 
without samples is very small and the results become unreliable. 
If the magnetic flux density is high, the "sensitivity constant", K.sup.2, 
can be large for small coils and small plates which yield low values of 
inertial mass. Therefore, the flat coil configuration used in this 
invention has the added advantage that no reduction in magnetic flux 
density results from the use of pole pieces or cores; the magnetic flux 
density is produced directly between permanent magnet poles. For best 
results, the ratio K.sup.2 /m should be high; an increase of K.sup.2 /m 
ratio from 10 to 100 times that of the apparatus described in U.S. Pat. 
No. 2,774,239 is possible in the present invention. 
In order to find both the magnitude and phase (and thus the in-phase and 
90.degree. out of phase components) of the I.sub.1 /E.sub.2 and I.sub.1 
'/E.sub.1 ' current to voltage ratios, a vector electrical impedance or 
admittance measurement method is needed, but there are many of these 
available. An electrical oscillator with quadrature outputs (exactly 
90.degree. out of phase) allows separate null balancing of the in-phase 
and 90.degree. out-of-phase components of the motional EMF relative to the 
driving current, for example. 
The velocity coils can alternatively be placed in one arm of the electrical 
impedance bridge circuit described in U.S. Pat. No. 2,774,239 and its 
dynamic electrical impedance determined for two known vector ratios of 
electrical current, I.sub.1 /I.sub.2, in the driving (current) coil and in 
the driven (velocity) coils, respectively, as described in that patent. 
Other electrical circuits and devices, phase meters, vector impedance 
meters, frequency response analyzers, etc. can be used to find the 
necessary current to voltage ratios. 
This invention provides a means for an absolute calibration of the 
electromagnetic transducer constant, K.sup.2, when values of the 
mechanical impedance of the suspended driving plate, Z.sub.Mp, are 
measured as a function of the frequency of vibration, f. That is, the 
90.degree. phase component of driving plate mechanical impedance is (Eq. 
9), 
##EQU5## 
and therefore, 
EQU .omega.X.sub.Mp =-S.sub.w +M.sub.p .omega..sup.2 (13a) 
At each frequency the current ratio I.sub.1 /E.sub.2 =Y.sub.12 * will have 
an in-phase and 90.degree.-phase component 
EQU I.sub.1 /E.sub.2 =Y.sub.12 *=G.sub.12 +iB.sub.12 (14) 
which, when multiplied by K.sup.2 are equal to the in-phase and 
90.degree.-phase components of driving plate mechanical impedance. 
Therefore, K.sup.2 B.sub.12 =X.sub.Mp, and 
##EQU6## 
From Eq. 15 above, it can be seen that if M.sub.p is known, the value of 
K.sup.2 can be found by plotting .omega.B.sub.12 vs. .omega..sup.2 to 
yield a straight line of intercept -S.sub.w /K.sup.2 and slope M.sub.p 
/K.sup.2. From the known mass of the driving plate, M.sub.p, and the 
measured slope, the constant, K.sup.2 can be calculated as can the support 
wire elastance, S.sub.w. 
With either the driving plate or the sample holder insulated above 
electrical ground (cf. FIGS. 2,3) a direct or alternating current 
electrical field can be applied between the sample holders and the driving 
plate during mechanical impedance measurements to ascertain possible 
effects of such electrical fields on the dynamic mechanical properties of 
materials. It is also possible to measure the complex dielectric constant 
and/or AC or DC resistivity of insulating materials while they are being 
vibrated in shear in this apparatus. 
It is obvious that by equal compression of the samples between the sample 
holders and the plate, the effects of compressive stress perpendicular to 
the vibration direction on dynamic mechanical properties can be found. 
Strips of sample materials can also be stretched transversely or in line 
with the vibration direction during dynamic mechanical measurements to 
study the effects of extensional stresses and/or elongations on dynamic 
mechanical properties and/or on dielectric properties. 
A simple variation of this invention to allow for measurement of 
extensional modulus or compliance is shown in FIG. 4. It is obvious that 
bars or strips of material fixed transversely to the driving plate 
vibration direction will allow for measurements of the flexural vibration 
properties of such bars or strips as indicated in FIG. 5. 
It is necessary to suspend the driving plate in the magnetic gaps of the 
permanent magnets by means of long, thin wires which keep it centered and 
fixed, but allow vibration in its longitudinal direction in order to 
measure liquids and soft gels. However, in measuring stiff, solid samples 
it is possible to hold the driving plate and coils in the magnetic fields 
of the permanent magnets by shear samples of equal thickness between the 
sample holders and the driving plate, for example. 
For stiff, solid samples such as metals and metal alloys, it is also 
possible to print coils, over an insulating layer, directly on a part of 
the material to be tested as shown in FIG. 6. In this case, the sample and 
driving plate are integral parts of the same piece, but the sample test 
section, of smaller cross section, will deform when vibrated with one end 
fixed while the parts of the driving plate where the coils are located 
will not deform. This scheme is particularly useful for determining the 
dynamic mechanical extensional modulus or compliance as illustrated in 
FIG. 6. The electromagnetic transducer constant, K.sup.2, and the support 
wire and driving tube mechanical impedance, Z.sub.Mp, are determined for 
each sample in this case before one end of the test section is clamped to 
give the additional sample mechanical impedance, Z.sub.Ms. If the stiff, 
solid material is a non-conductor, the current coil and velocity coil can 
be printed directly on the sample. 
The sample holder and/or clamping device can be rigidly mounted to the 
apparatus frame or housing as shown schematically in FIGS. 3 and 4, but in 
order to avoid possible coupling to vibrational modes of the entire 
apparatus, the sample clamps can alternatively be housed in a compact, 
massive structure which is isolated from the apparatus frame; because of 
its great mass, such a sample holder device will not vibrate except at the 
lowest frequencies where its vibration can be measured and taken into 
account in determining the sample mechanical impedance. 
In the preceding description of this invention, the single large coil shown 
in FIG. 2 was used as the driving coil, 15, in which an alternating 
electrical current was put in order to produce the vibrating force. The 
two small coils (13 and 14) in series but with reversed winding directions 
comprised the velocity-sensing coils needed to determine the mechanical 
impedance/admittance. However, it is clear that the coil functions can be 
interchanged; that is, alternating electrical current passed through the 
two small coils will produce a vibrating force while the single large coil 
can become the velocity sensing coil. In either case, the net inductive 
coupling between the three coils will be eliminated as needed for this 
measurement method. Further, to obtain exactly zero inductive coupling, 
the amount and phase of the current in each of the two small coils can be 
adjusted if necessary when they are used to produce the driving force. 
Conversely, when used as velocity sensing coils, the electromotive force 
(EMF) from each of the two small coils can be adjusted to bring their net 
induced, non-motional EMFs exactly equal to zero. 
It is also an important feature of this invention that vibration 
reduction/isolation means/materials are used to prevent the vibrating 
force produced by the driving coil or coils from causing any vibration of 
the permanent magnet holders or magnets producing the magnetic fields 
across the velocity sensing coil or coils; such vibrations cause 
corresponding vibratory motion of the magnetic fields and a spurious EMF 
in the sensing coil. 
A thin metal shield insulated from but extending across both sides of the 
vibrating force coil prevents capacitive coupling with the velocity 
sensing coil. 
From the measured values of complex modulus, it is also possible to 
calculate the relevant sound velocity and attenuation of materials. 
Further, with inductive and capacitive coupling between the vibration force 
coil or coils and the velocity sensing coil or coils eliminated, the 
automated measurement system illustrated in FIG. 7 allows measurement at 
each frequency within 1 to 20 seconds instead of within 10 to 20 minutes 
as in prior dynamic mechanical measurement methods and apparatuses. In 
this figure a functional diagram of one such automated system is displayed 
where a commercially available two-channel frequency analyzer, personal 
computer, X-Y plotter and printer are combined through the switch, S, with 
the shear transducer shown in FIGS. 3, 3A or a corresponding flexure 
transducer (shown schematically in FIG. 5), and a standard 100-ohm 
resistor in the resistor box. The frequency analyzer provides the 
energizing alternating voltage E=E.sub.0 sin .omega.t across the driving 
force coil 1 (15 in FIG. 2) and the standard, known resistance, R as shown 
in FIG. 10, producing an alternating current, I.sub.1, in coil 1. The 
frequency analyzer is capable only of determining the magnitude ratios and 
phase differences (i.e. complex or vector ratios) of two voltages, whereas 
the method hereinbefore described requires the complex or vector ratio of 
a current to a voltage. Therefore the voltage, E.sub.1 =I.sub.1 R shown in 
FIG. 10 is put into one input channel of the analyzer, and the motional 
voltage, E.sub.2, shown in FIG. 10 generated by coil 2 (13, 14 of FIG. 2, 
connected in series) is put into the other channel, and the complex (or 
vector) ratio E.sub.1 /E.sub.2 is determined. Then, 
EQU E.sub.1 /E.sub.2 =I.sub.1 R/E.sub.2, and I.sub.1 /E.sub.2 =E.sub.1 
/RE.sub.2(16) 
from which the necessary mechanical impedance is found as noted in Equation 
8. 
The magnitude, wave form, frequency, and frequency measurement intervals of 
the alternating voltage to be put across the force coil of the transducer 
are selected via the personal computer that runs the MS-DOS operating 
system. The frequency analyzer and plotter are controlled via a GPIB 
(IEEE-488) controller board. Numerical data summaries and data evaluations 
can be displayed on the computer monitor or printed; in every case the 
operator controls the data taking and display via the computer keyboard. 
Obviously many modifications and variations of the present invention are 
possible in light of the above teachings.