NATL INST. OF STAND & TECH Reference

NBS

Publications A 111 D b EbOlbl

NBSIR 83-2776

m *

7

t

' J «

Determination of the Viscoelastic Shear

Modulus Using Forced Torsional Vibrations

Center for Manufacturing Engineering

Automated Production Technology Division

U S. Department of Commerce

National Bureau of Standards

Washington, D C. 20234

Final Report

September 1 S83

-QC

100

.1156

33-2776

1983

)EPARTMcNT OF COMMERCE

'JAL BUREAU OF STANDARDS

NBSIR 83-2776

* *

NATIONAL BUREAU

OF STANDARDS UBRART

DETERMINATION OF THE VISCOELASTIC SHEAR

MODULUS USING FORCED TORSIONAL VIBRATIONS

OC/60

. UZ(=>

n°‘

Edward B. Magrab

Center for Manufacturing Engineering

Automated Production Technology Division

U.S. Department of Commerce

National Bureau of Standards

Washington, D.C. 20234

Final Report

September 1 983

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director

TABLE OF CONTENTS

Page

ABSTRACT a

INTRODUCTION 1

THEORY 1

DESIGN CONSIDERATIONS 5

1. General Requirements 5

2. Temperature Considerations 5

3. Fixture 5

4. Determination of J and fm 10

5. Specimen Mounting 13

INSTRUMENTATION 13

RESULTS 25

ACKNOWLEDGEMENT 31

REFERENCES 32

APPENDIX I 33

APPENDIX II 38

Torsion Spring Base 38

Accelerometer Mounting Arm and Accelerometer 38

Specimen Mounting Plate 41

Allen Screw Heads 41

Total Mass Moment of Inertia 44

APPENDIX III - Computer-Controlled Relay Switches 45

APPENDIX IV - Subroutine Names and Their Functions 48

APPENDIX V - Program Flow Chart 50

FIGURE TITLES

FIGURE 1

FIGURE 2

FIGURE 3

FIGURE 4

FIGURE 5

FIGURE 6

FIGURE 7

FIGURE 8

FIGURE A-l

FIGURE A

FIGURE A-3

FIGURE A-4

GEOMETRIC DESCRIPTION OF TORSION SPECIMEN AND ATTACHED SPRING

AND MASS.

FORCED TORSIONAL VIBRATION APPARATUS.

TRANSFER FUNCTION OF BOTTOM TORSIONAL SPRING AND ATTACHED MASS.

SCHEMATIC OF COMPLETE INSTRUMENTATION SYSTEM.

SIMPLIFIED VERSION OF A PORTION OF THE INSTRUMENTATION SYSTEM.

BACK-TO-BACK CALIBRATION FIXTURE.

SHEAR STORAGE MODULUS OF POLYURETHANE WITH 4% AIR AS A FUNCTION

OF FREQUENCY AND TEMPERATURE.

(

LOSS TANGENT OF POLYURETHANE WITH 4% AIR AS A FUNCTION OF

FREQUENCY AND TEMPERATURE.

DIMENSIONS OF TORSION SPRING BASE.

DIMENSIONS OF ACCELEROMETER MOUNTING AREA AND ACCELEROMETER.

DIMENSIONS OF SPECIMEN MOUNTING PLATE.

DIMENSIONS OF ALLEN SCREW HEADS.

ABSTRACT

A forced torsional vibration system has been developed to measure the shear

storage and loss moduli on right circular cylindrical specimens whose diameter

can vary from 2 to 9 cm and whose length can vary from 2 to 15 cm. The method

and apparatus are usable over a frequency range of 80 to 550 Hz and a temperatur

range of -20°C to 80°C.

INTRODUCTION

Many methods exist for the experimental determination of the viscoelastic

properties of materials. Some of these methods have been summarized in References 1 and 2 and those used more recently in a collection of abstracts in

Reference 3. The present method described herein, which uses forced torsional

vibrations, is an updated version of previous works (4,5). It was selected

because it most easily met the requirements placed on the sample's geometry

and dimensions, which was a circular cylinder whose diameter ranged from 2

to 9 cm and whose length ranged to 15 cm. This wide range of sizes is a consequence of the desire to use the same sample that was previously subjected to

a different kind of material properties’ test over a different (higher) frequency

range. The method and apparatus described subsequently is usable over the

frequency range from 50 - 550 Hz and a temperature range from - 20° to 80° C.

THEORY

Consider the forced harmonic torsional vibrations of a right circular

cylinder having a frequency dependent complex viscoelastic shear modulus

G*(f) = G'(f) + jG"(f), where G'(f) is the shear storage modulus, G”(f) the

loss modulus and f the frequency. If a mass with mass moment of inertia J and

a torsional spring of constant k are attached to one end of the cylinder as

shown in Figure 1 and the other end of the cylinder is subjected to a harmonically

varying torque, then the expression for the angular acceleration response ratio

of the top plane of the cylinder to the bottom plane is given by (4,5)

A

o

(Acc)

(Acc)

TOP

BOT

C^fi*sin(ft*) + cos(fi*)

-1

( 1 )

3

where

2J [ (f / f

2 - 1]

C = 5

1 4 4

irph(b - a )

ft* = x - jy

( 2 )

f

m

1_

2tt

x 2tt cos (8/2)

y

= 2 it

, 2, 2

pf h sin (0/2)

P =

[g' 2

+ g" 2

]

1/2

9

0 = tan"

1

(C’/G')

and p is the density of the cylinder, h its length, b its outer radius, a

its inner radius and G"/G* the loss tangent. The quantity f mis the natural

frequency of the attached spring-mass system with the cylindrical specimen

removed. The quantity r

t is the distance from the center of the axis of the

cylinder to the center of the top accelerometer and r^ is the distance from

the axis to the point of application of the applied force, which coincides

with the location of the bottom accelerometer.

(3)

(A)

Substituting Eq. (3) into Eq . (1) yields

A

o = R e

j<t> (5)
4

where

R

( 6 )

and

A = [C-jXsinCx) + cos(x) ]cosh(y) - C^ycos (x) sinh(y)

B = [C-jXcosCx) - sin(x) ] sinh(y) + C^ysinCx) cosh(y) (7)

Thus, if R and ^ are the measured amplitude ratio and phase angle, respectively,

and all the physical and geometric parameters of the specimen are determined by

other means, then G* and G” can be found using Eqs. (6) and (7). However,

because of the complexity of these equations, G’ and G" cannot be solved for

explicity. The numerical procedure used to obtain these quantities is described

in Appendix I.

As the excitation frequency approaches zero and f << fm

Eq . (1) becomes

( 8 )

where

(9)

O and G is a nominal value for p. The range of values for C

Q

, in N/m , for

5 Q

the experimental setup are approximately 2.5 x 10 C

Q

2 x 10 . Thus

7 2 for materials with a shear modulus of 3 x 10 N/m , A will vary from a

value slightly less than r t /r^ to a value of approximately (r t /r^)/8.
5

DESIGN CONSIDERATIONS

1. General Requirements

The general requirements are that (1) the specimens can range in size to

approximately 15 cm in length and 9 cm in diameter; (2) the shear moduli can be

as low as 3 x 10^ N/m^; (3) the method provides the shear modulus over a broad

frequency range from 50 to 550 Hz and temperature range from -20° to 80 °C.

2. Temperature Considerations

The temperature requirements place a limit on the overall size of the test

fixture, for it has to fit into a temperature chamber of reasonable size. It

also has to weigh a modest amount so that one could place it into a chamber

without the chamber requiring additional structural support. Lastly, in order

to operate the vibration exciters over the temperature range, the exciters have

9 to be air cooled/heated with room temperature air (20 °C) so that the springs

that sustain the exciter’s moving element retain their desired properties.

The temperature of the specimen is determined from the placement of a

thermocouple on the specimen’s surface and is read by a digital output device

with an accuracy of +1 °C.

3. Fixture

According to the theory used to determine the shear modulus from the

experimentally determined data, the specimen should be subjected to torsional

motion only, with no bending of the specimen. In addition, the fixture itself

must be free from structural resonances over a broad frequency range and for test

samples of varying length and diameter. The design chosen was a combination of

certain features of previous works [4, 5] and is shown in Figure 2.
6

FIGURE 2 FORCED TORSIONAL VIBRATION APPARATUS
7

To minimize bending the top and bottom torsional springs have an "x"

cross-section, which is very stiff in bending compared to its twisting resistance.

To show this consider a single force F acting on the spring through a moment

arm about the center of the spring of radius rg . The displacement due to

bending, Sg, is

and that due to rotation

2 FrfL

_ b S T

~ KG

where G and E are the shear and tensile moduli of the spring, respectively, L

is its length and I and K are the moment of inertia and the torsion constant of

the cross-section, respectively. The accelerations of the spring are proportional

to these displacements. Thus a good measure of their relative resistance to

this unbalanced torque is their ratio. Hence,

1 FL

3

S B 3 El

s,

s

T

B 1

6 (l+v)

where v is Poisson's ratio. For the ”x” cross-section shown in Figure 1,

3

( 10 )
8

Therefore

,

The bottom spring has the following dimensions: L = 8.26 cm, rg = 7.46 cm,

t = 0.32 cm and D = 10.16 cm. Using these values in Eq . (11) yields that

s b/ s t - 0.001. Thus, the bending displacements can be expected to be on the

order of a thousand times less than the desired torsionally induced displacements.

In the actual system two dynamic exciters are used, which eliminates most of

the unbalanced force. The output force of the shakers, however, is not

controlled, and it is assumed that the same input voltage to both exciters

yields approximately the same output force. However, because of this great

difference in stiffnesses, any small imbalance does not strongly couple to the

system. For the top spring the parameters in Eq . (10) are: L - 5.08 cm,

t = 0.0794 cm and D = 6.99 cm. Equation (10) now yields sg/s-p - 0.0002.

The experimental confirmation of the excitation portion of the fixture's

resistance to bending is shown in Figure 3. The input signal to the shakers

was broadband random noise. A commercial digital frequency analyzer was used

to obtain the time-averaged transfer function of the input voltage to the shakers

to the bottom accelerometer. The result is shown in Figure 3. The absence of

any resonance peaks until approximately 680 Hz, except the one directly related

to the first torsional resonance of spring-mass system, can be seen. This indicates that the useful frequency range of the fixture is as high as approximately

550 Hz. The torsional natural frequency of the bottom spring mass system is

of no consequence in the experiment because the magnitude of the input acceleration is kept relatively constant over the entire test frequency range.
9

(9P) aanindiAiv 3Aiivi3a

FIGURE

3

TRANSFER

FUNCTION

OF

BOTTOM

TORSIONAL

SPRING

AND

ATTACHED

MASS
10

4. Determination of J and f m

The top end of the top torsion spring is bolted to a movable solid steel

cylinder as shown in Figure 2. According to the theory developed previously,

this connection must be rigid compared to the torsion spring itself. The ratio

of the torsional rigidity of the solid cylinder, Kc , to that of the torsion

spring, K (given by Eq . (10)), is

Substituting in the appropriate dimensional values for the fixture shown in

Another important consideration is the range of values for J, the mass

moment of inertia of the attached mass. For a given natural frequency

of this system the specimen, when connected to J, must be able to influence

both the value of fm and, more importantly, its damping. An indication of

the magnitude of the parameters that influence these properties can be

approximately determined by assuming that the vibrating system consists of the

inertia J connected to two springs; one is the "tee" cross-section torsion spring

already discussed and the other is the spring formed by the torsional resistance

of the specimen, assuming that its internal damping can be ignored in this

portion of the analysis. The natural frequency of this new system, f^, is then

given by

Figure 2 yields K

c /K - 10^. Thus, the assumption of rigidity is a good one

( 12 )
11

where fm

is given by Eq . (3) and

C

2

jl / G

2

\

h

(b

4 -a

4

) /l\

(2nf ) 2

\ J

/

(13)

It is seen that in order for C2 to be greater than one, the combination of

h, fmand J must be chosen carefully for a given material. Unfortunately the

class of materials that are to be tested are very weak in torsion and typically

have a shear modulus of 4 to 15 x 10^ N/m^. Consequently the design requires

that J, h, and fm

be as small as possible. There are, of course, some physical

limitations in just how small J can be made. In the apparatus shown in Figure 2,

J = 1.8972 kg-cm^. The computational procedure to obtain J is given in

Appendix II.

Whereas J was computed, the natural frequency of the attached spring/mass

system, f m , without the test specimen attached was determined experimentally by

using a digital frequency analyzer in its transfer function mode. The excitation

was applied with a hammer to one of the "ears" (see Figs. 2 and A-2) of the top

mass. The noncontacting end of the hammer had an accelerometer mounted to

it. The resulting motion of the spring/mass was recorded by the accelerometer

mounted on one of the "ears." Using the zoom capability of the analyzer, fm

was determined to within +0*25 Hz, which, for natural frequencies fm > 100 Hz

results in an uncertainty of 0.25% or less. The results for two torsion springs,

one with t = 0.7938 mm and the other t = 1.588 mm (see Figure 1) and for two

thickness of the accelerometer mounting disk (see Figure A-2) are summarized

in Table 1. From the discussion in the preceding paragraph it is seen that

for the weaker class of materials the spring/mass combinations resulting in

the lower set of fmshould be used.
12

Values of

Spring flange

thickness (mm)

0.7938

1.588

Table 1

for Four Combinations of Springs and Inertias (J)

J = 1.8972 kg-cm^

(h^= 6.35 mm)

f-(Hz)

J = 2.3652 kg-cm^

(h^= 12.75 mm)

100.75 89.00

283.00 255.00
13

5. Specimen Mounting

In order that the specimen be mounted concentrically with respect to the

top and bottom springs and to eliminate any pre-twist of the specimen, the

following technique was employed. Two discs, one 12.3 cm in diameter that

attaches to the bottom "ears" and the other, 6.9 cm in diameter that attaches

to the top "ears" and which is included in the calculation of J, each have on

one of its faces a concentric raised disk 1.5 mm high and 12.7 mm in diameter.

These disks are mounted on their respective bell cranks with these raised

disks facing each other. Prior to mounting the cylindrical specimen, the

specimen is placed in a lathe to have each of its end planes turned smooth and

perpendicular to its axis. In addition, a concentric cylindrical depression

is turned on each of the end planes that is 1.59 mm deep and 12.7 mm in diameter.

Epoxy is applied to each end plane of the cylindrical specimen and the specimen

is placed onto the bottom disk. The top spring mass assembly is then lowered

onto the top of the specimen and the epoxy is allowed to cure.

INSTRUMENTATION

The instrumentation system used to measure the accelerometers ' amplitude

ratio and phase angle is shown in Figure 4. The electrodynamic vibration

exciters are connected in parallel and receive their input voltage from an

amplifier with the capacity to provide 15A rms into 1 ohm. The input to the

amplifier is connected to an oscillator. The exciters themselves have an

impedance of approximately 2 ohms and require 5A rms to obtain their rated output force. The oscillator's output voltage amplitude and frequency are under

computer control.
14

FIGURE

4

SCHEMATIC

OF

COMPLETE

INSTRUMENTATION

SYSTEM
15

The accelerometers have a sensitivity of nominally 10 mV/g and have a

unity gain preamplifier built into them. The output signals from the accelerometers are then amplified 30 dB and passed through two 2 Hz bandwidth tracking

filters. The tracking frequency is provided by a second output signal from

the oscillator which remains constant at 1 V. This 1 V signal is amplified to

meet the tracking filter’s requirement of 3.5 V. The 30 dB gain given to each

accelerometer signal provides better use of the dynamic range of the tracking

filters and uses the digital phasemeter in a voltage range in which its accuracy

is better, namely, at levels above 50 mV rms . To maintain control over the

voltage levels throughout the electronic measuring system, the output voltage

of the oscillator is continuously adjusted so that the output voltage of the

bottom accelerometer remains approximately constant at 2.5 mV rms over the

entire test frequency range.

The signals from the output of the tracking filters are sent directly to

the digital phasemeter and the digital voltmeter, both of which are under

computer control. However, the signals to the voltmeter are read sequentially

with the aid of a computer controlled switching setup described in detail in

Appendix III. The phasemeter has an autocalibration feature which is employed

every sixth measurement to insure that the phase measurements are as precise as

possible. The digital voltmeter is used in its autoranging mode, since the

output voltage for the upper accelerometer differs widely over the frequency range.

The purpose for the attenuator (labeled ATTN) is discussed in detail

subsequently.

To accurately measure the acceleration ratio of the top accelerometer to

the bottom one and the phase angle between them, one must remove from the

measured data the influences of any electronics inserted between the electrical
16

signal at the output of the accelerometers and the recording instruments: the

digital voltmeter and the digital phasemeter. The ultimate accuracy of the

measurement depends, of course, on the accuracy of the voltmeter and the

phasemeter. In this experiment the voltmeter has an accuracy of approximately

+0.2 dB for rms voltage above 100 mV and +0.3 dB for voltages less than 100 mV

and greater than 100 pV over the frequency range 1 to 2000 Hz. The phasemeter

has an accuracy of jjD.l 0 for rms voltages greater than 50 mV and +0.2° for rms

voltages as low as 1 mV from 50 Hz to 50 KHz.

The effects of the intervening electronics are removed in the following

manner. Consider the simplified equipment diagram given in Figure 5. Let

Hj(f) be the transfer function of the accelerometer's built-in impedance

converters, H^j the transfer function of the 30 dB fixed gain amplifiers, and

Hpj the transfer function of the tracking filter, where j = 1 refers to the

bottom accelerometer channel and j = 2 to the top channel. The true output

signal from the accelerometers, Vj , are related to the signals appearing at the

inputs to the digital voltmeter and phasemeter, Vjy , as follows:

(14)

Solving Eq. (14) for the acceleration ratio A0 = V2 /V 1 yields

(15)

where C is a complex quantity given by

( 16 )
17

LJJ Zz<xo

CLo

Hc\i

I! • swH

cn

C/2 ZOMHCHZW

Cxi H

cnZ

Eda;H

faozowH

cti O

fa<

faozow

c/2 ad

Ed>o

EdM

faMM

fa2H

C/E

LO

Ed

cxi zOM

fa
Thus, the amplitude ratio and phase angle are, respectively,

*21 *R21

"

( 17 )

where | Vj^2 / Vri | is the ratio of the values read from the digital voltmeter and

b^2 i is the value read from the digital phasemeter. The quantity |C~^| is the

correction that must be isolated and measured in order to remove the influence

of the electronic components on the amplitude measurement and <p c is the

quantity that must be isolated and measured to remove the influence on the

phase angle measurement.

To remove the effects of C at a given temperature, we proceed by first

removing Hqj and Hpj from the measurement chain and perform a back-to-back

calibration of the two accelerometers. Thus, V

2 = V^eJ (since one accelerometer

is upside down with respect to the other) and

H„ V „ 2 R2

H-, V

1 R1

. *21 ' *BB + ” ' (18)

where |Vr2/Vri! is determined directly from the reading of the digital voltmeter and

directly from the digital phasemeter. A picture of the back-to-back calibration fixture is shown in Figure 6.

We now disconnect the accelerometers from the measurement chain and replace

them by two signals at the same frequency that differ only in amplitude and not

phase. Thus = Vp and V2H2 = At Vp where Vp is the amplitude of the input voltage

to both channels and 0 < At < 1. The attenuation At is introduced because of the
19

FIGURE 6 BACK-TO-BACK CALIBRATION FIXTURE
20

high degree of amplitude and phase nonlinearity of the tracking filters. The

introduction of the attenuator allows the input signals to each channel to be

approximately equal (to within +2.5 dB) to each accelerometer's output signal.

In the actual test procedure the attenuator At can be introduced to either

channel depending on whether or not the accelerometer's output signal ratio is

greater than or less than 1. If the At is switched, then At is simply replaced

by 1/At in the subsequent results. Equation (17) then yields

H

G2

H

F2

= A

V

R2

h

gihfi

t V

R1

21 (19)

where |Vr2 /VriIi is determined directly from the readings of the digital voltmeter

and directly from the digital phasemeter. Using Eqs. (18) and (19) yields

the desired result

"I V

2 V

R2 A VR2 V

R2

V

1

V

R1

n. t V

R1 B-B

V

R1 I

-J

^21 ^R21 ( 20 )

The test procedures used to determine the quantities in Eq . (20) are done under

computer control using a set of 16 relays. The details of this switching system

are given in Appendix III.

In the actual testing of the accelerometer's back-to-back response it was

found that at a given temperature it could be assumed that the accelerometers

relative amplitudes are constant over the frequency range of interest and that

the phase difference was essentially zero. In other words |Vr2/VriIbb = C 3 , a.

constant and <{>33 - 0 in Eq. (18). The constant C 3 is determined at each

temperature at a nominal frequency of 200 Hz.
21

The errors associated with the measurement of the amplitude ratio and

phase angle cannot be easily related to the resulting errors in the determination of the shear modulus because of the highly nonlinear nature of Eqs. (5)

to (7). Consequently, to get an estimate of the accuracy of the numerically

evaluated G* and G", the following procedure is used. Equation (1) is evaluated

over a frequency range for an assumed value of G' and G". This results in a

set of amplitude ratios (AQ

) and phase angles ( 4>) as a function of frequency.

The set of AQ and <|> are then altered by the accuracy of the digital voltmeter

and phasemeter, and the inverse problem is considered. Thus, at a given frequency we change the amplitude ratios by up to +0.2 dB and the phase angle by

up to +0.3° and solve Eq. (11) for G' and G” . The results are shown in Tables

2 and 3 for J = 1.9 kg-cm 2

, f m

= 100 Hz , p = 1000 kg/m3

, G' = 20 x 10 6 N/m2

,

G"/G* = 0.4, h = 10 cm, b = 5 cm and a = 0. Since the technique only works in

the region of the systems resonance, frequencies above 175 Hz were not considered.

It is seen from the results presented in Tables 2 and 3 that the percentage

errors in G

f and G"/G’ are asymmetrical with respect to the errors in the

amplitude and phase at frequencies away from fmand large and unequal at

frequencies at or near f m , with the largest errors occuring in the loss tangent.

Another inaccuracy in the determination of the values for G’ and G" are

errors due to the calculation of J and the determination of f min Eq. (2).

Using a procedure similar to what was done above gives the effects of J and f m

shown in Table 4. There are no effects of the errors in J and f mon the

ratio G"/G’. Thus, the results in Table 4 apply to both G' and G" , individually.

The total error in the determination of G* and G" are the sum of the errors

appearing in Tables 2 and 4, and 3 and 4.
22

TABLE 2

PERCENTAGE DIFFERENCES IN COMPUTED STORAGE MODULUS

AS FUNCTION OF ERRORS IN AMPLITUDE AND PHASE

50 Hz 75 Hz

PHASE

ERROR AMPLITUDE RATIO ERROR (dB> AMPLITUDE RATIO ERROR ( dE > (Deg.

)

-.2 -. 1 0.0 . 1 -.2 -. 1 0.0 . 1 . 2

-.3 5.6 2.3 -1 . 2 -4.8 - 8.6 9. 4 3.9 - 2.2 - 8.8 -16.2

-.2 5.9 2.6 -.8 -4.4 - 8.2 9.9 4.5 -1.5 - 8.0 -15.2

-. 1 6.2 3.0 -.4 -4.0 -7.7 10.4 5. 1 -.7 -7. 1 -14. 1 0.0 6.5 3.4 0.0 -3.5 -7.2 11.0 5.7 0.0 - 6.2 -13.0

. 1 6.9 3.7 . 4 -3. 1 -6.7 11.5 6 .

.7 -5.3 - 12.0

. 2 7.2 4. 1 . 8 - 2.6 -6.3 12.0 7.0 1 . 5 -4.4 -10.9

• «-» 7.5 4.4 1 .

- 2.2 -5.8 12.6 7.6 2.3 -3.5 -9.8

1 00 Hz 125 Hz

PHASE

ERROR AMPLITUDE RATIO ERROR (dE) AMPLITUDE RATIO ERROR ( dE > (Deg.

-. 1 0 . 0 . 1 . 2 a -. 1 0.0 . 1 . 2

-.3 - 12 . 1 -3.8 4.2 11.6 18. 1 - 1.2 -.3 . 7 1 . 6 2.5

-.2 -14.8 -5.8 2.3 10.5 17.3 -1.4 “ a 5 . 4 1 .

2.3

-. 1 -17.5 -7.7 1 .

9.5 16.5 -1.7 -.7 . 2 1 .

2 . 1 0.0 - 20.

-9.6 0.0 8.4 15.7 - 1 .

- 1 .

0.0 . 9 1 .

. 1 -22.9 -11.5 -1.4 7. 4 15.0 - 2.2 - 1.2 -.2 . 7 1 .

. 2 -25.6 -13.4 -2.7 6.5 14.3 -2.4 -1.4 -.4 . 5 1 .

. 3 -28.3 -15.2 -4.0 5.5 13.6 -2.7 -1 . 7 -.7 . 3 1 . 3

150 Hz 175 Hz

PHASE

ERROR AMPLITUDE RATIO ERROR ( dE > AMPLITUDE RATIO ERROR (dE)

(Deg.

-.2 -. 1 0.0 . 1 . 2 -.2 -. 1 0.0 . 1 . 2

-.3 . 4 . 3 . 2 . 2 . 1 1 . 1 . 6 . 1 -.4 -.9

-.2 . 3 . 3 . 2 . 1 . 0 1 . 1 a 6 . 1 -.4 -.9

-. 1 . 3 . 2 . 1 . 0 -. 1 1 . 0 a 5 . 0 -.5 - 1.0

0.0 . 2 . 1 0.0 -. 1 — O 1 . 0 . 5 0.0 - . 5

- 1.0

. 1 . 1 . 0 -. 1 -. 2 -.2 1 . 0 . 5 -.0 -. 5

- 1.0

a ^ . 0 -. 1 -.2 -.2 -.3 . 9 . 4 -. 1 - . 6 -1 . 0

. 3 -. 1 -.2 -.2 -.3 -.4 . 9 . 4 -. 1 -. 6

- 1.1
23

TABLE 3

PERCENTAGE DIFFERENCES IN COMPUTED LOSS TANGENT

AS FUNCTION OF ERRORS IN AMPLITUDE AND PHASE

1 50 Hz | 75 Hz

PHASE

ERROR AMPLITUDE RATIO ERROR (dB) AMPLITUDE RATIO ERROR (dB)

( D e g . ) -.2 -. 1 0.0 . 1 . 2 -.2 -. 1 0.0 . 1 . 2

-.3 10.5 7.9 5. 1 2. 1 -1 . 2 19.0 14.3 9.

2.9 -4.

8.9 6.2 3.

-3.0 16.3 11.5 6.0 o a •-> -7.

1 7.3 4.6 1 . 7 -1 . 4 -4.8 13.7 8.7 3.0 -3.5 -11.1

0.0 5.7 3 . 0 0 . 0 -3.2 -6.6 11.0 5.9 0.0 -6.7 -14.6

. 1 4. 1 1 . 3 -1 . 7 -4.9 -8.

8.3 3.0 -3.0 -10.0 -18.0

. 2 2.5 _ O• -» -3.4 -6.7 -10.3 5.6 -6.0 -13.2 -21 .

o . 9 -2.0 -5. 1 -8.5 -12.1 3.0 -2.7 -9. 1 -16.4 -25.0

100 Hz 125 Hz

PHASE

ERROR AMPLITUDE RATIO ERROR ( d B ) AMPLITUDE RATIO ERROR (dB)

(Deg.

)

- , 2 1 0.0 . 1 9 -.2 -. 1 0.

. 1 . 2

-.3 -58.

-32. 1 -14.3 -.7 10.2 -7.5 -4.6 -1.7 1 . 0 3.6

-.2 -49.8 -26. 6 -9.6 3.5 13.9 -6.9 -4.

-1 . 2 1 . 5 4. 1

1 -43.3 -21 . 1 -4.8 7.8 17.7 -6.3 -3 . 4

- . 6 2. 1 4.7

0.0 -36.

-15.6 0 . 0 12.0 21.4 -5.7 -2.8 0 . 0 2.7 5 . 2

. 1 -30.2 -10.1 4.8 16.2 25.2 -5. 1 -2.2 . 6 3.2 5.

•l

• cL -23.7 -4.5 9.6 20.4 29.

-4.5 -1.6 1 . 2

r, w* a KJ 6.3

. 3 -17.1 1 . 0 14.4 24.6 32.8 -3.8 -1.0 1 . 8 4.4 6.9

150 Hz 175 Hz

PHASE

ERROR AMPLITUDE RATIO ERROR ( dB

)

AMPLITUDE RATIO ERROR (dB)

(Deg.

)

-.2 -. 1 0.

. 1 . 2 -.2 -. 1 0 . 0 . 1 . 2

-.3 -3.0 -1.6 -. 1 1 . 3 2.7 -1.4 - . 5 • 5 1 . 4 2 . 4

-.2 -3.0 -1.5 -. 1 1 . 3 cL • 7 -1.6 — 6

o a 1 . 3 2 . ?

1 -2.

-1.5 - . 0 1 . 4 9 o -1.8 __ o a 2 1 . 1 2. 1 0.

-2.9 -1.4 0.0 1 .

Cm m W -1.9 -1.0 0 . 0 1 . 0 1 . 9

. 1 -2.8 -1.4 . 1 1 .

2.

-2. 1 -1.1 __ -^1 . 8 1 . 8

2 -2.3 -1.3 . 1 1 .

2.9 -2.3 -1.3 _ o a 6 1 . 6

. 3 -2.7 -1.3 1 .

3.0 -2.4 -1.4 _ c

a . 5 1 . 4
TABLE 4

24

PERCENTAGE DIFFERENCES IN COMPUTED STORAGE MODULUS

AS FUNCTION OF ERRORS IN J AND Fm

50 Hz 75 Hz

INERTIA

ERROR

<*>

NATURAL FREQUENCY ERROR

-1.0 -.5 0.0 .5 1.0

NATURAL FREQUENCY ERROR C/.'f -1.0 -.5 0.0 .5 1.0

-1.0 3.9 2.5 1.1 -.3 - 1.8 7. 1 4.2 1 .

- 1.6 -4.6

-.5 3.3 1 .

.5 -.9 -2.3 6.5 3.6 . 6 -2.3 -5.3

0.0 2.8 1 .

0.0 -1.4 -2.9 5.9 2.9 0.0 -3.0 -5.9

. 5 2.3 . 9 -.5 -2.0 -3.4 5.3 2.3 - . 6 -3.6 - 6.6

1 . 0 1 . 8 . 4 -1.1 -2.5 -4.

4

6 1 .

-1.3 -4.3 -7.3

< J

it 100 Hz 125 Hz

INERTIA

ERROR NATURAL FREQUENCY ERROR <>:> NATURAL FREQUENCY ERROR <*>

-1 . 0 -. 5 0.0 .5 1 . 0 -1 . 0 -.5 0.0 . 5 1 . 0

- 1.0

-.5

0 . 0

. 5

1 . 0

- 12.2

-12.3

-12.3

-12.4

-12.5

- 6 . 1 - 6.2

- 6.2

- 6.2

- 6.2

0.0

0.0

0.0

0.0

0.0

6 . 1 6.2

6.2

6.2

6 .

12.3

12.4

12.4

12.5

12. 5

-1 .

- 2.2

-2.5

-2.9

-3.3

-.5

-.9

-1 .

- 1.6

- 2.0

. 7

. 4

0.0

-.4

-.7

2.0

1 . 6

1 .

. 9

. 6

3.2

2.9

2.6

2.2

1 .

150 Hz 175 Hz

INERTIA

ERROR NATURAL FREQUENCY

-1.0 -.5 0.0

ERROR

.5 1.0

NATURAL FREQUENCY

-1.0 -.5 0.0

ERROR (>.)

.5 l.O

- 1.0 - . 5 . 2 . 8 1 .

2 . 1 . 0 . 5 < q 1 .

1 .

-.5 -.9 -.2 . 4 1 . 1 1 .-. 4 . 0 . 4 . 9 1 .

0.0 -1.3 -.7 0.0 . 7 1 .

-.8 -. 4 0.0 . 4 , 9

. 5 -1.7 -1 . 1 -. 4 . 3 , 9 -1.3 -.9 -. 4 -.0 . 4

1 . 0

- 2 . 1 -1.5 -.8 -.2 . 5 -1 .

-1 .

-.9 -. 4 -.0
25

The entire measurement system is under computer control. The details of

the interactive program’s logic is given in Appendix IV.

RESULTS

The apparatus and computer controlled test procedure were used to determine

the complex shear modulus of an inhomogeneus polyurethane material with 4% of

its volume containing air bubbles. The sample was 5.08 cm in diameter and its

original height was 10.16 cm. Several different combinations of measurements

were made to get an estimate of the variability of the complex shear modulus

values as of function of slight changes in specimen temperature, choice of f m

and the height of the specimen. To determine the latter effect the original

10.16 cm high specimen was cut into two pieces, one 4.8 cm high and the other

4.9 cm high. The results are tabulated in Tables 5 and 6.

Comparing the results of the data in column 1 with those in column 2, 3

with 4, 5 with 6 and 7 with 8 in both tables it is seen that the repeatibility

of the data is excellent; typically better than 1% for both the storage modulus

and the loss tangent. Comparison of columns 3-6 of both tables indicates that

the effect of a choice of fmalso yields a repeatibility of better than 1%

for the storage modulus and 8% for the loss tangent. The large variation in

the less tangent is consistent with the results implied by Tables 2-4.

Comparing columns 1 and 2 with columns 3 and 4 and with columns 7 and 8

it is seen that the change in length of the specimen yields results that are

within 4% of each other. This small deviation is probably due to the unequal

distribution of the air bubbles within the material.
26

IT*D

rH

33 H

<yao

4-4O

>>

XI

to

as

co

r^.

sO

m

co

CM

O'

ON

CO

co

00'T

ao

aooO

coo

o

COo

o

o

O'

00

o

as

oo

aooo

coo

o

coo

o

vO

CMCM

CO

CMcm

CM

CMco

CM

as

CM

CM

<rCM

\0

ooo

o o

CM

r^r>*coaOiOCM<TvrssO aoascocoin^mmco iO vO

cm r-* O' CM 00 (O N

H (O

Nooas^oo^ooasH«jON hn h

(0^<0<f'M'J(0(0^'flflOO'H«OCNin

00'OiA'^iOiA>niniAiriifli/3yO'O'O>O

\DMna'OfMcocj^^(N^o<^C'fo

oO'Oin^inioiftininiAir^^'C^o

(MS'JH'J(ONNinyONN*CmOOsO incMCOO'HHHNfOM^COffS'JNasM

OcO'JN(0(MCMN*J'O^N‘f<OMsOO' 'OnOOffSHHHCMfl'J^COas^NO'JO

lOm'T^vomiOioiniovomiOsOvONor^

r^NOCOcosXr^^HmcMco^HOCMcor^cM^T iAO^'J'OOsO'nNcdON^H'J^'O

COOCOHNH^fO'JasHOH'^'T'jrs NHHinaOHOO'U'OHlO'ONlCl^N

incncO'OO'O^HNfO ^•mcoiop^oocM^iovn

sa

aoc

<u

<0u

0) 3H

<u u

to a

a

3H

5oa

c

0) 3

aQjU

lu

ooooooooooooooooo OHCNcO'jinvCrNCCOsOHCjncin^ HHHHHHHHHHtMCMtMMCJNN

Test

Site:

NBS

NSRDC
Length

(cm)

10.16

10.16

4.8

4.8

4.8

4.8

4.9

27

CTs

'O’ O

|S<fOnncN(NNCNCNCNN(N«N(NCNCNN

o>

-3* O

no'M»^(S(nOcsHHO(ncoa'0

CNCNCNCNCMCNCNCMCNCNCNCMCMCNCMCM

Cu

4)

c/d

o

o

a

a

C/D

CO^HOrs'JvOlOrsnH^lNNHO nrscn'3, >j^vffon<r>j'Tvf'Tinin MC^MNNNNCS<NNMCN(SCNMM

ON

GO

CMcn

CM00

<T* <r oO'JHOcoooHincoH'n'tn^'O 'JCOOH(SHCNHHHHfn*T>J'T(nN HhCNNNCSMCSCNCN(SN(N(NNCN|N

<3% co a

4)

C/D

<r o o o co o o h o n /) in h co co

,—(CMCNCNCNCMCMCNCMCNCNCNCNCNCMCN

o

o

Q0)

CMO

CnOON^OCNCnCNONvOpHP^CNr^unrHOO rsH'C'finininin'Tinr'coOcors cois* rH<T*>CNCMCNCNCNCNCN<NCNCNC'OCNCNCNCN

C/3 032

OO

Ou

a)

C/3

v£> CO

ON

o>ocNOCN>^-vO>mr>-o^r^cn^cooON ^OvOOLntnLnmLn'<rvon^coOCNr^cor^

CMCHCOCMCMCMCMCMCMCMCMCMCOCMCMCMCM

o

o

CLV

C/3

00

CM

(NjrsvOisfn'O^fncooo NNCSCOvOin»JCV|(N>J

CNCMCNCNCNCMCMCNCNCN

O

o o

>3- CN

CM

00

cuV

CO

^flNOOnnH^'tn n n n o n n ^ m n <

CM CM CM CM CM CM CM CM CM CM

§ 2

4) H

>%O

c

4)3

CT

4)U

&L

OrHCMcn<rmvOr^coaNOrHcsjm<ftn^o ooooooooooooooooo

HHHHHHHH HHCM^JCMCMCMCMN

4) HC/D
28

Comparing columns 7 and 8 with column 9 shows the change in values of

the moduli due to the change in the test site. Although the loss tangent seem

to remain unchanged the shear storage modulus inexplicably decreased by about 7%.

The apparatus and test method produce data that are self-consistent. In

addition when the results are compared to values obtained previously from

other tests at much higher frequencies, but on the same sample, these values

are what are expected.

The 4.8 cm high specimen was tested over a range of temperatures from -10°C

to 40°C with f m = 100.8 Hz. The results are shown in Figures 7 and 8 where

they exhibit the desired properties of high loss tangent at decreasing temperatures and of being relatively frequency independent. Unfortunately, from a

rheological point of view this material does not exhibit the property of timetemperature equivalence and therefore cannot be reduced to a composite master

curve that would effectively produce shear moduli data at a given temperture

over a much greater frequency range than shown.
29

<S> ©)

©)

©)

©)

@>

©)

©)

©)

©)

©)

©)

©)

a

r a

LO

LJ LJ LJ CJ LJ LJ

O O O o o o a a o o a

1 o C\J CO

1 n II ii II II ii o ©) + * X o - o

+

+

xo

+ *

+ =& * X

+ =fc * xo

+ * X) + * XJ + =fc * >o

+ * XD

+ * XD CD + * XO - o

+ =& * o OJ + * XD

+ =& * XO

+ =8= * xo

=fc * xo

* >o

o

o

co N

HI

>-

LJZ

LU

ZD

a

LU

QL

U_

a

o

- o

z

o

M

H

CJzZ

Z

CO

<3

adM

<3

-j- ZH

Edz

H

EdZ

z

O

(X, u.o

COz

XJz

a

o

asw

o

3

o

H

COa

<3

EdZ

co

CdZ

z

o

CD

r^

o

co oLO

o o

CO

o

C\

a o

( 2 -w/n g.oi)

snmaow aovyois 3V3HS

OF

FREQUENCY

AND

TEMPERATURE.
oo

LOOOoomoo

CMooo

30

Nm

>-

CJz

LU

ID

CD

LU

QL

U_

Cbo2owH

CJ22

Cb

C/3 <otf M<K'3- Hi—

W22HW

a;2

>*Jo2uCboH2WO2CH

C/3 C/3 o2

00U3 3i 2OM

Cb

1N33NV1 SSCH

FREQUENCY

AND

TEMPERATURE.
31

ACKNOWLEDGEMENT

The assistance of Mr. William Penzes, for his design and construction of

the switching network, 30 dB amplifiers, and various power supplies and connectors,

is gratefully acknowledged.

Mr. R. Reitz of the U.S. Naval Research and Development Center obtained the

data presented in Figures 7 and 8.
c 32

REFERENCES

1) I. M. Ward, Mechanical Properties of Solid Polymers , John Wiley and Sons,

London, (1971), Chapter 6.

2) L. Cremer, M. Heckl, and E. E. Unger, Structure-Borne Sound , SpringerVerlag, Berlin, (1973), pp 189-216.

3) "Measurement of Dynamic Moduli and Loss Factors of Visoelastic Materials,"

Session J, 99th Meeting: Acoustical Society of America, J. Acoust. Soc. of

Amer., Suppl. 1, Vol. 67, Spring 1980, pp S23-S25.

4) W. J. Gottenberg and R. M. Christensen, "An Experiment for the Determination

of the Mechanical Property in Shear for a Linear, Isotropic Viscoelastic

Solid," Int . J. Engng. Sci., Vol. 2, (1964), pp 45-57.

5) J. H. Baltrukonis , D. S. Blomquist, and E. B. Magrab, "Measurement of the

Complex Shear Modulus of a Linearly Visoelastic Material," Technical Report

#5, Department of Engineering Mechanics, Catholic University of America,

Washington, D. C. (May 1964).

6) D. W. Marquardt , "An Algorithm for Least Squares Estimation on Nonlinear

Parameters," J. Soc. Ind. Appl. Math, Vol II (1963) pp 431-441.

7) J. C. Nash, Compact Numerical Methods for Computers: Linear Algebra and

Function Minimization, John Wiley and Sons, New York, (1979), Chapter 17.
33

APPENDIX I

Marquardt’s Minimization Procedure

The values of G’ and G" are obtained from Eqs. (6) and (7) using Marquardt’s

methods [6, 7]. The method will be outlined below and the specific functions

for our particular case will be given.

The Marquardt procedure is an efficient means of minimizing

m

S (x) = E [f.(x)T = f f (I“ i=l

1

where

f

i

(x) = g

i (x)-g^ (1-2)

and gi(x) are the m (nonlinear) functions of the parameters x(x^, x£, . .Xn)

f and g^ are the measured values. Equation (1-2) states that we are minimizing

the sum of the squares of the deviation of the data from the functions used to

fit these data. In our case, m = 2 and n = 2. The Marquardt procedure says

that a good next guess for the value of the parameter x, denoted x + q,

can be determined from an interactive solution to

(J J + ADia[

T

J] )q = -jT

f

where J is the Jacobian matrix whose elements are defined as

3^7

f i

( ?)

J

(1-3)

J. . (1-4)
34

The superscript T denotes the transpose of the matrix and Dia [ . . .

denotes

that the matrix has all zero elements except along its diagonal. The left hand

side of Eq . (1-4) must be positive definite and, therefore, X must always be

chosen so that this side keeps its positive definiteness. The positive-definiteness

is determined in the solution to Eq . (1-3) by using the Choleski decomposition

of the matrix on the left hand side and checking to see that each diagonal term

of the decomposed matrix is greater than zero, which indicates positivedefiniteness. If they are not all greater than zero, the value of X is increased

by a factor of ten.

The interative solution itself is straightforward. Starting with a value

of X = 0.1, X is reduced by a factor of 2.5 before each step in the solution if

the preceding solution for q has given

S(x+q) < S(x)

If

S(x+q) >_ S (x)

then X is increased by a factor of 10. The process is repeated until certain

convergence criteria have been satisfied.

For our particular case we have from Eqs. (6) that x]_ 3 G* , X2 “ G",

gl = D', and g2 = <j>, where, for convenience, we define D' = (r^/r t )D. Since

D' and <J> are functions of x and y, which in turn are functions of G" and G', it

is easiest to use the chain rule for partial differentiation to obtain the four

elements of the Jacobian matrix. Thus
35

J

11

J

12

J

21

J

22

3D’ 3D’ 3x -+ 3Dliz_

3G' 3x 3G ' 3y 3G'

3D' 3D' 3x

- +

8D ' 3G" 3x 3G"1 9y 3G"

34> . _3<j>_ 3x 3i 9y

3G f 3x 3G ' 3y 3G’

3<j>

. 3x

, 9^ iz_

3G" 3x 3G" 9y 3G"

where

3D 1 3x

3D'

3y

3£

3x

[A

[A'

+ b

2 r3/2

+ b

2 ]- 3/2

+ B

2 ]' 1 [A

[A 1

+ B

' A f

+ B

3B

] 3x

Hi

3y

3B r iAi

ix

" B

3x

3A n [A

2 + B2 ]" 1 [A |f

- B

A and B are given by Eq . (7) and their deviatives are — = [(C, - l)sin(x) + C, xcos (x) ] cosh (y) + C,ysin(x) sinh (y)

3x 1 l L — = [ C, xsin(x) + (1 - C ) cos (x) ] sinh y

- C

± y cos(x) cosh (y)

3y 1

3B

= _ 3A

3x 3y — = [C..xcos(x) + (C - 1) sin(x) ] cosh (y) + C^sin (x) sinh (v)

3y -L
36

The remaining partial derivatives are obtained from Eqs . (4). Thus,

3x

3G

[xG’ - y G"]

2p^

1^=1 [xG" + yG' ] 2p

3y _ 3x

3G* 3G"

3x

3G

Returning to Eq . (1-3) we can now write the solution for q^ and q2 in terms

of the elements of the Choleski decomposed matrix L^j as

q 2 v

1

/L

22

q

l = (V

1 - L21 q2

)/L

ll

where

V

1 = b

l

/Lll

V

2 = (b

2 " L

21

V

1 )/L

22

L n VA

ii

L

21

A2lWA

ll

L

22

A

22

A

21 /A

ll
37

and

b

b

1

2

11

12

22

21

J ll

f l + J

21

f 2

J

12

f

l + J

22

f

2

(i + x) (j^ + j

2

21

J

11

J

12 + J

22

J

21

(1 + X) (J 12 + J

22

= A

12

)

)

When L22 > 0 the matrix is positive definite.
38

APPENDIX II

Computation of the Mass Moment of Inertia J

The total mass moment of inertia, J, is comprised of the following parts:

(1) the base of the torsion spring; (2) the accelerometer mounting arm, including

the accelerometer and the counter balance; (3) the specimen mounting plate;

and (4) the Allen screw heads. Referring to Figures A-l to A-4, the following

formulas and numerical values are obtained:

Torsion Spring Base

The mass moment of interia, Jj), using Figure A-l, is

T - 1T7 2 J

2 -

2 Vd

where, Wp is the weight of the base and is given by

where p is the density of the base material. For steel, = 0.09548 kg and

Jp = 0.5825 kg-cm^

Accelerometer Mounting Arm and Accelerometer

The mass moment of inertia, J g , using Figure A-2 is

where

i
39

FIGURE

A-l

DIMENSIONS

OF

TORSION

SPRING

BASE
40

E

E

o

h--

c\i E E E E E E

E

E

T— E E E E E E o

CO C\l CO co to ID

<T> CM ID CD CM o h- CO

c\i csi n! T- cm CD

CO CM ^T r- T“

II II II II II II II II o

k_ kl” -C

k- *o

CM O Q _l

<

JZ

FIGURE

A2

DIMENSIONS

OF

ATTACHED

MASS

AND

TOP

ACCELEROMETER'S

MOUNTING

POSITION.
41

and W^cc is t ^ie weight of the accelerometer (2.5 gm). For aluminum, =

(6.190 x I0”^)h^ kg and W

Q = (6.013 x 10 ^)h^ + 0.0025 kg. Then for h^ =

6.35 mm, Jg = 0.4680 kg-cm^ and for h^ = 12.7 mm, Jg = 0.9360 kg-cm^.

Specimen Mounting Plate

The mass moment of inertia, J^, using Figure A-3, is

J

M

* T

( Vt + Vb>

where

and

W = P\ur

t

w

b

“ c Vr

b

For aluminum, W

fc = 0.03307 kg and = 0.05355 kg. The J^ = 0.7308 kg-cm^

Allen Screw Heads

The mass moment of inertia, J^, using Figure A-4, is,

J

H

“ 4[W

b

(f

r

c + d 3> + Vl(r

c + rd>

+ d

3 )]

where

W = phirr

b c

and

W = ph ir(r - %

For the large screws, = 0.00149 kg and Wt = 0.00105 kg. For the small

screw, W^, = 0.000621 kg and Wt = 0.000345 kg. Then, for the large screws

Jr = 0.08408 kg-cm^ and for the small screws Jr = 0.03185 kg-cm^. The total

is Jr = 0.1159 kg-cm^.
42

E E E E

E E E E

in CO CO CO

05 T— i—

CO CO

CO

II II II II Qu. k- £

QQ

-C

FIGURE

A-3

DIMENSIONS

OF

SPECIMEN

MOUNTING

PLATE.
43

_l E E E

< E E E

00 00 o> 00 iq

c/d T“ 04 CO

E E E E

LLl E E E E

00 00 h- 00

<3 T-| CO CO iq

DC CO oi CO

<

_l

II II II

04

II

sz O1*. O

CO u

-o

FIGURE

A

4

DIMENSIONS

OF

ALLEN

SCREW

HEADS.
44

Total Mass Moment of Inertia

The mass moment of inertia is equal to

J = JD + + JM + JH

For the two accelerometer mounting plates of thickness h^ we have:

h^ = 6 . 3 5 mm

h^ = 1 2 . 7 mm

J = 1.8972 kg-cm2

J = 2.3652 kg-cm2
APPENDIX III

Computer-Controlled Relay Switches

The schematic diagram of the switching functions and connection to the

computer-controlled switching box is shown in Figure 4. The BNC connectors to

the switching box are depicted as two concentric circles in the figure and the

label that appears next to the BNC connector on the relay switching box is als

given in the figure. Each of the relays is controlled by one of the 16 lines

available on the output of the HP 98032A 16-bit interface, which is connected

from the computer to the switching box.

Following the nomenclature of the 16-bit interface manual, the lines

denoted DOXX are connected as follows:

D00 Sets attenuator to 5 dB

D01 Sets attenuator to 10 dB

D02 Sets attenuator to 15 dB

D03 Sets attenuator to 20 dB

D04 Sets attenuator to 25 dB

D05 Sets attenuator to 30 dB

D06 Connects accelerometer #2 to fixed gain amplifier of channel til when

external switch is in test position, otherwise it connects directly to

phasemeter and the digital voltmeter.

D07 Connects oscillator to fixed gain amplifier of channel til when D015 is

’low* or connects the output of the attenuator to the fixed gain ampli

fier of channel til when D015 is 'high*.

D08 Connects accelerometer til to fixed gain amplifier of channel til when

external switch is in test position, otherwise it connects directly to

phasemeter and the digital voltmeter.
46

D09 Connects oscillator to fixed gain amplifier of channel //I when D015 is

'low' or connects the output of the attenuator to the fixed gain amplifier of channel #1 when D015 is 'high'.

D010 Connects accelerometer //I or tracking filter output of channel #1 to

the digital voltmeter.

D011 Connects accelerometer // 2 or tracking filter output of channel //2 to

the digital voltmeter.

D012 Connects oscillator output to the digital voltmeter.

D013 Connects oscillator output to the amplifier.

D014 Sets attenuator to 0 dB

D015 Reverse D07 and D09: Connects D07 to attenuator output and D09 to the

oscillator output.

If N is an integer value of a 16-bit binary number that activates the

appropriate lines to control the relays and XX in the denotation DOXX indicates

the N-l bit location of the 16-bit binary number, then the following list shows

the correspondence between N and the lines set "high." (See the main text for

a discussion of the test protocols used.) As a last remark it is mentioned

that the attenuator’s impedance is such that it loads the oscillator’s output

to a value that is approximately 6 dB less than if the oscillator were to be

connected to a high impedance device. This fact is taken into account in

subroutine ’Instr' (see Appendix IV).

N Lines 'High' (switch closed)

9536 D06, D08 , D010, D013

10560 D06

D08 , DOll, D013

4096 D012
47

9216 D010, D013 (for back-to-back accelerometer calibration only)

10240 D011, D013 (for back-to-back accelerometer calibration only)

1664+M+S D07, D09, D010, see below

26884M+S

where

D07, D09, D011, see below

M Lines 'High' (switch closed)

16384 D014

1 D00

2 D01

4 D02

8 D03

16 D04

32

and

D05

S

0

-32768 D015

In Figure 4 the relay symbols have their corresponding DOXX value adjacent

to them. The two pairs of relays (D06, D07) and (D08, D09) and the triplet of

relays (D010, D011, D012) are each connected such that only one of each pair or

triplet of relays can be closed at a time.
48

APPENDIX IV

Subroutine Names and Their Functions

Accplo Plots measured acceleration levels versus frequency.

DVM Controls and reads the Ballantine digital voltmeter.

Freqax - Draws 'tic' -marks and labels the logrithmic frequency axis.

Freqmn Determines upper and lower limits of the plotted frequency axis

based on the minimum and maximum test frequencies selected.

Freqsel Queries user for starting and ending test frequencies and calculates

all test frequencies in between these limits.

Graph - Plots experimentally determined acceleration levels and phase angles

as a function of frequency. [Calls Subr. - Accplo, Freqax, Freqmn,

Header, Phaplt, Phasis, Vertax, Vrange].

Guess - Before entering subroutine ’Mark' for the first time it completes

an initial estimate of G’ and G" from the measured acceleration

ratio corresponding to the lowest frequency. The phase angle is

set equal to zero.

Header - Prints headings containing all pertinent test data prior to printing

all tabulated and graphical output.

Instr - Runs the complete measurement protocal for either experiment:

back-to-back calibration of accelerometers or determination of

shear modulus. [Calls Subr. - Dvm, Osc, Pmeter, Switch.]

Mark - Performs nonlinear least squares fit to each pair of acceleration

ratio and phase angle at a given frequency. Special subroutine

’Ftn* is part of this subroutine to speedup the computation process.

[Calls Local Subr. - 'Ftn'.]

Osc - Sets frequency and amplitude of Rockland frequency synthesizer.
49

Para

Phaplt

Phaxi.s

Pmeter

Scan

Switch

Table

Vertax

Vrange

Queries user for all pertinent test parameters prior to the start

of the test.

Plots the measured phase angles as a function of frequency.

Draws ’tic* -marks and labels the vertical axis for measured phase

data.

Controls and reads the Dranetz phasemeter.

Measures only the acceleration ratio over any frequency range,

frequency increment and excitation amplitude. [Calls Subr - Dvm,

Osc, Switch.]

Controls sixteen relay switches to perform the various measurement

protocals and directs the accelerometer signals to the digital

voltmeter and phasemeter.

Tabulates all experimetally obtained and calculated results.

[Calls Subr - Header.]

Draws ’tic' -marks and labels the vertical axis for measured acceleration levels

Determines the upper and lower amplitude limits for the acceleration

level plot
APPENDIX

V

50

Program Flow Chart

Main Program: "SHRMOD "
51
Subroutine: ' Instr*
53

Exit
54

Compute new value for

oscillator's attenuator

setting

©-
55

I

Next L

Exit
DISTRIBUTION LIST

1)

Commander (2 copies)

Naval Sea Systems Command

Code 55N

Washington, D. C. 20362

2)

Commander

Naval Surface Weapons Center

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White Oak

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3)

Officer in Charge

Annapolis Laboratory

David Taylor Naval Ship R&D Center

Code 2842 (John Eynck)

Annapolis, Md 21402

4)

Commander (6 copies)

David Taylor Naval Ship R&D Center

Code 1905.2 (Wayne Reader)

Bethesda, Md 20084

5) Commander (1 copy)

David Taylor Naval Ship R&D Center

Code 1965 (Bruce Douglas)

Bethesda, Md 20084

6) Edward Hobaica

Electric Boat Division

General Dynamics Corporation

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7) Commander

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8) Commander

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Attn: Robert Ting

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NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 83-2776

2. Performing Organ. Report No

738.00

3. Publication Date

September 1983

4. TITLE AND SUBTITLE

Determination of the Viscoelastic Shear Modulus Using Forced Torsional Vibrations

5. AUTHOR(S)

Edward B. Magrab

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. N00024FQ02013 (NavSea 054)

8 . Type of Report & Period Covered

Final 1980-1982

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

U.S. Naval Ship Research Development Center

Bethesda, Md 20084

10.

SUPPLEMENTARY NOTES

N00024RQ02013 (NavSea 05H)

MIRP N00167-81-MP-00009 (DTNSRDC)

MIRP N00167-82-WR2-0504 (DTNSRDC)

| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gn ifi cant

bi bliography or literature survey, mention it here)

A forced torsional vibration system has been developed to measure the shear

storage and loss moduli on right circular cylindrical specimens whose diameter can vary to 9 cm and whose length can vary from 2 to 15 cm. The method and apparatus

are usable over the frequency range 80 to 550 Hz and a temperature range of

-20°C to 80°C.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

shear modulus; torsion; vibrations; viscoelastic

13. AVAILABILITY

| X[ Uni imited

| | For Official Distribution. Do Not Release to NTIS

r~~l Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

IX) Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

61

15. Price

$ 10.00

USCOMM-OC 9043-PS0