A STUDY OF THfi VIBRATING REED AS A
DEVICE FOR THE DETERMINATION OF

VISCOELASTIC PROPERTIES

Them for ”'00 Doqru of M. 5.
MICHIGAN STATE UNIVERSITY

William Wai Tung Seto

1961

 

 

 

 

This is to certify that the

thesis entitled

A STUDY OF THE VIBRATING REED AS A
DEVICE FOR THE DETERMINATION OF

VISZOELASTIC PROPElTIES
presented by

WILLIAM WAI TUNG SETO

hes been accepted towards fulfillment
of the requirements for

MASTER OF SEIENCE degree in AEELIED MECHANICS

Mljor professor

Date May ’4; 1961

0-169

 

 

 

 

ABSTRACT
A Study of the Vibrating Reed as A Device

for the Determination of Viscoelastic Properties

by William Wai Tung Seto

When cantilever specimens of different
lengths are subjected to various values of frequency
of vibration by an impressed forced vibration at the
clamped end, the ratio of the amplitudes of oscillation
of the free to the clamped end at the steady state
condition can be found. If the phase lag of the free
end behind the clamped end is also measured, the vis—
coelastic complex modulus can be calculated from the

mathematical relation derived.

In addition to the internal damping of the
material of the cantilever specimens, there is inherent-
ly airbdamping when the cantilever specimens are oscill-
ating. The effect and significance of this airhdamping

on the vibrating reed test of a material are also consi-

dered.

A brief outline of the simple linear vis—

coelastic theory is also included.

A STUDY OF THE VIBRATING REED AS A DEVICE

FOR THE DETERMINATION OF VISCOELASTIC PROPERTIES

By
William Wai Tung Seto

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Applied Mechanics

1961

 

Acknowledgement

The author wishes to acknowledge the support
given.to this work by the Division of Engineering Research,
Michigan State University --—- Mr. John W. Hoffman,
Director; also to express his sincere thanks to Dr.
Clement A. Tatro of the Department of Applied Mechanics,
Michigan State University for his invaluable help and

his kind permission for unlimited use of his equipment.

Major Professor

Dr. George E. Ease.

TABLE OF CONTENTS

CHAPTER PAGE
I. INTRODUCTION .......................... I
II. OBJECT .................... ..... ....... 2
III. REVIEW OF PREVIOUS EXPERIMENTAL W0 ” .. 3
IV. EXPERIMENTAL WORK ..................... 7
V. MATHEMATICAL ANALYSIS ................. 9
VI. AIRrDAMPING .................. ..... .... 13
VII. DISCUSSIONS ........................... 19
VIII. APPENDIX .............................. 21

BIBLIOGRAPHY .................. ..... ... 25

Chapter I Introduction 1

Most engineering materials when subjected to
loads do not behave as perfectly elastic solids. The
assumptions of homogeneity, isotrop , and time-indepene
dent elasticity are in direct contrast to the phenomenon
of fatigue, time and temperature-sensitive cohesive
strength and creep behavior of real materials. There-
fore, the analysis of the mechanical behavior of such
materials requires information obtained through various
studies concerned with the degree to which real materials
differ from the ideal Hookean solid. In particular, for
example, many of the physical "constants" related to
stress analysis of a given material are sensitive to
loading rates and thereby lead to a consideration of the
so called 'dynamic constants". A method which has been
used widely in the determination of such constants is
the so-called vibrating reed test. It is with a study

of this test that the following investigation concerns

itself.

Chapter II Object

The primary purpose of this investigation is
to study analytically and experimentally the use of the
vibrating reed as a means of determining viscoelastic
properties. The experimental portion of the work was
carried out using methyl methacrylate (commercially known
as plexiglas or perspex) as the test material. Various
influences such as the effect of airbdamping, excitation

frequency, and reed geometry were considered and included

as part of the investigation.

Chapter III Review of Previous Experimental Work 3

A review of the literature dealing with the
determination of the dynamic physical constants reveals
a number of experimental and instrumentation techniques

appropriate to the present work.

Among the many different driving mechanisms
used as the driver to impart a sinusoidal oscillation to
one end of the reed, a phonograph recording head was used
by an early investigator, M. Horia of M. I. T. (1)“ For
vibrations with low and moderate damping, the "electros-
tatic method" for driving is preferrable, and has been
employed in the determination of "inelastic losses" in
some high polymers as a function of frequency and temp—
erature. It makes use of a metal foil fixed on the free
end of the reed. The metal foil is attracted by an al—

ternating potential in the air-gap of a strong magnet.

Some investigators attain the required sine

usoidal oscillation of the reed by subjecting it to an

impressed force, supplied by an audio oscillator through

“Numbers in brackets refer to bibliography at the end of

this report.

4
a mechanical linkage between a permanent magnet speaker
and the centerline of the reed. Sustained oscillations
of the reed can also be achieved by suitable adaptation
of reciprocating apparatus designed for other purposes;

such was the case taken by the present work.

As far as the measurement of the amplitude
of vibration of the free end of the reed is concerned,
some investigators used a stroboscope, synchronized with
the frequency of oscillation of the reed, for easier
viewing of the maximum points. A. W. Nolle (2) used a
small telescope with crosshairs on it for this purpose;
While L. E. Nielsen (3) employed a differential trans-

former to convert amplitude of mechanical oscillations

to electrical potentials.

Another possible means of measuring the
amplitude of vibration is by measuring the capacitance
across two plates with a fixed piece of dielectric
material attached to the reed and passing between the

two fixed plates with the vibration of the reed thus

causing a change of capacitance.

In the "electrostatic method" mentioned

5
above, the vibration of the end of the reed is observed

by an optical method. The end of the reed interrupts a
light beam. When the reed vibrates, this light beam is
modulated, and in falling on a photo-cell, creates an.

A. 0. signal proportional to the amplitude of vibration.

This signal is then amplified by conventional means and

fed into suitable recording apparatus.

Although it has only indirect connection with
this work, it is interesting to note that based on vis—
coelastic behavior, a method has been developed for
mathematical treatment of a sinusoidal rate of loading
of fibers (4). Here the vertical displacements of a
Weight attached to the end of single filament of fiber
is recorded by a movie camera, with a ground-glass as the
screen and a synchronous motor as the clock. Initial
displacement is given by means of an electromagnet, and

temperature as well as humidity are controlled through-

out the whole experiment.

On some occasions, the evaluation of the

dynamic constants of a material can also be made from the

measurements of the velocity of transmission and the

attenuation of sound in the material. however, this has

6

been chiefly applied to ' low loss " materials, such as
metals and some plastics. It consists of a signal gene-
rator, crystal driver, pickup, amplifier, and scope (5).
This experiment is elaborate, extensive, and expensive.
In addition, the material used for this investigation
does not come under this " low loss " catalogue, and

hence it was not used.

Chapter IV Experimental Work

The vibrating reed apparatus consists of two
main components: namely a driving clamp to impart a
sinusoidal displacement to one end of the reed, and a
recording apparatus to measure the displacement of the

free end.

In this work, the "calibrating beam", de-
signed and built by the hichigan State Highway Depart-
ment, was used as the driver to impart sinusoidal osci-

llations of various frequencies to the clamped end of

the reed.

With regard to the measurement of the ampli-

tude of vibration of the free end of the reed, much
effort was spent in careful consideration and prelimi-
ary attempts to build and use apparatus such as the
rotating drum, the photo—cell, capacitors, movie camera,
and telescope with cross—hairs, all of which have been
listed as methods of suitable merit in the previous
section of this report. It was found, however, that with.
"visual observation"

some care and operator experience, a

method provided data of nearly equivalent to the indi-

Oated accuracy with a great saving of effort. In this,

8
a straight edge mounted on a tripod-stand was brought

close to the end of the vibrating reed for determining
the value of the corresponding displacement. This is

possible because the amplitude of vibration of the free
end will reach a constant steady state magnitude after

the transient motion of vibration has died out.

The ratio of the amplitudes of oscillation
of the free to the clamped end at three different fre-
quencies, namely 270 c.p.m., 435 c.p.m., and 645 c.p.m.,
were measured for different lengths of reed, the material
of which were assumed to be statistically homogeneous and

to have constant lateral dimensions.

I - Frequency of
Vibration (c.p.m.) Lengths of reed

Width of reed (1 inch) 18 ’ 17 '16 5 15 f 14 13

f 276 €5.6 9.7 -4.8- 2.9 1.8 1.3
i 425 ;5.9 6.1 4.9 3.2 2.9 1.5 .
3 645 '5.4 4.8 4.3 4.0 3.1 2.3 ‘

\.—_ ._--

Width of reed (E inch)f .1 .. 1_“
“’*'“ {18 17 16 - 15 14 13

_—.___..._ A.--a - .

270 [4.6 9.6 -5.4 3.5 2.3 1.7
425 '3.7 4.8 3.2 2.2 1.8 1.4
645 2.2 .3°1 §2.7 1.8 1.4 1.1 j

__1_ _1i.,."_11._1__1J._1-_11L-_-----L1_1-.1

Chapter V Mathematical Analysis

The problem involved here is to obtain the
response of a viscoelastic cantilever beam undergoing
sinusoidal oscillations at the clamped end. Because of
the time-dependent boundary conditions, it is convenient
to employ the Mindlianoodman procedure (6) to solve the
associated elastic beam problem and then by means of the
elastic-viscoelastic correspondence principle to convert
the results to obtain the solution for the viscoelastic

beam. Thus for the beam shown in the following diagram,

a sin.wt 1:;irh 3 7" 't‘ "' . 1-..;

Boundary conditions

 

y(O,t) = a sin wt ........ (l)
(%§x=o=o (2)
(;.§)x=l=o (3)
(3:1)x=l=0 (4)

Initial conditions

y(0) = O oooooooooooooooooooooo (a)

§(O) ...... (b)

O ......OOOOOOOOO.

10

The differential equation for lateral vibration of beam

is ~.2 « 4
;‘1- + 112.11 = O and n2 =~——E--~---I- 5—-
.) t- fl 4 A f
C x
where E = Young's modulus of elasticity,

I = moment of inertia of cross-section,
A = cross-sectional area,
V”: weight of material per unit volume
Assume the solution
y(x,t) = e(x,t) + f(t) g(x) = e(x,t) + a sin wt g(x)

Therefore, the beam equation becomes

2 14 :1 2 _ _ 2 .-4 2
n (:3 i ) + ,\ §2_- n a sin Wt, g, + gaw sin wt

The boundary conditions become

e(O,t) :2 a sin wt (1 - 3(0)) ........ (la)

Ligfio’fil=;— a Sin Wt 8(0) cocoa-000000 (23)

 

w) 1
'e(1.t
;;__H;E-R= _ a sin Wt g (1) 00.000000... (3&)
A e(1,t)
-«v = — a Sin Wt g (l) ooooooooooo (4a)
’; I}

If g is
g =.E sin B: + F 008 Bx + G sinh Bx +4H cosh Bx

w
where B - n

It can be shown that the boundary conditions on e become

zero if g is such that

11

E

II
I
a)

- E sin Bl - F cos Bl + G sinh Bl +.H cosh Bl = O
- E cos Bl + F sin Bl + G cosh Bl + H sinh Bl =:O
after solving the above equations,
3(1) =>G (sinh Bl - sin Bl) + H(cosh Bl - cos Bl)
+ cos Bl
When.the transient motion has died out, 1.8. e(x,t)——'~0,
the steady state response of the beam will be
y(x,t) = a sin wt 3(X)
Therefore,

y(l,tl_ cos Bl + cosh Bl
y(O,t) = 8(1) = l + cos Bl cosh Bl

which is the ratio of the amplitudes of vibration

of the free to the clamped end, and

2 %

13,33”,
\EI/

\‘

 

W a frequency of oscillation,

m mass per unit length,

E = Young‘s modulus of elasticity.

II R denotes the ratio of the amplitudes of oscillation

of the free to the clamped end, and Q the phase lag of the

free end behind the clamped end, then
_ 1,4 cos B1 + cosh Bl

R e = *-~w-~~-
l + cos Bl cosh Bl

12

The solution for a viscoelastic material is given by re-

placing the elastic modulus E by the corresponding visco-

elastic complex modulus. This is best obtained by the

graphical method as shown here.

Sample calculation:
when Bl a 1.00, cos B1 = O. 5403

cosh B1 = 1.5431

_ 1, 0.5403 + 1.5431 2.0834
R e = —-—--— , - = .
1 + 0.5403 x 1.5431 1.8337
.2 i
R‘éit I//
i ,
1 i
!
: /.//
O L////--H >-—
0 l

13
Chapter VI Air - Damping

For a cantilever loaded by its own weight,
the deflection of the free end is
_W_1_:._
d = 8 EI
where W = weight of beam per unit length,
E = Young’s modulus of elasticity,

I a moment of inertia of cross-section,

l = length of beam.

It can be shown that the fundamental natural

frequency of free vibration of the cantilever beam loaded

by its own weight is

.— H--...__ .... ....—
-
-

w 14
Let ‘1 , w2 be the natural frequencies of

two cantilever beams made of the same material, having the

same thickness, but different both in width and length.

But for rectangular cross-sections,

b h3 b2 h3
1 I _ 1

_‘—

12 2- 12

 

 

I1:

14

For the same material,

 

 

 

2
Therefore, .. __1_. i
f 8 E h3blg 8 E h2g
w = l __ -_ - -...-- -1, . __ = ._
l J 7'hblli ‘ 7 1:
3 fi1i_______1 __
8 E h b2g 8 E h g
w = ~—--— =
2 , -5 4 , 4, 4
4 hb212 J .12
Now if 11 = 12, then
w:L = W2 1. e. the two cantilever beams will
9

vibrate freely at the same natural frequency. In other
Words, cantilever beams of the same length, thickness, and
made of the same material will vibrate freely at the same
frequency even though their widths are different. This

is true when there is no damping force other than their
own internal damping. However, if the amount of air—
resistance is negligible compared to the amount of inter—
nal damping present in the system, the system, as a whole,

Will not be affected much.

The following table gives the Values of
natural frequency of free vibration of cantilever beams of

various lengths and widths; they are, however, made of the

15
material, methyl methacrylate, and are of the same thick-
ness. These experimental results were obtained through
the use of SR - 4 strain gages, Universal Strain Analyzer,
and pen - and - ink oscillograph.

 

 

 

 

Length of Width of , Width of
Cantilever Cantilever l f Cantilever %
17 in. 5.11 c.p.s. 4.67 c.p.s.
16 in. 5.71 c.p.s. ’ 5.42 c.p.s.
: 15 in. 6031 0013080 5093 copes. l
1 14 in. '1 7,34 c.p.s. 6067 Cop-.8. :
b _.-_ __..__, - _ _ 1 . _ _. . .. --.._. . - __ .1
, ; I i
| ‘-~-];3_ 1n. 1 8034 CopoSo i 7075 c.p.s’ ;

These results indicate that air-damping cannot be ignored

in this vibrating reed test of this particular material.

Another way for the determination of the

influence of air-damping on the vibrating reed is the
"logarithmic decrement" which is the logarithm of the two

consecutive amplitudes of displacement in a decaying curve

of vibration.

As has pointed out in the previous part of

this section, L1 = L2 is a condition for the reeds to

vibrate freely at the same frequency. In other words,

whenLl = L2 the values of the logarithmic decrement

determined from two decaying curve of free vibration

should be the same.

Let D be the logarithmic decrement,
therefore D = ln(Al/A2) = ln<An/An+1)
OrD=ZWQJ%a?_

n

..-_.

16

In this manner, the following table was obtained.

7F~tenéthudt7hf width of- _ 7 Width of
’ reed : reed 3/4 inches g reed 1 inch.
. 17 in. E D‘; 0.28.- f I D-e‘0.31‘

Wis—1n. ' D = 0.3:) 7 i L" "15—: dia—
77777 — 15 in. D = 0.31 E D = o.28
‘7 -714 ih.- u = 0.30 2 D = 0.29
’~_TI‘ L = 0.31 i D = 0.32

l} in.
Sample calculation:
When the reed length is 13 inches,
Al/AS = 1204/39? =3 3035 to... (l)
Al/AS = 1004/302 = 3017 000.. (2)
3024 Our... (3)
3026 00.00 (4)

111/115 = 14.6/4.5
10.4/3.1

Al/A5
The average til/A.5 = 3.25

4 D In 3.25,

D = 0.32

17
The results show that reeds of different
widths have different values of logarithmic decrement,
i.e. the reeds are vibrating at different frequencies.

This agrees with those obtained in the last section.

When air-damping is not negligible, the
system becomes one with elastic constant k, internal
damping CI’ and air-damping CA' By solving the differ—
ential equation,

mx + (CI + CA) x + kx = 0

it can be shown that the frequency of the system is

 

__ 2
f =F } ‘ "“i
f; " Elf g4 .
x 2m '
D .. (_:_I_+ Ci r
2m
Therefore,
D " 21ml CAm \ f.
m \. 2i“v'“,’lh
...-... __ g _ _., m 1.- ’-
D n, (CCIn * CAnj
If there is no air-damping, CA = Q, then
D... _ “13:9,???sz .fa
Dn. g\CIn/2m16fn

For a constant length reed, this ratio should be a straight

line for various widths.

18

When air-damping is not negligible, CA £.O

K

 

1“
.C + C y(W )/ 2m
-Dm*.-:_I““ A -_ _’ “/f \l
.f’” - { .x‘ '”—i
Dn LCIn + CA y(W ?/ 2mn Ki’n“
{/Dm\ f fmx
The graph Of\C_‘fl versus 5f ;is no longer a
n n

straight line. Therefore, the amount of airbdamping

present can be interpreted from this deviation.

If the vibrating reed test were carried out
in a vacuum space, the effect of air-damping can be
completely ignored. Or if the exact amount of internal
damping of the test material is known from other source,
the amount of airedamping on the reed can also be deter—
mined. An experiment, which includes little of the effect
of airbdamping, is to displace air by helium in.a confined

space Where the vibrating reed test is being carried out.

19

Chapter VII Discussions

In a previous section of this report, it has
been shown that the viscoelastic complex modulus of methyl
methacrylate can be determined when the ratio of the ampli-
tudes of oscillation of the free to the clamped end at
steady state and the phase lag of the free end behind the
Clamped end are both known from measurements made by the
vibrating reed test. In the experimental portion of this
work, the ratios of the amplitudes of oscillation of the
free to the clamped end at steady state for various
lengths of cantilevers at three different values of fre-
quencies were measured. With the values of the phase lag
of the free and behind the clamped end, the viscoelastic
complex modulus can thus be determined directly from the

graphical solution presented.

From the experimental results of the free
vibration of the cantilever beams, it is observed that
cantilever beams of the same length, thickness, and made
of the same material but different width exhibit different
values of natural frequency of vibration which is in
direct contrast to the results given by the mathematical

analysis neglecting airedamping. This implies that air-

20
damping does play a significant part in the overall damping
of this particular system, and would therefore be of im—

portance in practical applications.

Assuming that air-damping is proportional to
the area of beam exposed, the cantilever of width 1 inch
would encounter more airbdamping than that by the can-
tilever of § inch width; and consequently would vibrate
at a slower pace. This is in agreement with the experi-

mental results obtained.

21
Chapter VIII Appendix

Linear Viscoelastic Theory

Basically, this theory assumes that both
elastic and viscous elements are involved in the resis-
tance of a real body to deformation under load. Thus the
response of a given material to various load conditions
has associated with it the response of a mechanical model
comprised of a suitable arrangement of elastic springs
and Newtonian dashpots. A number of recent texts contain
excellent accounts of the theory as it has developed to
date ( 8,9 ). For the purpose of this study it is suffi—
cient to point out that the simplest models available are
those of a spring and dashpot in series (called kaxwell),
and a spring and dashpot in parallel (called Kelvin). A
diagrammatic sketch of each of these models is shown in

Figure 2.

The stress-strain relations corresponding to

these are

,0\

_ ':/E + J/n (kaxwell model)
= g E + n: (Kelvin model)

where '3 = unit stress,

unit strain,

t?)
It

Young's modulus of elasticity,

22
n = coefficient of viscosity,

T = stress rate, or rate of change of stress,

7‘ .\ .

= strain rate, or rate of change of strain,

When either of these models is subjected to
a sinusoidally oscillating force of radian frequency 3- ,
it is convenient to define a "complex modulus" having an
inrphase component and a component 90 degree out—of—phase.
The inrphase component corresponds to the elastic charac-
teristic of the material, and the out—of—phase component

is related to the dissipation characteristic of the mater—

ial.

The presence of a free dashpot in the Maxwell
model indicates that it would best represent a material

having "true flow" characteristic which produce residual

deformations upon loading. Since plexiglas was observed

not to be of this nature, the Kelvin model representation

was chosen. For the kelvin model the complex modulus is

given_as
Y (1;) = E + inta

It may be observed here that this complex modulus has a

damping component proportional to frequency Q] and n.

23

 

SR—4 gages recording
driver “7:55 ' :3 7 ' ‘ edge
4 .L
i
i
J
strain f - pi 2_ Fstrain
analyzer 7’ “ recorder

Figure l a. SCHEMATIC DIAGRAM OF EXPERIKENTAL WORK

VJ~V ~ ‘3 i read

Figure 1 b. POSITION OP SR-4 STRAIN GAGED ON REED

Figure 2.

./'\

——.—-———m.—_...
. I
o
.
h...

“..--..HL ..

"MT -- a
}

‘k

(a)

24

 

 

3‘
A
1
I I
plan} n
LJJ.,
f_m_ i
i
I
Y
(b)

SPRING DASHPOT KCDELS OF LINEAR VISCOELASTIC

MATERIALS:
(a) MAXWELL MODEL,

(b) KELVIN MODEL.

P)
U1

Bibliography

Horio, M. & Onogi, S. "Forced Vibrations of Reed as

A method of Determining Viscoelasticity", Journal of
Applied Physics, Vol. 22, pp. 977-981, 1951.

Nolle, A. w. “Methods of measuring Dynamic Mechanical
Properties of Rubber-like Materials“, Journal of
Applied_Physics, Vol. 19, pp. 753—774, 1948.

Nielsen, L. E. " A Recording Torsion Pendulum for
the Measurement of the Dynamic Lechanical Properties
of Plastics and Rubbers" Rev. Sci. Instruments, Vol.
22, pp. 690—693.

Rossmassler, S. A. & Eyring, H. " The Flow Preperties
of Fibers", Textile Research Journal, Vol. 19, pp.
810-815, 1949.

Whitte, R. S. &.Mrowca, B. A. & Guth, E. "Propagation
of Audio Frequency Sound in High Polymers" Journal of
Applied Physics, Vol. 20, pp. 481—485, 1949.

Mindlin, R. D. & Goodman, L. E. "Beam Vibrating with
Time-Dependent Boundary Conditions", Journal of App-
lied Mechanics, Vol. 17, pp. 377-382, 1950.

Perry, C. C. & Leissner, H. R. " The Strain Gage Primer"

Mc-Graw Hill Book Co. Inc., New York, Toronto, London,

l955~

8. Alfrey, T. & Guruee, B. D., Chapter ll, Vol. I of
"Rheology" edited by Eirich, F. R., Academic Press
Inc. New York, N. Y., 1956.

9. Bland, D. R. "Linear Viscoelasticity" Pergamon Press,

New York, N.Y., 1960.

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