OFFICE OF NAVAL RESEARCH

CONTRACT Nonr-1866(24)

NR - 384 - 903

TECHNICAL MEMORANDUM

No. 49

DYNAMIC MEASUREMENT OF THE

HARDNESS OF PLASTICS

Robert A. Wolkling

MAY 1963

ACOUSTICS RESEARCH LABORATORY

DIVISION OF ENGINEERING AND APPLIED PHYSICSHARVARD UNIVERSITY - CAMBRIDGE, MASSACHUSETTS

Office of Naval Research

Contract Nonr-l866(2U)

Technical Memorandum No. U9

DYNAMIC MEASUREMENT OF THE HARDNESS OF PLASTICS

by

Robert A. Walkling

May 1963

Abstract

This report describes a device for measuring the force-displacement

relations prevailing when a small, rigid, spherical indenter slides over

the plane surface of a piece of plastic material, A sinusoidaj .:,v varying

force, superimposed on a static bias force, is applied to the Indenter by

a magnetostrictive, bimetallic, laterally vibrating bar. The resulting

displacement of the indenter into the material is measured by a small

seismic pickup. The pickup is capable of detecting peak displacements as

small as 5O angstroms at frequencies above 100 c/s, even in the presence of low-frequency displacements more than 100 times as large.

The effective stiffness, defined as the ratio of the variational

force to the variational displacement, was measured for a blank, seven-

inch phonograph record composed of a vinyl chloride-acetate copolymer. _

Measurements were made, with bias loads ranging from 0 to 3 a record speed of 1 rpm, using a 1-mil radius, sapphire indenter vibrating

at 500 c/s. Subsurface yielding appears to begin near an indenter load

of I50 mg, and plastic yielding at the surface begins at an indenter load between 1.0 and 1.6 gm. The measured value of the stiffness varies with

position around the record, probably due to the residual stresses trapped

in the plastic material by the molding process. The stiffness varies

with bias load approximately as predicted by the Hertz theory of elastic

contact for intermediate loads, but not for light or heavy loads.

Acoustics Research Laboratory

Division of Engineering .and Applied Physics

Harvard University, Cambridge, Massachusetts

TM 1+9 TABLE OF CONTENTS

Page

ABSTRACT . 1

TABLE OF CONTENTS.i11

LIST OF FIGURES. T

LIST OF TABLES.v11

SYNOPSIS. ix

Chapter

I. INTRODUCTION. 1

H. THEORY BEHIND THE MEASURING DEVICE. 8

2.1 Mechanical and electrical design considerations . . 8

2.2 Magnetostrictive drive and vibrating bar theory . . 14

2.3 Seismic pickup and condenser-diode detector theory . 2k

III, EXPERIMENTAL EQUIPMENT. 32

3.1 Design and construction of the measuring device . . 32

3.2 The basic scheme of measurement. $6

3.3 Associated electronic equipment. 37

3.L Special features of the measurement system. 1+0

3.5 Signal-to-noise ratio. M

IV. CALIBRATION AND MEASUREMENT PROCEDURE.

1+.1 Seismic pickup calibration..

1+. 2 Measurement procedure. 5°

V. EXPERIMENTAL RESULTS AND CONCLUSIONS. 57

ill

TM 1+9 Iv

Appendix A.

Appendix B.

Appendix C.

REFERENCES

Page

REDUCTION OF THE EQUATION OF MOTION FOR A LOADED, FREE-FREE VIBRATING BAR.73

ANALYSIS OF THE CLAMPED-LOADED VIBRATING BAR .... 79

THE DRIVING-POINT IMPEDANCE OF A PIVOTED, LOADED VIBRATING BAR.83

_ , , ..90

TM 1+9

LIST OF FIGURES

Figure

1. Magnetostrictive bimetallic bar, "the eagle," and the eagle shoving seismic pickup.follows page 12

2. Uniform rectangular bar...

5. Condenser-diode detector circuit.25

4. Photograph of eagle.follows page

5. Scale drawing of eagle.follows page 31*

6. Photograph of eagle showing support and counterweight.follows page %

7. Photograph of experimental setup.follows page 36

8. Block diagram of electronic equipment.follows page 36

9. Eagle drive circuit and phase shifter.follows page 40

10. Block diagram of pickup calibration system .... follows page 48

11. The initial calibration curve for the seismic pickup at 200 c/s.follows page 48

12. A typical eagle calibration curve at 5OO c/s, used for obtaining the values of the eagle constants m and K for a value of 6 equal to zero.follows page 52

a

13. Diagram of the relative positions of the indenter during one vibration cycle. ..55

14. Mean and standard deviation of the real and imaginary parts of the effective stiffness of a vinyl copolymer phonograph record, as a function of indenter bias load.follows page 60

15. Diagram of an RCA 7" phonograph record, showing area traversed to obtain the data for Fig. 14 . . follows page 60

16. Mean and standard deviation of the real and imaginary parts of the effective stiffness of two other vinyl copolymer phonograph records, as a function of indenter bias load ..follows page 60

V

:* «d jiuiifimirjf ptirU» oí Uic eidcecivo «• J..M1080 oT a via¿l oopol^sior pbcnofmph V ¿ó, for different indou ver bino lo?rU#

: .mjwiun o, jfOeition U2v.«u2t1 tórs rco^rd . . , follow« pt!.p.e 60

. it . i«A i. ci:n iü&vUiKii^ ;,erí o 02 ' ’'iic ‘3*feoUre. • «:«öc- of ; wlfiÿl cOivul¿r.^r ¿Uxui-iuÿh, record ovei- two different nuctors of tlio record, aa a :.vi.v! ’.on of iudontur bine loed . . . , . , ... follow« p«ee

•c of the recorded deta ixw; which the curvee

60“

sr ¿lit, 17, uid lo wore obteineu ...... followe page oO

¿•i.».-. xct.-h of record w&ri> cauacd by rciAiaUen « 0 *a ¡íiduA.1 «troAfies .............. follow« page 60

1...M dnd iiafeinary part« of the effective «tlff- n;¿.- -jf a vlfl/1 copolimr jphoáograph record at i v : location«, plotted ae a function . c>f 1 .1 :«3¿cdlAtel/ after the turntable «topped

, , . . , , , , ,^. , , , « ... ... , p

A r.iv.vlod, ‘letf-ded vibro tin/; bar .-,. . . .-. ,1. , . . ¢5

*.

USi Oir IAi31£S

I. '(♦'hie o!* Laplace iTansforao........................

i'abie 11. Kepreaanlatlv* values of £. and £- ea a function of Kt.................... ... ... f .. .

be.-e

. 13

,

iL

TM 1(

SYNOPSIS

Numerous methods are available tor measu.rjnp the material proper¬

ties of viscoelastic materials. Some of these methods are static in

nature, while others measure the dynamic properties of the material. In

most cases, it is the bulk properties of the material that are obtained.

The results obtained from these measurements are often strongly

influenced by the measurement method employed. Some theory has been

developed for interrelating these measured results and for expressing

the material properties in forms most useful in dealing with different

practical situations. In most cases, however, either the procedure is

quite cumbersome or the evaluation is possible only through approximation

techniques. Thus it is still advisable to approximate as closely as pos¬

sible the real situation under study in order to obtain an appropriate

measure of the relevant material properties.

The present investigation is concerned with the dynamic measure¬

ment of the force-displacement relations prevailing when a small, rigid,

spherical indenter slides over a plane surface. The indenter load is of

sinusoidal form superimposed on a static bias load of equal or greater

magnitude, and the plane surface has a linear velocity with respect to

the Indenter. The measurements consist of applying known forces and

observing the resultant displacements of the indenter into the material.

The ratios of these two quantities, force/dieplacement, give the effec¬

tive dynamic stiffness of the material under the indenter.

In addition to describing the basic aim of the present research,

Chapter I provides a review of the literature related to and leading up

ix

TM U9 X

to the present research, most of which deals with the problem of phono¬

graph tracing distortion.

A review of existing measurement techniques revealed that no

device was currently available to perform the needed measurements under

the desired conditions. These conditions are discussed in Chapter II

and can be characterized as follows: There should be no contact between

the device and the record surface except for the indenter itself. The

device should not be mounted rigidly above the recording surface, but

should be free to follow any long-wavelength suriace variations which

might be present. The dynamic mass of the device should be as small as

possible. It should be electrically and mechanically stable over long,

periods of time so that a fixed calibration can be maintained. The

applied forces must be variable over a range extending from less than

10 milligrams to more than 1 gram, and the pickup must be capable u±

measuring displacements at least as small as 0.03 microns. The remainder

of Chapter II presents the theory for the design of two devices to be

used in making the desired measurements. The first is a bimetallic, lat¬

erally vibrating bar for applying a sinusoidally varying force superim¬

posed on a static bias force to an indenter which is attacking the plane

surface of a piece of record plastic. The theory of a free-free vibrat¬

ing bar with mass and spring load is developed, and the equation describ¬

ing the operation of the measuring device, called "the eagle," is

obtained. The second device is a seismic pickup for measuring the dis-

placement of the indenter into the record material as a result of the

applied forces. The analysis of the clamped-loaded vibrating bar is con¬

tained in Appendix B, and Chapter II concludes with the analysis of the

condenser-diode detector circuit for use with the seismic pickup. This

I'M h1 J xi

analysis provides us with limits on the values of the circuit elements

for optimum operation of the pickup.

Chapter III describes the experimental equipment used for measur¬

ing the effective stiffness of the record material. The equations

derived in Chapter II are used as a guide ..’or the design of the eagle,

and the construction and mounting of the eagle are described. The

related electronic equipment consists of an audio frequency source and a

dc bias source connected in series to supply the driving current for the

eagle, an rf oscillator to supply the required rf voltage for the seismic

pickup, several fixed-gain amplifiers, a variable twin-T filter tuned to

eliminate the frequency of the fundamental resonance of the seismic

pickup, a phase shifter for providing a variable phase reference signal,

a wave analyzer, a phase meter, and a two-channel graphic recorder to

record the magnitude and relative phase angle of the seismic-pickup out¬

put voltage. Each of these units is described in tenas of its character¬

istics and its function in the measurement circuit. The eagle drive sys¬

tem is considered in detail, including the problem of finding connecting

wires for the eagle which are flexible enough to pemit the freedom of

motion which the eagle requires. A brief discussion of the noise problem

follows, and the signal-to-noise ratio when making measurements is given

as approximately 30 dB,

In Chpater IV, the calibration and measurement procedures for the

eagle are set forth. The seismic pickup was calibrated by means of an

electrodynamic "shaker," and the pickup output voltage was found to vary

linearly with displacement over a considerable range of amplitude, A

secondary shaker standard, consisting of a modified loudspeaker, was

.."'»IS*.. .1.

TM

/

ihereaiTer employed. A measurement setup procedure is followed each

the oqulenient is turned -on í\ár - . ani-rement run, in order to ensure

standard operating conditions. The eagle constants are rechecked before

nach series of runs by means of an added-mass calibration procedure, and

the measurement runs are then carried out. All the measurements were

obtainou wloh a 1-mil radius, an. ; hirt indenter vibrating at ffiü c/a.

Different bias loads, ranging from 0 to * 0n, are achieved by changing

the number of small steel bails constituting the eagle counterweight.

Seven-inch blank records, composed of n vinyl chloride-acetate coco 1,,1..or

with small amounts of carbon black and stabilizer added, were the only

material samples used. A brass ring holds down the outsl : edge of the

recomí to guarantee that the record is lying flat against the turntable.

A turntable speed of 1 rpm allows sufficient time during each revolution

to get ready to add the next counterweight, and at the same time is fast

'■sough to ensure that at each Instant the i rid enter Is operating on virgin

am be rial. The data are transcribed from the chart of the graphic recor¬

der in a form which can be processed by an IBM 7090 computer in order to

obtain the real and imaginary parts of the dynamic stiffness of the

Indenter-material contact.

Results typical of those obtained are shown In Chapter V, and pos¬

sible implications of the results are discussed. It is observed that

plastic yielding at the surface begins at an indenter load in the range

from 1.0 to 1.6 0Ü. Indenter loads above this range are characterized

by a visible track which the indenter leaves on the record, and by a

gradual leveling off of the ’/alues of the real part and increasing values

of the imaginary part of the stlffneea. Several other features of the

TM b9 xiii

stiffness curves are pointed out, and their possible causes are discussed.

In this context, it is suggested that subsurface yielding probably begins

near an indenter load of I50 rag. An anomaly in the behavior of the

stiffness for indenter loads from 3OO to tOO mg is ascribed to the

release of residual stresses which may have been established during the

molding process. Variations in the value of the stiffness with position

around the record and with radial distance from the center of the record

are indicated. The adequacy of elastic theory in describing the experi¬

mental results is discussed, and the added complication of a possible

skin effect is also mentioned. Examples of the change in the measured

value of the stiffness as the record stops rotating are exhibited.

These emphasize the necessity for constantly providing fresh material to

the indenter in order to avoid these effects. Finally, the areas which

this device has opened for investigation are Indicated briefly.

BLANK PAGE

iiM ,pplpll(^•>|f^|l■^|»|¡lWPl", ..11 "NPil^l|,,

I*' ''"2? ._ _

TM 49

Chapter I

IHTROttJCTlON

The study of the behavior of solid materials under stress is

normally divided into two parts, The first part is confined to the

linear, elastic region, where stress and strain are related by Hooke's

law. The behavior in this case can be described in fairly simple terms.

Beyond the elastic limit, however, Hooke's law is no longer valid;

plastic flow results, and the relationship between stress and strain

becomes considerably more complex. A large group of materials exists,

however, in which there is no sharp division between these two regions.

In such materials, any stress produces a response with both elastic

deformation and viscous flow components, the magnitudes of which depend

not only on the mechanical properties of the material, but also on the

length of time during which the stress is applied. These materials,

called viscoelastic materials, Include many of the modem plastics.

A rapidly growing body of knowledge concerned with viscoelastic

materials is becoming available. Much of it is of an experimental

nature, and carpiste mainly of measured mechanical properties. A brief

review of the various experimental methods normally used has been

written by Ferry [l]. Unfortunately, the results of these measurements

are often strongly Influenced by the particular method employed. Seme

theory has been developed for interrelating tlees measured results and

1. J. D, Ferry, Chap. 11 In Rheology. Theory and Applications,, Vol. 2,

Frederick R. lirich, Editor (Academic Press Inc., New York, 1958).

-1-

TM 49 -2-

for expressing the arterial properties in forms most useful in dealing

vith different practical situa ticas; however, in most cases either the

procedure is quite cumbersoae or the evaluation is possible only through

approximation techniques [2, p. 4]. In addition, the size and shape of

the sample, as veil as any preliminary preparation which the sample

might require prior to the actual measurement, can affect the results

through voit hardening or changes In the surface structure. For this

reason it is still advisable to approximate as closely as possible the

real situation under study in order to obtain an appropriate measure of

the relevant material properties.

The present investigation is concerned vith the dynamic measure¬

ment of the force-displacement relatione prevailing vhen a small, rigid,

spherical ladenter slides over a plane surface. The indes*ter load is of

sinusoidal font superimposed on a static bias load of equal or greater

magnitude, and the plane surface has a linear velocity vith respect to

the Indenter. The measurements consist of applying knovn forces and

observing the resultant displacements of the indenter into the material.

The ratios of these tvo quantities, force/displacement, give the effec¬

tive dynamic stiffness, or vhen divided by the angular frequency of the

applied load, the mechanical impedance of the material under the Inden¬

ter. Since the phonograph playback proceas is one of the primary areas

for the practical application of these mea but aments, the experimental

dimensiona and ranges vere chosen so as to approximate the phonograph

playback conditions as closely as possible.

2. I. H. Lee, Chap. 1 in Viscoelasticity. J. T. Bergen, Editor (Academic ss, lev York, I960) ; see also Chap. 2 in this book.

TM 1+9 -5-

The interest in this prob len can be traced back to the earlier

analysis of phonograph tracing distortion, DiToro [3], Pi orce and

Sunt [1+], and Levis and Hunt [5], in order to simplify the analysis, all

assumed rigid groove vails, Kornei [6] , and subsequently Miller [7],

extended the analysis by applying the Hertz theory of the contact

betvean elastic solids [8J to the elastic reaction between the groove

vail and the stylus. Miller's values for the elastic moduli vere

obtained from measurements of the properties of bulk samples which had

been obtained from records prepared by cutting away the surface until

all traces of the record grooves had been removed. Miller also made a

few indentation tests In. order to de termine the elastic limits of the

materials used. In these testo he placed a standard 3-ail sapphire

stylus (or in seme cases a 1/64-inch steel ball) on the plane surface of

the record material and applied the desired load. The record was then

moved slowly under the stylus^ the stylus was removed; and the width of

the resulting track was measured through a microscope.

Hunt [9] ; in extending Miller's Indentation testa, used the eame

procedure with both a 3-mil and a 1-mil sapphire stylus. Hunt's results

3. M. J. DiToro, J. Soc. Motion Picture ïugra. ££, 1+93-509 (1937).

1+. J. A. Pierce and F, V. Hunt, J. Soc. Motion Picture Ingrs. 31, 157-186 (1938).

5. W. D. Lewis and F. V. Hunt, J. Acoust, Soc. Am. ¿2, 3I+8-365 (1941).

6. 0. Koraei, J. Soc. Motion Picture lagrs. 569-390 (1941).

7. F. G. Miller, Doctoral Dissertation, Harvard University, 1950. Besults summarized in F. Y. Hunt, Acústica 4, 33-35 (1954).

8. H. Hertz, Journal ffir die reine und angewandte Mathematik 92. I56-I7I (1882).

9. P. Y. Hunt, J. Audio Ing* Eoc. 3, 1-17 (1955). See also F. Y. Hunt, J. Appl. Phys. 26, 85O-856 (1955).

TM 1*9 -k-

vere far fim concluaive but gave sufficient evidence for him to postulate

the existence of a site effect similar to the effect that has often been

observed in fine metal whiskers. In these whiskers, the observed yield

stress, or elictic limit, is often maiiy times greater than the yield

stress of the «une material in bulk. Hunt explains that the yielding of

material under a spherical indenter occurs when the shear stress exceeds

some critical salue. If the stress is concentrated over a small enough

volume, there is a finite probability that this volume will not enclose

any "flaws" and hence will exhibit a larger yield stress which in the

limiting case approaches some theoretical salue. This, Hunt argues, would

explain the anomalously low wear rates which he and Fierce had observed on

both sty 11 and records during their early investigations of tracing dis¬

tortion.

Max !10] has questioned another apparent anomaly in phonograph prac¬

tice. Computations of stylus pressure are usually based on elastic theory

wnri give values varying from 25 000 to 100 000 pel. These values are then

compared with the ultimate strength of record plastics, which is 16 000 psi.

Max points out that it is not the stylus pressure on the record surface per

se which causes failure, but the shear stresses produced in the material

by these pressures. When maximum shear stresses are then computed, they

are found to be still in excess of the 16 000-psi limit. Max suggests

that this discrepancy is due to the Inadequacy of elastic theory when

applied to viscoelastic materials because of the time-dependent effects

which occur, apd he makes a plea for tho accumulation of more inowlftdge

about the dynamic properties of these record plastics.

10. A. M. Max, J. Audio Eng. Soc. J, 66-69 (1955).

TM 1+9 -5-

Barlow [llj bas objected »o Hunt's hjpothesia on the ground that

under any stressed condition there is always an elastic component, whether

or not plastic flow occurs. This elastic component recorers when the load

is removed, and hence makes the resulting trace smaller than if there had

been no recovery. For large indentations the effect is negligible; but,

Barlow claims, for small indentations, when the stresses are near the elas¬

tic limit, the reduction in trace width would produce precisely the effect

which Hunt has observed. Barlow further emphasizes that yielding begin®

in the subsurface regions, and that it can occur there with little if any

effect on the surface Itself. On this basis he has set limits on stylus

loading which are determined by the onset of surface yielding [12].

Finally, he has shown some data [ij] obtained in a way similar to the

method used by Miller and Hunt, except that he first applied a thin alumi¬

num film to the record surface. This film is scraped away by the stylus

wherever it makes direct contact with the surface, facilitating subse¬

quent microscopic measurements. Unfortunately most of the work was done

with a 1-mm ball, and the results were reduced to the "equivalent'' effect

of a 1-mil stylus, Barlow does not discuss possible effects of the alumi¬

num film on the apparent hardness.

Flom and Huggins [lU] have pub 1 shed some data on Plexiglas for

indenters of different size with different loads and different loading

times, but neither the material, the loads, nor the loading times are in

the area of primary interest here.

11. D, A. Barlow, J. Audio Eng. Soc. 4, 116-119 (1956).

12. T. A. Barlow, J. Audio Eng. Soc. £, IO9-II7 (1957).

IJ. D. A. Barlow, J. Audio Eng. Soc. 6, 216-219 (I958).

14, D. G. Flom and C. M. Huggins, J. Audio Eng. Soc. 2, 122-124, 128 (1959).

TM 49 -6-

Moêt recently, Walton |15j has discussed the factors Inrox/ed in the

stylus-grooT# relationship, and he has Included a phenomenon vhlch he calls

"surf-board action," a lifting action on the indenter which arises when the

surface Telocity exceeds some critical Talus. This action results in the

fact that the applied load is supported only partially by the elastic reac¬

tion of the naterlal, the other part of th« load being supported by the

dynaaic lift. Consequently the apparent elastic limit is higher under an

indenter whose relatiwe Telocity is aboTe the critical Talus than it ij

for one whose relatlTe Telocity is below the critical Talus. A consider¬

able number of experimental curres are shown which seen to rerify this

cenclusion and indicate that the critical Telocity is in the riclnlty of

the Telooltles found in current phonograph practice. Sons other results

were also obtained which Walton has not yet been able to explain. The

actual forces inrolred in the stylus-groore relationship, especially those

due to the acceleration of the effect!re mass of the stylus, are then con¬

sidered [l6l, and a pickup is designated for operation "without any groore

deformation at all." This design became impractical from Walton's point

of Tlsw and was altered so that the pickup would merely produce deforma¬

tion which naoothed the surface irregularities of the material but did not

distort the recorded wareform. Sosie rery good optical and electron micro¬

graphs of groore walls are shown which may be Interpreted as confirming

the adequacy of his design specifications; at least, the plastic flow

produced by the tracking of his own pickup is clearly less than that pro¬

duced by a similar coomerclal pickup.

15. J. Walton, Wireless World 62, 353-357 (1961).

16. J. Walton, Wireless World 62, 407-413 (1961).

TM k9 -7-

Many questione, however, still recnain unanswered0. Is the site

effect a legitimate hypothesis for explaining the experimental results

referred to above? Is there any relationship between an analysis based on

elastic theory and data taken beyond the elastic limit? What is the elas¬

tic limit for these materials under a spherical indenter? Is it dependent

upon indenter radius or material velocity? Does the manufacturing process

produce any kind of "skin” which could make the surface hardness greater

than the bulk hardness of the material? Does subsurface yielding really

leave the surface undeformed? Can the various results of the above inves¬

tigators be reconciled into one composite theory?

In order to seek answers to some of these questions, a device was

developed for applying a sinusoidally varying force superimposed on a

static bias force to an indenter which is attacking the plane surface of a

piece of record plastic. Another device was designed to measure the dis¬

placement of the indenter into the material as a result of these forces.

The effective dynamic stiffness of the material, obtained as the ratio of

the force to the displacement, can then be converted into the dynamic

material moduli by the use of elastic or viscoelastic theory.

It was hoped that this method might also provide another useful

for the non-destructive measurement of the mechanical properties of

materials other than those used for phonograph records. Indentation tests

using a hardened steel ball on various samples of steel plate have been

made by Davies [if), but his results are subject to many of the criticisms

brought against the indentation tests referred to above. More will be said

of this application when the results of the present investigation are discussed.

I?. E. M. Davies, Proc. Roy. Soo, (London) A197, ^16-^32 (19^9).

TM 49

Chapter II

THEORY EKHIHD THE MEASURING DEVICE

2.1 Mechanical and electrical design considerations

Before the design of the measuring device can proceed, a reasonable

estimate of the expected forces and displacements must be obtained. Hunt

0?' P- 4j, In discussing his Indentation tests, shows a microphotograph of

the Indentation track made In unfilled rlnyllte by a 1-mil stylus under a

2-gram load. The track width Is 9« 5 microns (|i ) and the depth of pene¬

tration Is given as 22 mlcrolnches, which Is 0.55 p or 55OO angstroms, approximately the wavelength of green light. Since these conditions

obviously cause the material to be stressed beyond the elastic limit,

defozvations which are wholly elastic should be smaller by at least an

order of magnitude. According to Hunt [9, p. Ij], simple theory predicts

that yielding begins with only 11 milligrams on a 1-mil stylus. This load

would produce a penetration of only 3 millimicrons, or 1/200 of the wave¬

length of light. It has been assumed here that the effective stiffness of

the Indenter-material system is constant, with a value of 3. 5 x 10 ‘ dyneq/cm.

Measurements of the stylus-groove resonance frequency tend to substantiate

this value at least In order of magnitude. Hence, a reasonable design

specification would set the effective stiffness of the record material at

3xl07 dynes/cm and would allow for dynamic displacements ranging from

0.003^ to 0.03^1.

In order to avoid the possibility of any influence due to repeated

application of stress to the same volume of material, the record must have

a linear velocity great enough so that new material Is constantly being

-8-

TM 49 -9-

preeented to the Indenter. Unfortunately, the surfaces of phonograph

records are not Ideally flat; in fact, they are not even optically flat in

most instances. Cellulose nitrate lacquer recording "blanks have a mirror¬

like finish, hut measurements with an optical flat reveal surface varia¬

tions of 0.6 ^ and greater, while the plane surfaces of many commercial

pressings do not even appear to he flat when examined In reflected light.

Hence, dynamic displacements at the measurement frequency with amplitudes

less than O.OJjj normal to the record surface must he measured in the

presence of fluctuations considerably greater than 0.6 ja . Naturally, any

record warp would further aggravate the problem. Cutting, shaving, pol¬

ishing, or otherwise preparing the surface to make it more flat would cer¬

tainly alter the surface properties we are trying to measure, so the meas¬

urements must he made on the unretouched surface. The only solution to

this difficulty lies in the use of a measuring device which allows the

indenter to ride over these large-scale surface variations without alter¬

ing the applied load by more than a negligible amount. Fortunately, by

proper choice of record velocity, the wavelengths on the record correspond

lag to the frequencies of the applied loads can be made small compared

with the lateral dimensions of most of the surface variations, which are

of the order of millimeters, and mechanical and electrical filtering can

then easily separate the desired pickup response from the undesired noise

occurring at lower frequencies.

Numerous commercial recording heads were examined to determine their

suitability for making measurements of the dynamic stiffness of record

materials, but not one was found that was entirely suitable. Although

they can all be used to apply a known sinusoidal force to an indenter,

many of than have no facility for measuring the resultant displacement of

TM 1*9 -10-

the Indeater Into the material. Those that do have this facility (e. g.,

the Vestrez recorder) tend to he extremely heavy and require some sort of

advance hall mechanism In order to maintain a constant depth of cut during

the recording process. The advance hall consists of a roller or sliding

felt pad which "feels" the record surface and allows the mass of the

recorder to follow the surface variations. Naturally, the advance hall Is

also applying stress to the material, either directly In front of the sty¬

lus or Immediately to one side. This working of the surface, plus the

fact that surface fluctuations can occur between the advance hall and the

stylus, makes this type of recorder also unsuitable for the Intended meas¬

urements.

Since no c esmere la 1 equipment was available to make the desired

measurements, It became necessary to construct a suitable device. The

performance requirements were the following: There should he no contact

between the device and the record surface except for the Indenter Itself.

The device should not he mounted rigidly above the recording surface, hut

should he free to follow any surface variations which might he present.

The dynamic mass of the device should he as small as possible. It should

he electrically and mechanically stable over long periods of time so that

a fixed calibration can he maintained. In addition, as has been Indicated

previously, the applied forces must he variable over a range extending

from less than 10 milligrams to more than 1 gram, and the pickup must he

capable of measuring displacements at least as small as O.OJp. A fairly

wide frequency range of operation would also he desirable, hut this Is of

minor concern.

Two types of electromechanical transducers had to he selected from

among those ccmmonly available; one to apply the driving force and the

other to serve as the displacement piclrup. Hunt [Í8, p„ 6j conveniently

divides electromechanical transducers into two broad classes: 1) those

which operate by means of electric fields, and 2) those which operate by

means of magnetic fields. In order to avoid croeatallc between the drive

and pickup circuito, the two tranoducere should not be of the same

class, and this was imposed as a further design conditior.

Of the transducers in the first class, perhaps the best known is the

electrostatic transducer, the most common examples of which are the elec¬

trostatic loudspeaker and the condenser microphone. This transducer has

distinct advantages in terns of small mass and small dimensions, can be

made very stable, and is quite sensitive, being able to detect extremely

small displacements from its static position. It is essentially a dis¬

placement measuring device, and was therefore considered to be ideal for

the displacement pickup.

The driving force transducer, therefore, had to be chosen from among

the transducers in the second class. Unfortunately, many of the trans¬

ducers in this class, Including the moving coil system, veil known for its

reliability and accuracy, rely upon heavy magnets for their operation.

Because of weight limitations, these all had to be excluded. One trans¬

ducer was found, however, which seemed to be suitable: a magnetostrictive

device consisting of a rectangular bimetallic strip or bar (Fig, la). One

half of the bar is magnetostrictive in the positive sense and the other

half is magnetostrictive in the negative sense, so that an applied alter¬

nating magnetic field produces a distributed bending moment which gives

18. Frederick V. Hunt, Electroacoustics (Harvard University Press, Cam¬

bridge, Massachusetts, and John VIley and Sons, Inc., New York, 19^)•

TM 49 -12-

rlB« to lateral vitiation. The alternating field can be applied by means

of a coll wrapped around the bar and In Intimate contact with It, while

the necessary bias field can be applied through the same coll or through a

second coll wound along with the first.

Three basic end conditions exist for laterally vibrating bars,

clamped, simply supported, end free. The requirement of non-rlgld mount¬

ing prohibits the use of the clamped end condition. The simply support ;

end condition is extremely difficult to obtain In practice, so It was

decided to use the free end condition. The original plan for supporting

the bar used a pivot at the midpoint, with the indenter attached to one

end and the other end hanging free. This was soon changed, however, to

the fora shown In Fig. lb, In which a very light spring support is attached

to the midpoint of the bar, the Indenter is also attached to the midpoint,

and both ends are left free. As the bar is driven sinusoidally, the ends

vibrate, producing at the midpoint an Inertial reaction force sufficient

to drive the Indenter Into the record surface. Because of the distinct

similarity between this device In operation and a bird flapping its wings,

It was very quickly dubbed "the eagle."

The displacement pickup must be able to detect the minute displace¬

ments produced by the driving force, but must remain relatively insensitive

to the large displacements caused by gross variation« In the record sur¬

face. Since the frequencies corresponding to these surface variations can

be made considerably lower than the driving frequencies employed In the

measurements, the task of separating the two responses is greatly simpli¬

fied. A device which makes Just this separation is the seismic Instrument

[19, p. 6f], a mass-spring system which responds to the amplitude of

19. J. P. Den Hartog, Mechanical Vibrations (McGraw-Hill Book Company,

Inc., Hew York, 193^).

FIG le

"THE EAGLE," SHOWING SEISMIC PICKUP

blank page

.*.

¿S'i'Äö"""- ''S' ■ -"it“.ægiBiKai

■ -#■-!

>

TM 1*9 -13-

vibraticais above its resonance frequency, and to acceleration instead of

amplitude below its resonance frequency. The low-frequency accelerations

caused by record surface variations are extremely small, so the instrument

is relatively insensitive to these frequencies.

The seismic pickup is mounted at the midpoint of the magnetostric¬

tive bar, directly above the indenter, as shown in Fig. 1c. As the bar

vibrates, the seismic mass m remains stationary, so that the relative

motion between the bar and the mass is precisely the motion of the inden¬

ter as it is being driven into the record surface. The bar and the mass

are electrically insulated from each other and form the two plates of a

small capacitor. The value of this capacitor varies with the plate sepa¬

ration, and hence it is directly related to the motion of the Indenter.

The capacitor plates are used to vary the coupling between a fixed-ampli¬

tude rf generator and a double-diode detector circuit, whose rectified

output voltage is proportional to the Indenter displacement. This ampli¬

tude modulation system was chosen over frequency modulation, in spite of

the freedom of the latter system from certain types of noise, because of

the lack of long-term stability of many FM detection schemes and the con¬

siderable complexity of the others. Dc bias, such as Is most conaon In

condenser microphones, was not used primarily because of the necessity of

installing a cathode follower Immediately adjacent to the condenser ele¬

ments. This would have added unnecessary bulk and mass to the measuring

device. The availability of subminiature and microdiodes allows AM detec¬

tors to be even more simple and compact than was possible with vacuum

diodes, and very good stability can also be obtained.

TM k9 -1Â-

2.2 Magnetostrictlyo drive and vibrating bar theory

The general treataient of the lateral vibrations of bars is so well

known that it seens hardly necessary to reproduce the basic derivation of

the differential equation here. The reader is referred to a few selected

works which show soma of the various approaches that can be made ¡19-26].

The usual procedure is to consider a uniform rectangular bar of density p,

elastic modulus X, and length l, which is large in comparison with the

dimensions of the cross-sectional area S (see Fig. 2). The lateral dis-

Fig. 2. Uniform rectangular bar

placement of an element of the bar at position i and at time t is y(x,t).

Ve assume that the displacements are email, and we neglect losses and

rotary inertia of the bar elements. If K is the radius of gyration of

an element of the bar taken perpendicular to the plane of vibration

20. Lord Rayleigh, The Theory of Sound (Dover Publications, lew York, 19h5), aid ed.

21. Horace Lamb, The Dynamical Theory of Sound (Dover Publications, Inc., Hew York, I960), 2nd ed.

22. Harry F. Olson, Elements of Acoustical Engineering (D. Van Hostrand Company, Inc., Hew York, 19^7), 2ad ed.

23. Philip M. Morse, Vibration and Sound (MoGraw-Hill Book Conçany, Inc., Hew York, I9MÎ), aid ed.

2k. Warren P. Mason, Electromechanical Transducers and Wave Filters (D. Van Hostrand Company, Inc., Hew York, I9W), aid ed.

25. Ein H. Tong, Theory of Mechanical Vibration (John Wiley and Sons, Inc., Hew York, 19o0). - ----

26. H. W. MoLachlan, Theory of Vibrations (Dover Publications, Inc., Hew York, 1951).

TM 1*9 -15-

of vibration), the equation of motion is given as

(2-1)

the net force with which the adjacent elements act on an element of the

bar, while the right-hand side represents the mass of the element times

its acceleration.

If a load or constraint is applied to one of the ends of the bar,

it is generally incorporated into the boundary conditions. In our case,

however, we wish to apply the constraint to the midpoint of the bar and

leave both ends free. The usual methods for handling Eq. (2-1) make no

provision for constraints applied elsewhere than at the ends, and the

numerous approximation methods available which do provide for constraints

applied at arbitrary points are generally concerned only with vibrations

at resonance ( see for eaample re£ 25). For this reason, an extension was made

of an operational calculus method of solution used by McLachlan ¡36,pp. 117-32¾.

It might be noted that we assume a uniform bar, whereas we intend to

work with a bimetallic bar. In general, p and E will not be the same for

the two metals employed. This could cause considerable, though not insur-

mountable, difficulty if we were interested in a numerical solution to the

equation. However, even for a uniform bar, the solution is so formidable,

as will be pointed out later, that all hope for a numerical solution was

abandoned. Thus, being content with just the functional form of the solu¬

tion, we can continue to pursue the analysis of a uniform bar and consider

p and E as some sort of average between the values pertaining to the two

metals employed.

nii'lMiittil'iii! riiiib'jiui.iiiii!

TM 49 -16-

For the moment, let ue he general and assume that a nass M and a

spring load k are attached to the bar at arbitrary points x ■ hp hg,

respectlrely. These loads are acted upon by the force represented in the

left-hand side of 1¾. (2-1) and thus are simply added to the right-hand

side of that equation. The fact that they are attached at points instead

of orer finite areas can be handled very easily by use of the Dirac

¿-function, vhich is defined as S(x) ■ 0 for x / 0, and J*6(x)dx ■ 1

[of. 27, p. 122], The terms to be added are therefore MS (x- h^ÿdx and

k 5 (X - hg)ydx. It can be seen that these tvo tenis are dimensionally

correct, since their integrals taken respectively from b^-6 to h^+£ and

hg- £ to hg+£ are Mfr and ky. Equation (2-1) then becomes

- XSK 2dxà V/Ò x^ - pBf dx + MS(x- h^ÿdx + kô(x-hg)ydx . (2-2)

Ve assume that the motion is sinusoidal and that it can be expressed

in the form y(x,t) » ^xje^“1, where w ■ 2*f and f is the frequency of

vibration. Substituting this into (2-2), we obtain

- 18 K 2dx d\i/*x* - - w2pSudx - w 2M¿ (x - h^)udx + k 6 (x- hg)udx .

2 If we then transpose all terms to the left-hand side, divide by -IS K dx,

define a ■ K.2, ß ■ k/SB K 2, and ■ u^p/E K 2, we finally

obtain

d^u/dx^ - A + aS (x -h^u + (3<5(x-hg)u - 0 . (2-3)

27. Philip M. Morse and Berman Feshbach, Methods of Theoretical Physios (McGraw-Hill Book Company, Inc., Hew York, 1953).

TM 1*9 -17-

ïquation (2-3) 1b now solved by taking its Laplace transform^

inserting the boundary conditions, and then taking the inverse transformo

The various required functions and their respective transforms are given

in Table I, along with the definition of the Laplace transform its

inverse. The entries in the table have been obtained from Morse and

Feshbach [27, pp. I579-I582], with certain changes made in the variables

to make the table more suitable for the present analysis.

Using Table I, we obtain for the Laplace transform of Eq. (2-3)

(p1* “ + ao"phlu (^) + (3e_ph2u(h2) - p\(0)

-P2u'(0) - pu"(0) -u"'(0) - 0 . (2-h)

The factors u(0), u'(0), u"(0), and uM,(0) are related to the displacement,

slope, bending moment, and shearing force of the bar at 1 - 0. The last

two of these are specified directly as boundary conditions, while the

other two are evaluated later in terms of the boundary conditions at x » 1.

At the free end of a bar, the bending moment and shearing force

must vanish, since by definition there is nothing attached to produce

either of these quantities, normally, this implies that u" ■ u"' - 0.

This is not true, however, in the presence of magnetostriction forces in a

bimetallic bar. Schenck [28J has derived the bending equation for this

type of bar and has found that in general the bending moment equals

Ciu"-C2, where and Cg are constants and Cg is related to the applied

magnetic field and the magnetostriction constants. If the banding moment

28. I. A. Schenck, Internal Memorandum, Acoustics Rese rch Laboratory, Harvard University, April, 1962.

TM 49 -18-

TABLE I

Table of Laplace Transforma

If ^(P)

^ 00

e~pIu(x)dx,

then u(z)

fC+loo

1 Sel e^YÍP^Pí

Jc-loo

C real and ^ 0.

Original Function Laplace Transform

Au( z)

4^u/dz^

AY(p)

p^Y(p) - p5u(o)

- P2U'(0) - pu"( 0)

- u'»(0)

u(z-a), z »a 0, z ca

<5 (z-a)u(z)

■ln(az)

coa(az)

alnh(az)

ooah(az)

e'apY(p)

e^uia)

a/(p2+ a2)

p/(p2+ a2)

a/(p2“ a2)

p/(p2-a2)

where A and a are conatanta.

TM U9 -19-

ie set equal to zero, we find that u" * Cg/^i* This ratio we shall call

D, since it is a driving function representing the magnetostrictively

induced curvature of the har, and it will be one of the boundary condi¬

tions at X * 0,1 . Since D is not a function of x, the other boundary

condition remains u"' • 0.

Inserting these boundary conditions at x * 0, we obtain from

Eq. (2-4)

(pU-\4)Y + ae"phl uUjj + (3 e‘ph2 u(h2) - p5u(0) - p2u'(0) - pD - 0 ,

which we solve for After expansion of l/(p^ —into partial frac¬

tions, the result is

y -[ae"pl11 uih^ + pe'ph2 u(h2)J 2 2 p + ^

|pu(0) + u'(0)J 2av2 p + \

Using Table I and taking the inverse transform, we assume that

O^h^hg-cl and thus obtain three expressions:

u^x) » I u(0) (cosh \x + cos \x) + u'(0) (sinh \x + sin \x)

+ (cosh \x - cos \x) O-ÄXäsln ,

vlJx) - u (x) - ofu(h

Uj(x) ■ u2(x)-^ pu(h2) jjainh k(x - hg) - sin X.(x - h2)J hg^xasi . 2\

L) jainh k(x- - sin \(x- h^jj h^ x *hr

TM 1*9 -20-

g mal nation shows that these expressions are smoothly connected at

* " hl'h2‘

The reduction of Kq. (2*5) consists of the elimination of u(0) and

u'(0) In terms of the boundary conditions at x « l, and finally the elimi¬

nation of the additional unlmowna u(h^) and u(hg) in terns of the expres¬

sion as a whole. This reduction is extremely complicated. A considerable

simplification results if we particularité Eq. (2-5) to the specific situ¬

ation under investigation by specifying the point of application of both

the mass and spring as the midpoint of the bar. Setting h^ ■ hg ■ i/2, we

obtain from (2-5)

u(x) ■ (cosh \x + cos \x) + U'gj^ (sinh \x + sin \x)

+ —- (cosh \x - cos \x) x^ i/2, 2k

u(x) - (cosh Ax + cos \x) + (sinh \x i sin Ax)

(2-6)

+ —5 (cosh Ax - cos 2 A

Ax) - u(|)[sinh A(x-|) - sin A(x-|)j

t/2Sx£t , 2A-

and in the interest of further simplicity, we can now define b ■ u(i/2)

and K' - (a + p)/2A5.

The reduction of (2-6) is not so complicated as the reduction of

(2-5), but it still requires considerable algebraic manipulation and

ingenuity in order to put the circular and hyperbolic functions into con¬

cise form. Since we are concerned here only with the motion of the bar at

its midpoint, the general solution has been carried out in Appendix A, and

a discussion of it will be found there.

TM 1*9 -21-

From Appendix A, we find that the displacement b of the midpoint of

the bar is given by

(2-7)

where

A( 1/2) - sin^- - sinh— ,

B(l/2) * coah^ cob^- + x ,

(pe m cosh-jr sin” + sinl cos^- .

Solving ( 2-7) for K', we obtain

AU/2) P .

1/2) ” B(</2) '

K* was previously defined as

r -(»Sit k

2\5SSk2 '

(2-8)

and since ■ u^p/EK^, we have h^ESK2 » u^pS/k, which can be substi¬

tuted into the above to yield

- (¿M+ k ki

u)2pSl 2 (2-9)

Here pSl represents the total mass of the bar and will be designated

Defining a new quantity K (unprimed) as

K = - to2 M+ k ,

TM 1*9 -22-

we Insert this into (2-9), equate (2-9) and (2-8), solve for K, and obtain

I, u\(pe (2io)

Equation (2-10) is the result we have been seeking, since K represents all

the constraints applied to the midpoint of the bar in terms of an effective

stlffhess, and embraces the unknown effect of the record plastic, as well

as the indenter-pickup system and the support which holds the eagle in

place.

It should be added here that K can also include resistive damping,

since the Inclusion of a velocity-dependent damping term would simply add

to Eq. ( 2-2) a term of the form ry times the appropriate ó-function.

This term would carry through as Y " Jwr/ESK^ in Just the same way that

a and jî> carried through the whole analysis, with the end result that K is

put in the general form

K - -wSi + Jwr + k .

The original intent, as has bean stated previously, was to approach

Eq. (2-10) numerically. This intent was thwarted, however, due in part to

the paucity of good magnetostrictive data for the materials used and in

part to the unknown effect of the finite site of the constraints added to

the bar. Since some sort of experimental calibration would be necessary

in any case, if only to check the accuracy of the numerical computations,

it was quickly decided to forego the large amount of computation which

would be involved in evaluating (2-10) and use it simply as a functional

relation. Needless to say, the frequency dependence of K is somewnat

TM 1+9 ■23-

in volved, since frequency occurs not only explicitly in to and tut also

in the argument of every one of the transcendental functions in A(f/2),

B(l/2), and (|)e. However, for any one given frequency, every factor is

constant except K, D, and t. For this reason it was decided to calibrate

and measure only at discrete frequencies, in which case Eq, (2-10) can he

expressed with considerable simplification as

K - m' ? + G 0 (2-11)

Another simplification can also be made: Although K includes all

constraints added to the bar, some constraints are permanently attached

and never change, such as the mass of the indenter-pickup system and the

eagle support. F^1" convenience, these can be removed from K and trans¬

ferred to the other side of the equation, where they can be incorporated

into G. When this modified G is defined as -K,., i:a. (2-11) becomes

Kr = m' £ - Ka , (2-12)

where Kr represents only the load on the eagle presented by the phonogrc a

record, and is therefore zero when the eagle is hanging free in the air.

Furthermore, in order to make the measured quantities explicit, we

recall that the driving function D is proportional to the magnetic field,

and hence to the ac drive current I. In addition, the pickup output volt¬

age Y is proportional to the indenter displacement b„ These proportional¬

ities can be expressed by the relations

D = CpI ,

b » CvV . b

TM 49 -24-

Insertiiiß these relations into ( 2-12), we can absorb the ratio into

m* by defining m - m'Cp/C^. Making this substitution, we finally obtain

the expression

K m: K. ( 2-13)

where m and K are now constants of the apparatus to be determined by a a

calibration procedure, I and V represent the driving current and pickip

voltage, respectively, and represents the effective stiffness of tL©

record material, which is the quantity we desire to measure.

2.3 Seismic pickup and condenser-diode detector theory

Turning to a consideration of the seismic pickup, we note that it is

included in the category of the clamped-loaded vibrating bar. The analysis

of the clamped-loaded bar, rather than being included here, has been

carried out in Appendix B, since its value in this application unfortu¬

nately turned out to be negligible. Numerical attacks on the solution,

which is a transcendental equation, became entirely too cumbersome to be

worthwhile. Consequently, a much simpler approach was used, which will be

described in the next chapter.

The seismic pickup mass serves as one plate of a variable capacitor,

the body of the eagle serving as the other plate. This capacitor is rep¬

resented by C1 in the circuit shown in Fig. 3- In practice, the generator

produces an rf sine-wave voltage which is modulated by and detected by

the remainder cf the circuit. The result is an output voltage V, which

consists of a dc component proportional to the amplitude of the rf voltage

and an ac component proportional to the amplitude of both tho rf voltage

and the modulation. We assume a square-wave voltage input in order to

TM b9 -25-

Fig. 3. Condeneer-diode detector circuit

Bimplify the analysis of this circuit, but the results do not differ

appreciably from those obtained in prectice through use of a sine wave.

Let the rf square-wave voltage have a peak magnitude E and frequency

f, with period l/f. The following aseumptione are made:

1) the forward resistance of is small enough so that charges

completely to E, leaving zero voltage drop across at the end of each

charging cycle;

2) the forward resistance of D2 is small enough so that Cx dis¬

charges to C2, leaving zero voltage drop across D2 at the end of each

discharging cycle;

3) negligible charge leaks off C1 through the back resistance of

4) in considering the rate at which charge leaks off C2, the back

resistance of Dg can be incorporated into Rg and thus be considered

infinite:

5) the stray capacitance across each diode is negligible.

During each half cycle that the generator voltage is -E, capacitor

becomes fully charged to « C^E, since diode Dx holds point A at

ground potential. When the generator voltage reverses, point A swings to

a potential of +2E, J)1 cuts off, Dg conducts, and C1 loses charge

TM 1*9 -26-

. ¢^21-Y0). Eaving received Aq1 from Cp capacitor Cg iß now at

its maximum potential Vo and con ta ine charge q2 = c2Vo' Thl9 charee 18

being loat through H2 at a rate auch that the total amount of charge lost

after a time t is given by

Aq2(t) » q2|l - exp(-t/R^2)J ,

which in one complete rf cycle anounts to

Aq2(l/f) - q2[l - exp(- l/m£r^ .

At the end of this time interval, Cg receivee another quantity of charge

from C1 and the decay procese begins agiin, repeating itself f times per

second.

In equilibrium, Cg must in each cycle lose as much charge as it

gaina, which is to say Aq2 - Aqr Equating these two quantities yields

C^SK-Y^ - C^jl-expi-l/fRgCg)] ,

which, when solved for V0, becomes

__2E_ (2-ll*)

Vo “ 1 + (C^fl-expi- l/fR^g^l '

It must be stressed that Yq is the peak of the instantaneous output volt-

age^ with the Instantaneous voltage over one rf cycle given by

V « Vo exp(- t/R^S 2) 0 < t < 1/f ( 2-15)

If the time constant R¿D2 is very long in comparison with one rf

period, so that fRgCg» 1, the exponential factor in Eq. (2-11+) can be

TM k9 -2?.

expanded in series form. If only the first significant tem is retained,

the expression becomes

2E 2EfR2C1

0 1 + (l/fRgC^ ‘ 1 + fR2C1 ’ (2-16)

which is independent of Cg. Two interesting applications of this circuit

can be pointed out here: When is muc^ greater than one, we find

that V0 <= 2E, which is the condition for the use of this circuit as a

voltage doubler rectifier as employed in some power supplies, or alter¬

natively, as a peak-to-peak reading voltmeter. If on the other hand

fR^31 is much loss than one, we have the result that Vo ■ 2EfR^^. In

this case the output voltage is a linear function of frequency and the

circuit can be used as a counting ratemeter. These applications imply the

availability of a peak-reading voltnu-ter, but by Eq. ( 2-15) ^ readily

be shown that as long as fR2C2»l,

TaTe ■ Tn.e ’ V0 [l - ( i/ai-P 2)]

to the second order of approximation, and this can be considered approxi-

matoU “l“1 40 V If modulation is now applied to capacitor C^, what is the effect on

VQ? Taking the derivative of Eq. (2-16), we observe that

5 dC

2EfR.

1 (l+fRg^)2 Cl(1+fR2Cl)

(2-17)

The first part of Eq. (2-17), rewritten in the form

2E fR2Cl

C1 (1 + fR^)2

TM 1*9 -28-

Bhows that for and E fixed, dVo/dC1 Is maximum when fRgC^^ » 1, which is

intermediate between the two extreme cases discussed above. In this

Instance, we find from Eq. (2-16) that Vo « E, and since we have specified

that fR^Pg»!, we also have the implication that Cg»^. Furthermore,

it can be seen that the value of dVQ/dC1 is symmetrical about its maximum

and is relatively Insensitive to changes in fRgÿ in fact, fi^i

change by a factor of 6 in order to reduce àY0/àC1 to one half of its max¬

imum value.

When the modulation is simple harmonic, can be represented by

C1 " Clo + Clmc0s V " Clo(1 + m 008 Wmt) ' (2_l8)

where is the static value of (^, is the peak of the modulation

component, is the angular modulation frequency, and m - C^/C^ is the

modulation factor. This can then be substituted into Eq. (2-16) in order

to determine the effect of simple harmonic modulation on V0. However, if

we mke the assumption that m^ 1, terms of higher than first order in m

can be neglected, and the result is the same as the one obtained by taking

the derivative. Thus we may simply substitute and into Eq. (2-1?)

in place of dC, and dV0 respectively, where represents the peak of the

modulation component of V0. The second part of (2-1?) then gives us the

detection sensitivity

jxa 1 ^Im Yo “ ! + fRgC1 C1 ’

which, when fR^ - 1, becomes

Yom 1 ^m

Yo " 2 C1 *

TM 49 -29-

Since all of the relatione derived above are independent of Cg, it

would eeem desirable to make Cg as large as possible so that the approxi¬

mation made in deriving Eq. (2-16) would be most accurate and so that the

rf ripple amplitude given by Eq,, (2-15) would be reduced to a minimum.

Unfortunately, two factors arise which place limits on the sire of Cg.

The first of these limitations is the fact that the time constant

RgCg must be small enough to allow the decay voltage given by (2-15) to

follow at each instant the shape of the variations in Vo required by

changes in C^; otherwise, distortion will result. The rate of change in

Vo produced by varying ^ is given by

dV dV^ dC. V . dC. _o o 1 _0 1_1 dt “ dC1 dt “ C1 i + fR2ci dt * ( 2-19)

Frcm Eq. ( 2-l8), we have the fact that

dC^dt - -mwmCl08int^t,

which upon substitution into (2-19) gives

- Vu m _ o m c

. ^ sin w t lo _m

! 1 + fPgC1

This can be written

dV -mu V sin w t _o _m o _m

dt “ (1 + fOT ) (1 + mcosu t) c. X m

On the other hand, the decay of V0 as given by Eq. (2-15) is

T0 - y 8Ip(-t/R^2) , (2-15)

TM k9 -30-

and Its rat« of change is

cLV 7o' -Vo

dT “ " *

It is clear that distortion will result if ve do not hare

dt decay

Writing out the inequality gives us

mV w o m

(1 + fRgC^

sin u t a

( 1 + m cos w t ) ’ ' ZU

which heccnes

“aRPC2

(1 + fRgCjJ (1 + mcosw^t)

m sin w t m

(2-20)

The most stringent requirement is given hy the minimum of the right-hand

side with respect to t. which occurs for cos u t » -m. This indicates m

that sin « t » (l-m2)1/^, and hence (2-20) "becomes in

"«¥2 1 + fRpC, p 1/2 -=-= (l - m2)1/2 . (2-21)

When m is small, (2-21) is not a very severe requirement; in fact, a

loss of modulation amplitude can occur long before there is any distortion.

The effect of this second limitation on Cg can best be observed by consid¬

ering everything to the left of Cg in Fig. 3 as a current source, deliver¬

ing a certain amount of charge per cycle. With modulation on 0^, this

TM 1+9 -31-

current, if we ignore rf ripple, is composed of a dc component which

passes through Bg and produces VQ, and a component at the modulation fre¬

quency f^. If the latter component passed only through R?, it would pro¬

duce V , but in practice, it sees shunted by the capacitor Co» This ODr <- c

shunt reduces the effective impedance of the combination, and thus reduces

the modulation voltage output below Vom. A convenient limit for the value

of Cg can be established by requiring that the impedance of Cg at fm be

greater than Rg, a statement which can be expressed by the inequality

w RrtCo < 1. This means that the effective impedance of the RoCo combina-

tion will be greater than Rg//2", and therefore the modulation voltage

output will be greater than V0Jsß, If a larger output than this is

required, a smaller Cg will then be necessary.

TM U9

Chapter III

EXPERIMENTAL EQUIPMENT

3.1 Design and construction of the measuring device

In the preceding chapter, we derived the equations which describe

in functional form the behavior of the eagle. We are now, therefore, in

a position to consider quantitatively Just how large the eagle should be,

The governing factor throughout this design is that the mass of the eagle

be kept as small as possible. The expression to be used in designing the

eagle is the denominator of Eq. (2-7), since we are less concerned with

the absolute value of the motion of the eagle than we are with avoiding

resonance, in the frequency range of interest. This range was originally

t-nV-wn to be approximately 100-400 c/s, so that it would fall above the

fundamental resonance of the seismic pickup but below the first resonance

of the eagle-phonograph record system. Since a major change from the

original eagle support has lowered the latter resonance, we have also

been able to work advantageously above the resonance and hence have made

most of the current measurements at c/s.

Setting the denominator of Eq. (2-7) equal to zero gives us

(Çe - K'B(i/2) - 0 . (3-D

Equation (3-1) is a transcendental equation, the solution of which is

best obtained by trial and error. There is no guarantee that any solution

obtained is the best one; but as long as a reasonable design is produced,

the solution will be considered satisfactory.

-32-

TM 1*9 -35-

In Eq. (3-1), CPe and B are functions of kt only, but \ is a func¬

tion of f, or in this instance the resonance frequency fa, p, and E

(see pp. lJ*fi' for an explanation of these symbols). K', on the other

hand, is a function of fo, M, k, a, v, I, and p, as well as kt. Since the

thickness a, the width w, and the length I are the quantities we are seek¬

ing, the remaining quantities must be specified,, The following values

were used: From previous published data on phonograph records, the stiff-

7 ness k for a 1.0-mil stylus and typical loading was chosen as 3 110

dynes/cm. Since it was expected that one of the two metals of the bime-

X

tallic bar would be nickel, the density p was chosen as 8.8 gm/cnr and the

elastic modulus E as 2xiOu dynes/cm . A reasonable value for the total

added mass M seemed to be 2 gm. If Iq. (3-1) is taken to represent the

first resonance, this value of M places an upper limit of 617 c/s on f0,

at which frequency the dimensions of the bar would vanish. In order to

obtain a reasonable size for the bar, therefore, fQ was chosen as )00 c/s.

Various trial values for a, w, and I were then inserted until a reasonable

set was found. The values finally obtained were a « 0.037 cm, w ■ 0, 5 cm,

and i ■ k cm; which when converted to English units, specify the bar as

15 mils thick, 3/16 in. wide, and 1.5 in. long.

Some long strips of the appropriate thickness and width were

obtained from Metals and Controls Corp„, Attleboro, Maes. The material

supplied is designated by the name TEUFLEX N4 and is a bimetal with one

half nickel, which has a negative magnetostriction coefficient, and the

other half 1*5 Permalloy, a nickel - 595& iron alloy with a positive

magnetostriction coefficient. To form the driving element of the eagle, a

1.5-in. length of this material was used, the edges were rounded, and the

TM 49 -54-

piece was wound with 200 turne of No. 32 Nyclad wire, A space wae left in

the center eo that the stylus and pickup could be attached,

The complete eagle is pictured in Fig* 4 and shorn to scale in

Fig. 5- An interesting point to r 9 is the use of nylon screws, which

completely eliminates the need for insulating washers and bushings. The

indenter is a sapphire Jewel, polished to a 1-mil radius, and mounted in a

straight, threaded shank. The threads serve two functions; l) the shank

is screwed into a threaded hole in the bottom plate of the eagle and then

cemented in place; and 2) tho calibration of the eagle is obtained by means

of known weights screwed onto the shank over the indenter tip.

The seismic pickup consists of a small brass block attached to a

phosphor bronze leaf-spring support by means of Eccobond solder 56C, a

conductive epoiy cement. The resonance of the clamped-free vibrator is in

the region of 30 c/b* We originally designed the seismic pickup by using

the theory of the mass-loaded cantilever presented in Appendix B, but we

were unable to obtain a low enough resonance frequency with a reasonable-

si sed block without having the cantilever become much too thin to be

mechanically stable. In addition, the first torsional resonance fell

approximately in the center of the frequency range of interest. In a

aomewhat successful effort to overcome these difficulties, the thickness

of the cantilever was increased and an elastic hinge was provided at the

support end by filing a transverse notch in the phosphor bronze (see

Fig. 5)* This notch has th® effect of converting the cantilever to a

lumped system, and was simply made deep enough to produce the resonance

frequency that was desired.

The triangular-shaped piece on the top plate of the eagle was

designed to increase the capacitance variation due to displacement between

i^.

i < • - •

Fig. 4. Photograph of eagle.

CONNECTION TO

RF OSCILLATOR

INSULATING SPACER

CONNECTION

TO DIODE D2

SUPPORT BAR

SEISMIC PICKUP MASS

TOP PLATE

2-56 NYLON

SCREWS

BOTTOM

PLATE

CONNECTION

TO DRIVE CIRCUIT

SIDE VIEW 0-80^

THREADS

2-56 NYLON SCREW

K-O.l

SEISMIC PICKUP

FIG. 5 SCALE DRAWING OF EAGLE

TM 1*9 -35-

the top plate and the pickup mass and to make the variation approximately

linear. The rf voltage is applied to the pickup mass, the bottom plate of

the eagle is held at ground potential, and the modulated voltage is

obtained from the top plate of the eagle„ The insulator between the two

plates is a piece of Teflon, chosen for its low dielectric constant in

order to minimize the stray capacitance between the two plates. The

detector diode D1 (see Fig. 3) has been placed in a cavity hollowed out of

the Teflon insulator and is connected between the top plate and the bottom

plate of the eagle. The other diode D2 is connected externally to a wire

leading from the top plate. The mean value of the pickup capacitance is

approximately 2 pF.

The eagle is attached to a bracket which occupies the same position

on the yoke of a 1930-vintage Scully recording lathe that a recording head

would normally occupy. This mounting enables the eagle to be raised and

lowered over the turntable. No horizontal motion of the eagle is required,

since in this form of lathe, a lead screw moves the turntable under the

recording head instead of moving the recording head over the turntable.

The eagle was originally attached to the bracket by means of a soft

leaf spring, which allowed vertical but not horizontal motion, and thus

allowed the eagle to ride over the surface warp of the records although at

the expense of a slight variation in bias load. The spring was made stiff

enough to support the eagle in the air, and yet at the same time it was

intended to be soft enough so that the variations in bias load caused by

the suri ace warp would not be significant. These variations were not sig¬

nificant for bias loads as large as one gram, but since the warp of even

the flattest records caused load variations of 55 mg during one revolution,

TM 1*9 -56-

meaBurementB In the 10-mg region were quite hopeleBB except at the highest

and lowest spots on the record. In order to eliminate this load ■variation,

a balanced eagle support was constructed with Jeweled pivots and with a

swinging cup for coun'terwelghts at the far end, as can he seen in Fig. 6.

The supporting pivot is placed two-thirds of the distance from the eagle

end; thus the counterweight must be twice as hea'vy as the eagle, hut its

dynamic mass at very low frequencies is only one-half that of the eagle.

With this arrangement, the bias loads are essentially unaffected by vertical

motion of the eagle (the necesaary wires do have a small residual stiff­

ness effect), and the load can be varied oy removing or adding counterweights.

5.2 The basic scheme of measurement

Figure 7 is a photograph of the eaq)erlmental setup showing the

Scully lathe (left) and the electronic equipment vised in making the meas-

uranents (right). A block diagram of this electronic equipment is given

in Fig. 8. This is not necessarily the optimum arrangement of -the various

conponentB, but it is the arrangement which gradually evolved over an

extended period of time.

The operation of the system can be described as follows: Both an

audio frequency source and a dc bias source are connected in series to

supply ■the driving current for ■the eagle. An rf oscillator supplies the

required rf voltage for the dlsplacanent-measuring seismic pickup, and the

demodulated output voltage from the pickup detector is amplified, filtered,

and applied to the input of a wave analyser. The wave analyser yields a

dc output ■voltage which is proportional ■to the amplitude of the signal

input voltage; and this dc voltage, which is directly rro'.ortloiKl to the

kill: V"Ptef- ‘> '. '■' r f-’nSr; ,,.,^ W:■ ■■> ''■ - .3

f''^^'y-’T^

JCUi

•H

c0oo

T3c«Vbop.Pt3«bOc

fH»o

-C

bOac

o£O,«bO

bOc

................................................................■/ '

14

.............../ .-

mk,It N

Pu5I)«

:s§ki

u.

4:«uuo♦*o

•Hb.

r'

WA

VE

AN

ALY

ZE

R

__

PH

AS

E

ME

TE

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7-

FIG

. 8

BLO

CK

DIA

GR

AM

OF

EL

EC

TR

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EQ

UIP

ME

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BLANK PAGE

TM k9 -37-

variational displacement of the indenter, is recorded by one channel of a

two-channel graphic recorder.

The phase of the input voltage to the wave analyzer is measured at

the same time by amplifying an ac voltage which is a filtered and phase-

locked replica of the signal input voltage to the wave analyzer, and apply¬

ing it to one input of a phase meter. The other input of the phase meter

is supplied by a reference voltage obtained from the eagle drive circuit

by way of a phase shifter and an amplifier. The phase meter provides a dc

output voltage which is proportional to the phase difference between the

two input voltages, and hence to the phase difference between the force

applied to and the displacement of the indenter. This phase difference is

then recorded by the second channel of the graphic recorder.

3.3 Associated electronic equipment

The Hewlett-Packard Model J>02A Wave Analyzer is the core of the

electronic instrumentation system. In addition to serving as a very

selective (7-cycle bandwidth) variable filter and voltmeter, it supplies a

dc voltage for the operation of a graphic recorder, it provides a restored

frequency output, and it includes the availability of an AFC mode of oper¬

ation. The AFC mode has the effect of locking the filter passband on the

frequency which ij being measured, and under this condition, the restored

frequency output is locked to the incoming signal in terms of both ampli¬

tude and phase. Thus the phase of the incoming voltage can be meas¬

ured without hindrance of noise or other frequencies which might be pres¬

ent. It is this restored frequency voltage which is amplified by an H. H.

Scott Type lUOB Decade Amplifier with gain set for 20 dB and then applied

TM U? -38-

to one input of an Advance Electronics Co. (now Ad-Yu Electronics Lab.,

Inc. ) Type 1*05 Precision Pbas^ Meter. The amplifier for the phase-

reference voltage is a laboratory-built Scott type amplifier with gain set

for 1*0 dB.

The graphic recorder is a two-channel (each 100 mV full scale) Texas

Instruments Servo-Riter recorder. This recorder produces a permanent inir

record of two variables side by side on two 1*. 5-inch grids. It is also

equipped with a marker pen, which we actuate by means of a microswitch

mounted In such a way that it is tripped once each revolution by a ridge

on the edge of the turntable. A minor difficulty was presented when the

phase meter was connected to the recorder. The variational signal voltage

available at the "external indicator" terminals of the phase meter was

much too large for the recorder and was offset from zero by approximately

three volts. It turned out to be more convenient for our purposes to make

directly available to the recorder the voltage drop across a resistor con¬

nected in series with the indicating meter. A simple voltage divider con¬

nected across this resistor then provided the proper range of variation

for operation of the recorder.

Since the detected voltage from the seismic pickup is quite low, a

Lowenstein, laboratory-built, low-noise amplifier has been inserted ahead

of the wave analyzer. This amplifier has an equivalent input noise of

V with the input shorted, and a bandwidth from 1 c/s to 110 kc/s. The

gain of hO dB has been well stabilized by means of a large amount of nega¬

tive feedback. A variable twin-T filter tuned to eliminate the frequency

of the fundamental resonance of the seismic pickup has been Inserted

between the amplifier and the wave analyzer. Since the eagle Is con¬

stantly being excited by the surface roughness of the phonograph record, a

TM 49 -39-

large voltage at this resonance frequency Is almost always present, and

the original pujóse of the filter was to prevent this voltage from over¬

loading a second amplifier wh'ch followed the filter. This second ampli¬

fier turned out to be superfluous, however, and was removed because its

gain tended to drift vith time, thereby changing the calibration of the

measuring system. A better location for the filter w-ald be immediately

before the Lowenstein amplifier, but the high impedance level at that

point would make such a filter much more difficult to build. The filter

has been left in its present location, since it permits easier visual

observation of the pickup output voltage on the Du Mont Type 208B Oscillo¬

scope and it also penults the wave analyzer to operate at a lower overall

input voltage level.

If we look at Eq. (2-13), the occurrence of the ratio l/y suggests

the use of a bridge type of instrumentation similar to that employed in

the measurement of transfer impedance or admittance. Since bridge meas¬

urements are usually more accurate than those obtained with a meter or a

graphic recorder, it might well be asked why one was not employed. Such

a bridge was actually set up and tested, using General Radio Company

Decade Resistor and Capacitor boxes and two toroid inductors for the ele¬

ments, with the wave analyzer acting as the detector. The bridge worked

well with the record stationary, except for the fact that a considerable

time was required to obtain an accurate balance because of the noise fluc¬

tuations that were present. With the reconi moving, however, not only was

the noise level higher, but both the amplitude and phase of the pickup

output voltage exhibited significant fluctuations from point to point around

the record. In the presence of these fluctuations, it was absolutely

1

TM 49 -40-

bopelesB to try to obtain a bridge balance, and bo the method vas aban¬

doned In favor of the arrangement vhlch has been described above.

3.4 Special features of the measurement system

The eagle drive system consists of a dc source, an ac source, and

tvo small resistors, all connected In series vlth the eagle drive winding

as shown In Fig. 9. The dc source supplies the magnetic bias field for

the magnetostrictive materials comprising the bimetallic bar. It Is a

laboratory-built supply capable of delivering at least one ampere to a

5-Í1. load vlth all ripple components suppressed by at least 60 dB. The

output Is Isolated from ground. The ac source, which drives the eagle,

consists of a Hewlett-Packard Model 200C Audio Oscillator and a labora¬

tory-built 12-watt power amplifier with an output transfomer capable of

taking the eagle bias current through Its secondary winding without being

seriously degraded In performance at the frequencies desired. The ampli¬

fier output transfomer is also isolated from ground. A voltage from the

oscillator Is applied to the y-axls of an RCA Oscilloscope for the purpose

of maintaining frequency stability by the use of a lissajous figure. The

z-axls was originally supplied with a 100-cycle reference signal from a

laboratory frequency standard, but In later work the reference was derived

from a local 400-cycle tuning fork oscillator.

The 0.1-Í1 resistor permits the Ballantlne Model 300 Voltmeter and

the Simpson 100-mV dc panel meter to measure the ac and dc eagle drive

currents, respectively. The ground connection has been made so as to be

closest to both the eagle winding and the ground side of the Ballantlne

voltmeter. This connection seemed to Introduce the least amount of hum

FIG. 9 EAGLE DRIVE CIRCUIT AND PHASE SHIFTER

TM 1+9 -41-

pickup. The 0.17-.Q. resistor provides a phase reference voltage propor¬

tional to the eagle drive current, and this voltage is connected to the

phase meter through the phase shifter also shown in Fig. 9« The phase

shifter is used for setting the phase reading to zero when the eagle is

unloaded. This procedure compensates for the phase shift produced by all

the apparatus, so that the resulting phase angle of the output voltage

from the loaded eagle is detenoined only by the load. In order to obtain

sensitivity in setting the zero, three variable resistors with values of

1 Mil (linear taper), 100 kil (logarithmic taper), and 1 kQ. (linear

taper) are connected in series.

The rf voltage for the eagle pickup is obtained from a 1-Mc crystal

oscillator, originally built for the laboratory by J. J. Faran. The major

requirement for the rf oscillator is that the noise affecting the rf volt¬

age be a minimum, since such noise would be detected along with the modu¬

lation from the pickup and would be a limiting factor in pickup sensitivity.

A crystal oscillator has been employed because, due to the high Q of the

crystal circuit, it has inherently less noise than oscillators which do

not employ a crystal element. This oscillator also incorporates an AYC

network to further stabilize the voltage amplitude. A separate power

supply is required, and a Krohn-Hite Model TIER 240 Power Supply has been

used. This supply is much larger than necessary for our requirements, but

it has the advantage of being able to provide a dc filament supply, the

use of which significantly reduces the 60-cycle hum component in the oscil¬

lator output voltage. The plate supply has also had an extra filter added

for the purpose of reducing oscillator noise.

TM 49 -42-

In building the detection circuit shown in Fig„ 3, numerous combina¬ tions of diodes were tried, with emphasis placed on subminiature and

microminiature types in an effort to obtain small size and weight. The

final combination chosen employs a II67A for diode and a Pacific

Semiconductor ezperimental type micro-diode for Turning to the rela¬

tions derived in the last part of Chapter II, we can use them as a guide

in determining the optimum values for Rg and Cg. If we apply the crite¬

rion that fR^L ft* 1, we find that since f is 1 me and C1 is approximately

2 pF, Rg should be approximately 0.5 Mil. The bounds on Cg are given by

the inequalities fR^Cg » 1 ajad w^RgCg 1. If we take fm as 5OO c/s,

Cg is thereby required to be much greater than 2 pF and much less than

640 pF, where the upper limit is not so critical as the lower limit. The

actual values found experimentally to be most satisfactory were Rg ■ 1 MU

and Cg ■ 1000 pF, shunted by the small capacitance of the cable leading

frem the diode to the detector. However, Rg is shunted by the back

resistance of the diodes, which at the operating signal levels could

easily be in the range 0.1-0.5 MA. This considerably reduces the effec¬

tive value of Rg, and thus brings the actual value of the product R^

more in line with the proposed upper limit.

One of the major problems encountered in setting up this instrumen¬

tation was that of connecting the eagle into the circuit. Two wires are

required for the drive winding and three more for the pickup, since one

is needed for the applied rf voltage, one for the output signal voltage

from the diode«, and one for the common ground. The difficulty arises

from the fact that few wires are flexible enough to penult the freedom of

motion which the eagle requires, a problem that was further aggravated by

TM 1*9 -ky

the change from the spring-mounted to the balanced eagle. Many samples

of so-called "flexible" wire were obtained in an effort to find the most

suitable arrangement. Unfortunately, all shielded wire had to be discarded,

and only the thinnest insulation could be tolerated. The connections to

the pickup were finally made by means of one loop of No. 1*1 uninsulated

wire for the ground connection and two loops of No. 1*1* enameled wire for

the other two connections, this enameled wire being the smallest, most

flexible insulated wire obtainable at the time. The connections to the

drive winding created more of a problem, since the wire had to be capable

of handling one-half ampere continuously for several hours. The solution

to this was finally obtained by using two pieces of size 7/1*0 Surprenant

No. BUB 7l*0U wire, with the insulation and four of the seven strands

removed from the last four Inches at the eagle end where the wire loops

from the eagle mounting bracket to the eagle itself. Since easily remov¬

able, lightweight connections to the eagle were desired, the center sock¬

ets and pins were taken fron several Amphenol Submlnax Series 27 coaxial

connectors, these being the smallest pins and sockets readily available.

The sockets are used to terminate four of the five wires to the eagle;

three of these sockets mate with corresponding pins, while the one for

the applied rf voltage mates with a small piece of No. 2k solid, tinned

wire which has been soldered to the pickup cantilever support. The ground

wire is simply attached by means of one of the eagle mounting screws, but

is quite easily removable. The residual effect of the stiffness of all

of these wires produces a variation in the indenter load of about 5 mg

for an indenter deflection of 1 mm.

TM 1*9 -1*1*-

3.5 Signal-to-nolB© ratio

Ab in moat problema of measurement, the major electrical hazard to

be dealt with is that of noise. The signal roltages obtained from the

pickup detector when measuremente are being made are approximately 3OO jaY

appearing as a modulation superimposed on a I5-V dc carrier. This ratio

indicates that the percentage modulation is extremely «nail, and that we

must therefore exercise great care in eliminating undesirable electrical

noise. The proper choice of rf oscillator and power supply, the connection

of proper ground wires, and the use of the narrow frequency selectivity of

the wave analyzer have all resulted in a noise voltage of 10 Y at the

same reference point. This value gives us a signal-to-noise ratio of

approximately 30 dB. It seems unlikely that further care in the electri¬

cal system could Improve this value, since the noise voltage seems to be

of about the same magnitude as the signal voltage produced by the response

of the pickup to building vibrations. In any case, the noise produced by

the surface irregularities of the moving phonograph record decreases the

signal-to-noise ratio by 5-10 dB. A considerable improvement in this

noise figure could be obtained by increasing the eagle drive current, but

one of the measurement conditions states that the TOriational displacements

must remain small compared with the displacements produced by the bias

loads. This condition would no longer be satisfied for the lighter bias

loads we have used if the eagle drive current were to be increased signif¬

icantly above the value chosen for these measurements. Some improvement

can be obtained, when necessary, by the use of BC filters at the two inputs

to the graphic recorder, so that the net signal-to-noise ratio when moving

measurements is still approximately 30 dB, but this value seems to be

iufficiemt for most of our purposes.

TM 1*9

Chapter IV

CALIBRATION AND MEASUREMENT PROCEDURE

Haying described the construction of the eagle and the arrangement

of the associated apparatus, we shall now turn to the measurement pro¬

cedure used to obtain the experimental results. In order to obtain the

dynamic stiffness of the phonograph record material from the eagle

drive current I and the resultant seismic pickup output voltage V, use is

made of Eq. ( 2-lj), which describes in functional form the operation of

the angle. This equation is

Er-m|-Ka , (4-1)

where m and Ko are constants whose values are to be debemined.

The first measurements with the spring-supported eagle were made at

200 c/s, sin-.e, as the theory predicted, the region around this frequency

was reasonably free from resonances. After the change was made from the

spring-supported to the balanced eagle, however, operation at 200 c/s

became extremely unreliable, due to the fact that the eaglo resonances

had been shifted to lower frequencies by the additional mass loading of

the balancing arm and counterweights (cf. p. 33). An investigation of

the frequency response of the eagle below 1 kc/s was therefore made, and

5OO c/a was found to be a suitable operating frequency. This frequency

is far enough removed from the various eagle resonances and their harmon¬

ics so that little difficulty is normally encountered during a measure¬

ment run.

-45-

TM k9 -46-

The chanfeg in the eagle support has also made necee sarj » slight

modification in Iq. (4-1). It will be remembered that the constant K fit

includes the effect of the riaas load attached to the center of the

bimetallic bar. This load consists of net only che mass of the eagle but

also the mass of the counterweights, dr counterweight mass were con¬

stant throughout a measurement run, Eq„ (4-1) would be correct as it

stands. Howerer, it is by remoring counterweights that bias loads are

applied to the indenter as it rides along the surface of the phonograph

record material. To include the effect of this reduction in the mass of

the counterweights, a term K ■ - <5 w2Mti must be subtracted from E .

In this expression, Mg represents the Talue of the bias load on the .

Indenter, in mass units, and 6 is the coupling coefficient relating the

effective static mass of the counterweight to its effective dynamic mass,

both quantities being referred to the eagle indenter. The equation which

describes the operation of the balanced eagle is therefore

E - m = - E r V a (h-2)

4.1 Seismic pickup calibration

The first step in the measurement procedure is the calibration of

the seimalo pickup, or the determination of the quantity in the

equation

(*-3) b

where b represents the variational displacement of the Indenter. A pre

else determination of is not absolutely necessary, since, in the

TM k9 -kl-

expreBBioa. for the etiffnesB of the phcaiograph record material given by

Eq. (4-2), has been included in the constant m and the value of m is

determined by a different means. Nevertheless, it is desirable to fcnov

the value of b at least approximately so that ve can guarantee that the

variational displacements are small with respect to the displacements

produced by the bias load. This condition can be stated in a more useful

form by saying that the applied variational forces df - bKr must be small

with respect to the applied bias loads. In either case, however, some

estimate of the value of C. must be obtained. D

A Ling Electronics Calidyne Model 6C Shaker was used for the initial

calibration of the seismic pickup. This shaker is capable of producing up

to 25 lb vector force output, which is considerably greater than that

needed to shake a 3-gm eagle at displacements of 2 p; but since it was an

available piece of laboratory equipment, it was pressed into service.

The shaker has the advantage of being equipped with a factory-calibrated

pickup coil which operates in conjunction with a permanent magnet to pro¬

duce a voltage proportional to the velocity of vibration. We are thus

able to monitor the velocity or displacement amplitudes produced. This

particular shaker is also equipped with a degaussing coll, which shields

the shaker table from the magnetic field of the driving coll.

A block diagram of the pickup calibration system is shown in Fig. 10.

The audio oscillator and amplifier used to drive the shaker are the same

ones normally used for driving the eagle, while the amplifier following

the ahaker is the Scott amplifier normally employed in the eagle phase-

measuring circuit. In this application, the Scott amplifier is connected

to the pickup coil of the shaker so that the vibration amplitude can be

....

TM U? -kQ-

Bonitorod with a Ballantine Model 3OO Voltmeter. The only new piece of

equipment employed in this system is a General Radio Company Type l^^U

Decade Voltage Dirider. The seismic pickup circuit was described in

Chapter III.

The calibration procedure was as follows: With the voltage divider

set at 1.0000, the shaker table was driven at an amplitude that was easily

observable through a microscope and the shaker pickup coil calibration

was checked. This was done at both 200 and 100 c/s. Since these measured

values agreed with the calibration supplied by the manufacturer, his cali¬

bration was therefore assumed to be correct. The frequency of the driving

voltage was then set at 200 c/s and a Lissajous figure was employed to

indicate any deviation from this value. With the voltage divider still

set at 1.0000, the shaker was driven at an amplitude which produced a

reading of 1.0 on the Ballantine voltmeter with the Scott amplifier set

for a gain of 100. The desired vibration amplitudes were then obtained

by changing the setting on the voltage divider, since this procedure

allowed for more precision than could be obtained by reading the voltmeter,

especially as the voltages approached the level of the noise voltage in

the pickup coll circuit.

The selo&lc calibration curves which were taken at 200 c/s are

shown in Fig. 11. In this figure, the amplified rma output voltage of

the selnlc pickup, as read on the wave analyser, is plotted against the

peak displacement of the pickup as measured by the shaker pickup coll.

It can be seen from the upper curve that the seismic pickup output volt¬

age varies linearly with displacement over a considerable range of ampli¬

tude. The lower curve shows additional data taken in order to expand the

:il. " K...1i Hi.|>"JllliiH»l -Il ï' ... ill.iiMiiil|l.mMilUUU

AM

PLIF

IED

rms

OU

TP

UT

VO

LTA

GE

OF

SE

ISM

IC

PIC

KU

P

FIG II THE INITIAL CALIBRATION CURVE FOR THE SEISMIC PICKUP

AT 200 C/S.

TM k9 -49-

lover left-hand comer of the upper curve. The noise voltage component

at 200 c/s is indicated and corresponds to a peak displacement of about

0.002 p, which is only 20 angstroms.

The above calibration was the only extensive calibration performed,

since it seemed reasonable to assume that the pickup is linear with

respect to variations in amplitude and that therefore only a few calibra¬

tion points at each frequency are actually necessary. In addition, the

pickup is mounted in a somewhat exposed position on the eagle, and it did

not seem unlikely that in the course of excessive handling the pickup

cantilever might become bent out of adjustment. This, in fact, did occur

several times, and since it was inconvenient to remount the eagle on the

shaker table for a recalibration of the pickup each time, a secondary

standard shaker has been employed. This shaker was constructed from a

6-inch speaker with a damaged cone. The cone and metal frame of the

speaker were completely removed and a brass slug was cemented to the coil

form in place of the dust cap. When in use, the speaker is connected into

the eagle drive circuit in place of the eagle drive winding, and the eagle

indenter rests on the center of the brass slug. The speaker was calibra¬

ted by first calibrating the seismic pickup on the original shaker at

each frequency of interest and then placing the eagle on the speaker and

reproducing the seismic pickup output voltages by means of the speaker

drive. These driving currents are easily reproduced and the mechanical

setup is extremely simple; thus the seismic pickup calibration can be

checked whenever it seems necessary. Seme loss in calibration accuracy

obviously results from this method, but, as pointed out above, extreme

accuracy is not necessary for this calibration and this method seems to

TM 49 -50'

gire »atlBfactory resulta. The yb lue b nov used for at 200 and 5OO c/b

are, reapectiYely, 2.02 and O.94 )i/V, where the unite are giYem In tema

of peak displacement and ms voltage.

4.2 Measurement procedure

When a measurement run is to he undertaken, the equipment 1b turned

on in advance and allowed to come to equilibrium bo that drift during the

measurement procedure is essentially eliminated. After a sufficient time

interval has elapsed, the following setup procedure 1b employed in order

to ensure standard operating conditions : We first adjust the zero fre¬

quency of .‘id wave analyzer and set the full-scale meter deflection to

correspond to a full-scale reading of 100 on the graphic recorder. The

rf oscillator voltage and the resultant dc voltage from the sei «aie pickup

demodulator are then checked by means of an Acton Laboratories Type 810

Vacuum Tube Voltmeter. We next demagnetize the bimetallic bar by see ding

an alternating current of one ampere at 200 c/s through the bar winding

and decreasing it slowly to zero. A standard magnetic bias field is then

applied to the bar by increasing the direct current in the bar winding

from zero to one ampere, decreasing it again to zero, and then increasing

it to one-half ampere. This procedure was found to produce the most

satisfactory bias field for the range of alternating currents employed.

We then superimpose on the direct current an alternating current at the

desired frequency. The amount of alternating current used in early meas¬

urements was 0.1 A, but this current was found to produce variational

forces on the indenter which were not small with respect to seme of the

applied bias loads, so that in more recent moasuraments the current has

i'W -51-

been reduced to O.OJ A. Finally, the phase meter full-scale reading is

adjusted so that l80° corresponds to a scale reading of 90 on the graphic

recorder, the phase reading oi the seismic pickup outuut voltage is set to

zero by means of the phase shifter described in the preceding chapter, and

the equipment is ready for making measurements.

Before the stiffness Kr of the record material can be obtained, we

must first determine the values of the eagle constants m and K in

Eq. (4-2) by means of an added-mass calibration. In performing this cali¬

bration, several masses ranging in value from 0 to 2. r; gm are screwed on

•.ho threadeu shank of uhe indenter and the mamitude and phase 0 .he

seismic pickup output voltage are recorded under uhe eagle driving condi¬

tions described above. In this instance, Mg in Eq. (4-2) is set equal to

zero and Kr is replaced by (1+6)^¾ , where M represents the value

of the mass which was screwed onto the indenter shank. The coefficient

1+Ô arises from the fact that the eagle balance must be maintained

throughout this procedure by adding appropriate counterweights, and the

net effect of mass plus counterweight, referred to the indenter, is l+¿

times the value of M (see p. 46). Since the phase angle of the pickup

voltage remains essentially at zero and is real, we conclude that m and

K are each real and contain no imaginary components. The ratio I/V is

then plotted against E^; the slope of this straight line gives the value

of m. while the intercept gives the value of ~Ka . A typical calibration

curve taken at 5OO c/s is shown in Fig. 12. For this calibration the

£ value of ¿ was taken to be zero. The curve gives for m the value - 34.0 x10

ohms dynes/cm, and for Kg the value - 34. 1 x 10^ dynes/cm. The last figure

TM k9 -52-

ln each of these relues Is not significant, but is retained for purposes

of odeputation. For other assumed values of 6 , these numbers can be

scaled accordingly. We follow this calibration procedure each time the

equipment is turned on before making a set of measurements, since ve are

thus able to make certain that no changes in the measuring system have

significantly altered the calibration values.

A blank record is then placed on the turntable and held down around

the outside edge by a brass ring. The records are samples supplied by

HCÀ Victor and are of the 7-inch, wide center-hole variety. The material

of these samples, designated V-3II, is the same as that commonly used in

the manufacture of Victor Becords, and is a vinyl chloride-acetate copoly¬

mer with nail amounts of carbon black and stabilizer added. A special

fora with a large center spindle has been made out of Fhenolic to support

these records on the turntable. A pin which mates with a small hole

drilled through each record sample near the Inner radius has also been

proTided. This pin ensures that the orientation of a record each time it

Is placed on the turntable is the same with respect to the turntable

ridge which operates the recorder marker pen. The use of the brass ring

to hold down the outside edge of the record guarantees that the record is

lying flat against the turntable and therefore will rot be able to retreat

when a load is applied to the indenter. If the record were able to move

under the indenter, the motion of the indenter would be greater than that

permitted by the stiffness of the record material, and hence the measured

value of the stiffness would be too small.

The measurement of the record-material stiffness E then proceeds r

as follows: After the turntable is moved into starting position and the

RATIO OF EAGLE DRIVE CURRENT TO SEISMIC PICKUP OUTPUT VOLTAGE I/y, mhos

FIG. 12. A TYPICAL EAGLE CALIBRATION CURVE AT 500 c/s ,

USED FOR OBTAINING THE VALUES OF THE EAGLE CONSTANTS

m AND Ka FOR A VALUE OF 8 EQUAL TO ZERO.

BLANK PAGE

TM 1*9

lead screw is set, the eagle Is lowered by means of the mounting bracket

until the indenter Just touches the record. This position is determined

either by observation of the eagle through a microscope mounted on the

lathe, or simply by observation of the moment at which the output voltage

fron the seismic pickup suddenly begins to exhibit a large amount of

chatter at the pickup resonance frequency. This chatter is caused by the

fact that the indenter is not in continous contact with the record material

but is alternately striking and leaving the surface once each cycle. A

scale on the movable bracket enables this position of the eagle to be

reset once it has been determined. With the eagle in this position, the

counterweights are then removed, the turntable is set in motion, and the

counterweights are added in small steps once each revolution of the

record. In this way a series of bias loads is obtained, starting with

the heaviest first. This method of operation has been used because it

takes a shorter time to add a weight than it does to remove one, and hence

less surface area of the record is wasted during the process of changing

the bias load. The counterweights consist of steel balls in three sizes,

1/16-, 1/8-, and 3/l6-inch diameter, of the type used in the manufacture

of bearings. Since the balls of each size are quite unifora, they make

excellent calibrated weights. At one stage during the measurement run,

the turntable is stopped and the accumulation of steel balls is replaced

by an equivalent brass weight screwed to the underside of the counter¬

weight cup, as shown in the photograph in Fig. 6. In this way the

counterweight cup is maintained in an upright position and the steel

balls are prevented from overflowing. Additional bias loads beyond the

total weight of the eagle can be obtained by screwing the calibration

TM 49 Rit¬

masses on the ináeater shank. In the present measurements, bias loads

ranging from 0 to 5 ga hare been employed.

By using a turntable speed of 1 rpm, v\ are allowed sufficient time

each re to lu tien to get ready to add the next counterweight, as well as to

watch the recorded data. At the same time, the speed is fast enough so

that the warelength of the tarlational indentation produced on the record

by the indenter at 5OO c/s is larger than the limiting diameter of the

circle of contact for a 1-mil indenter under the maximum applied bias

loads. This indentation wavelength, when the track is three inches from

the center of the record, is approximately 16 ji. Assuming purely elastic

defoxmation, the Hertz formula, which is referred to below, predicts a

contact-circle diameter of approximately 12 y. for an indenter load of

5 go. The measurements obtained by Hunt ¡j>, P.^ ell0w a track wläth of

10 ji for the same load. Harrower tracks, corresponding to smaller con¬

tact circles, would of course be obtained for lighter Æo&ds.

In our case, we are primarily interested in the effect of the much

snmller variational stress which is superimposed on the larger stress

produced by the bias load. At the maximum bias load of 3 gn, the peak

value of the variational load is only 0.1 gm; thus the total load on the

indenter varies cyclically from 2.9 to 3.I W- Although the stresses in

the material extend out somewhat beyond the edge of the circle of contact,

they are quite ana 11 at distances as large as half a radius beyond the

boundary of the contact circle. The situation is illustrated by Fig. 13;

which shows three successive positions of the Indenter, and its corre¬

sponding loads at these positions, as it slides along the surface of the

record. From this figure it can be seen that although a stress is pro-

TM 1+9 -55-

duccd in the material at location h by the maximally loaded indenter when

it is at location a, one half-wavelength away, this stress is quite «mu 11

Fig. IJ. Diagram of the relative positions of the indenter

during one vibration cycle.

compared to the stress that will be produced directly under the minimally

loaded indenter when it has reached location b. Since this is the

extreme case of load variation, it can be said that the indenter is

always operating on material which has not previously experienced any

stress as large as the one it is encountering at that Instant. We can

say, therefore, that the indenter is operating on virgin material. Since

the turntable lead screw has been set for a pitch of 220 lines/inch,

which is approximately II5 p/line, sufficient space has also been allowed

to ensure even more adequate protection against interaction between

adjacent indenter tracks.

The chart speed of the graphic recorder has been set for 3 inches

per minute, which provides three inches of chart paper for each revolution

of the record and gives adequate resolution of the data for most purposes.

The chart paper has 50 divisions full scale (marked from 0 to 10) for each

TM 49 -56-

channel, and vertical linee every tenth of an Inch. The distance between

adjacent vertical linee, therefore, corresponde to a rotation of the

record of 12 degrees. Data readings have been taken at every other ?ar •

tical line, thereby giving us 15 data points for each revolution, and

hence for each bias load, except for some Instances when one datum point

was omitted near the change in bias load and we used only Ik points.

This number of points is small enough so that the process of reading the

data does not become too tedious, and yet a reasonably good average value

for the stiffness Kr around the record can be obtained.

TM 1+9

Chapter V

EXPERIMENTAL RESULTS AND CONCLUSIONS

The experimental resulta presented here are intended to be illustra¬

tive, and in v.o sense do they cover the total range of measurement possi-

bllitiesj in fact, we have only scratched the surface. The design and

construction of this dynamic-stiffness measuring device, called the eagle,

has opened extensive possibilities for making a type of measurement which

it has not previously been possible to make, i.e., to make a point-to-

point examination of the elastic properties of a material on a microscopic

scale, as opposed to making bulk measurements of the same properties.

This type of measurement is not strictly limited to plastics, but is

applicable to any solid material as long as the material of the indenter

can be assumed to be hard in comparison with the material under test. In

order to establish reasonable limits on the extent of the present inves¬

tigation, the number of manipulated varie bles was held to a minimum. The

results, therefore, are simply intended to reveal the type of information

which can be obtained from this device, and in addition to provide a

small amount of insight into the rather difficult problem of the inter¬

action between the phonograph stylus and the record groove.

The experimental data are obtained in terms of an eagle drive

current I and a corresponding seismic pickup output voltage V, with mag¬

nitude I vl and relative phase angle <j>. This voltage is thus represented

-57-

-=)8- i'M i-

Tht above exT;res8ion Le then inserted into Eq. (^-2), which is

Kr " m? ' Ka “ ¿u)2mb • (1|_2)

in order to obtain the real and imaginary parts of the dynamic stiffness

Kr - kr' + Jkr”. Although the numerical computations are straightforward,

they are quite time-consuming. Therefore, since several other quantities

were desired along with K^, such as the variational indenter displacement

b and the variational indenter load df (or dMg, where F«Mgg, g being the

acceleration due to gravity), a FORTRAM program to make these computations

was written for an IBM 7090 computer and the computer was used to process

the data.

Before Eq. (h-2) can be employed, a value must be chosen for 6, which is the covyl'ng coefficient for the eagle support bar. This coef¬

ficient relates the change in the effective static mass of the ea?f.le

counterweight to the change in its effective dynamic mass, both quantities

being referred to the indenter. The value of this coefficient is strongly

dependent upon the frequency of operation. An expression for 8 in terms

of the counterweight mass, the bias load, and the constants of the support

bar has been derived in Appendix C. Computations described in this

Appendix indicate that at a frequency of 500 c/s, the error caused by

setting o equal to zero is less than other experimental errors for all

bias loads employed. For this reason the last term on the right in

Eq. (^-2) is discarded.

Typical results for obtained from one record are shown in Fir.

11, where the values of the mean and standard deviation oí’ k 1 and k " ' r r

rmãmmm fmmmÊiwmmwiMMMBÈÊim » h« irtWIif-- «ISiMIllWilPHIISW

TM 1*9 -59-

over one revolution of the record at each bias load are displayed as s

function of the bias load. In Fir. Ita. both scales are logarithmic, but

in 14b the abcissa scale is lorarithmic and the ordinate scale Is linear.

A diagram shoving the area of the record surface traversed in obtaining

these data is given in Fig. 15« The annulus has an inner radius slightly

larger than J" and a radial width of 0.17". The approximate location of

the 14 datum points used to obtain the average values of Kr around the

record are also marked on the diagram. Additional data from two other

records, taken under the same conditions as the data for Fig. 14, are

plotted on logarithmic scales in Fig. 16. These curves are similar in

all major respects to the curves shown in Fig. 14a. Thus ve can conclude

that the curves in Fig. 14 are representative of our experimental results,

and the detailed discussion of the results can he focused on these-curves

and on the record from which they were obtained.

Returning to Fig. 14, therefore, the most interesting features of

the curve of k ' are the two straight-line portions, shown in I4af The r slopes of the two sections are 1.50 and 0.25 for lighter and heavier bias

loads, respectively. The possible significance of these slopes, as well

as the significance of the knee of the curve between the two portions at

bias loads near I50 mg, will be discussed below.

For bias loads above 1 0a, the value of kr' increases at a slower

rate than it does for smaller bias loads, and gradually tends to level

off as the bias load increases. This variation is accasQwnled by a

gradual rise in the value of krN. If the record is examined under a

medium power ( 40X) microscope after a set of measurements has been made,

a visible track from the indenter is quite evident when the bias loads

TM -60-

are 1.6 on and larger, but no track is generally visible when the bias

loads are below 1.0 gm. It is thus apparent that at some indenter load

in tiie range from 1.0 to 1.6 gm, plastic flow begins on the surface of

the material. This flow is characterized by a gradual leveling off of

the value of k ' and an increasing value of k " r r ”

The knee in the curve of kr', at a bias load of IfO mg, can possibly

be explained in a similar way. Hunt [9] , Max [io], and Barlow [ll] each

point out that the maximum shear stress in the material under a spherical

1 na en ter occurs at a point slightly below the surface of the material.

It is at this coint, therefore, and not on the surface, chat plastic flow

will first occur. Hunt further states, as was pointed out in Chapter II,

that on tile assumption of elastic theory, this subsurface yielding will

begin when the indenter is loaded by as little as 10 mg. We are not

dealing with an elastic material, however, so It is quite possible that

due to strain-rate effects, subsurface yielding might begin under a higher

máenter load. The knee at I50 mg is one feature of the k 1 curve which r

could be caused by this onset of sub»!;rface yielding, since k • increases r

much less rapidly with above this load than it does below this load.

Axi examination of kr" at this bias load does not at present reveal much

additional infonnation, but more will be said of this later.

The few negative values of k^ at bias loads below 30 mg are inter¬

esting but probably not significant, since k^ is obtained as the differ¬

ence between two large quantities, and for small bias loads, then, quan¬

tities are of almost equal magnitude. Thus any fluctuations in the data

flue to noise in the recorded signal are considerably magnified and the

resulting accuracy of kr' is correspondingly decreased. Considering the

Pig. 14. Mean and standard deviation of the real and imaginary parts

of the effective stiffness of a vinyl copolymer phonograph record, as

a function of indenter bias load. Comparison with the Hertz formula

is also indicated. Data for this and all succeeding graphs obtained

with a 1-mil indenter vibrating at 500 c/s.

7777772

FIG. 15 DIAGRAM OF AN RCA 7" PHONOGRAPH RECORD,

SHOWING AREA TRAVERSED TO OBTAIN THE DATA FOR FIGURE 14.

ZZZZZZE

Fig. 16. Mean and standard deviation of the real and imaginary parts

of the effective stiffness of two other vinyl copolymer phonograph

records, as a function of indenter bias load.

E o>

a 2 Q

a

y) < CD

(£ UJ (- z IU o z

Fip. 17. RcpI and imaginary parts oí' the efiective stiffness of a

vinyl copolymer phonograph record, for different indenter bias loads,

as a function of position around the record.

EF

FE

CT

IVE

ST

IFF

NE

SS

Kr

- kr'+

j kr

* D

YN

ES

/cm

) ■ * 10*

—

SECTOR A

k r 0--0--0

k r"A—A--A

SECTOR B

kr" A—A—A

__ £>- no<r 0-0-^

--e""

1 _ U-

»-i •

//

//- //

- 1 / /u 1 / _

~TV~

/ !

\

N

V

M

_

A !

^ !

/ /

/ /

'-H- t

2

t /

/* /

-_ —i—i—i 4

* A

. A

i

1 Jr

1

>

a/' /

--4 1 l 1

ï-

Q

INDENTER BIAS LOAD Mb

Fig. 18. Mean real and mean imaginary parts of the effective stiff¬

ness of a vinyl copolymer phonograph record over two different sectors

of the record, as a function of indenter bias load.

SECTORS SECTORS

BA BA

FIO. 19 SAMPLES OF THE RECORDED DATA FROM WHICH THE CURVES IN FIOS. 14, 17 AND IS WERE OBTAINED.

-r'‘'u

% i

SimI8hVie

Ift*

1?

•t

ii

BLANK PAGE

TM h') -61-

data as a whole, it is extremely difficult to set limits of accuracy

which are valid for the entire range of experimental results, since multi¬

plicative factors, additive factors, and the arguments of trigonometric

functions all contribute differing amounts of error over different ranges.

The beet estimates which can be made are the following: For the absolute

accuracy of the measured results, the error in the value of k ' varies

from t 2.0 x 10^ at small bias loads to approximately 1 3. 5 x 10^

large bias loads, while the error in the value of is approximately

10^+0,3x10^. On the other hand, the relative accuracy of the measured

results, which is the accuracy in the measured variation of K^, from point

to point around the record and from one bias load to another, is much

better. Thus the relative error in the value of k ' varies from + 2.0x10° r -

at small bias loads to i 0.6x 10^ at large bias loads, which amounts to

an error as small as 2. and the relative error in the value of k " can r

be represented as 3$ i 0.3 xi0° over the whole range of bias loads.

One striking feature in the curves of Fig. 14 has not yet been men¬

tioned, namely the maximum in the value of kr" near a bias load of 300 mg.

The cause of this maximum is still not entirely clear. It is interesting

to note, however, that this maximum occurs at a bias load above which a

marked increase appears in the spread of values of k^' around the record.

It is quite possible that there is some correlation between these two

phenomena.

In order to pursue this problem further, we- must first point out

that most of the spread in the values of K is not caused by noise fluc- r

tuations in the recorded data, but is due to significant fluctuations in

the values of from point to point around the record. Figure I7 makes

TM b9 -62-

thie facii abundantly clear. In this figure, we have plotted the indi¬

vidual values of k ' and k " as a function of position around the record r r

for nine different bias loads. The pccition numbers correspond to the

numbers on the diagn in Fig. 15.. an(i are the lh datum points used in

obtaining the mean values plotted in Fig. lh. It can be seen that not

only do the values of k ' and k " vary considerably from point to point r r

for any one bias load, but also the maxima in k^' occur at different

values for different positions. For example, at position 9, the maximum

value of kr' is 17* 10^ dynes/cm, while at positions 1 and b, the maximum

value of kr' is approximately dynes/cm, all occurring at a bias

load of 2.U8 gm. It can also be seen that no maximum of kr" occurs over

a certain portion of the record for bias loads near JOO mg.

In order to display the extreme differences which occur in the values

of k ' and k ", we have plotted separately their respective mean values r r 7

over positions 1+-6, labeled Sector A, and positions labeled Sector

B. These curves, plotted in a form similar to Fig. Ih, are shown in

Fig. 18. Looking at the dashed curves, representing Sector A, we note

that the maximum in k " at the JC)0 mg bias load is accompanied by a sudden r

increase in the values of kr', On the other hand, the solid curves, rep¬

resenting Sector B, exhibit no such maxinum in k^", and no corresponding

increase in the values of kr'.

The discontinuity in the curve of kr' for Sector B is not signifi¬

cant for our present purposes, although it does represent a significant

variation in the value of kr'. It reflects an extreme case of the varia¬

tion of K along a radial line, and its presence is due to the manner in r

which the data were obtained: The measurement run was begun with a bias

TM 49 -63-

load of 0.277 @n, and this was decreased to zero. We then made an

adjustment in the counterweight, began with a bias load of l.Ot gn, and

decreased this to 0. 260 gm. Thus the overlapping points on the two seg¬

ments of the curve represent portions of the record separated by almost

the full width of the annulus covered in the measurements. Each one of

the measurement runs followed this same procedure, and the same discon¬

tinuity is present in Figs. 14 and 16, although it tends to be smoothed

out when the data taken over one revolution are averaged. For greater

radial differences, however, or for different sides of the record, even

the average value of can show significant variations. The four crosses

on the curves in Fig. 16a represent mean values taken over one revolution

at inner and outer radii on both sides of that particular record, for a

bias load of 1.Oh gm. Thus it can be seen that the value of Kr is depen¬

dent not only on the sector of the record, but also on the radial dis¬

tance from the center of the record.

Returning to Fig. l8, it is apparent that the phenomenon which

causes the maximum in the value of kr" for Sector A also gives rise to

the large spread in the values of k ' between the two sectors at bias r

loads above 3?0 mg. This spread is reflected in Fig. Ih as an increase

in the standard deviation of k^', which was referred to previously. The

explanation of this maximum in kr" and the rapid rise in kr' is still

open to conjecture, although it is clear that it is dependent upon the

record material and not upon the measuring apparatus, since the effect is

definitely localized over certain regions of the record. Samples of the

recorded data fron which Figs. lU, I7, and l8 were obtained are shown in

Fig. 19, and these samples show this localization quite clearly. In each

pair of curves, the upper curve represents the magnitude and the lower

curve the relative phase angle of the seismic pickup output voltage. The

locations of the data points are also indicated.

One possible explanation for the effects described above can be

baaed on the rather exhaustive study of the deterioration of sound record¬

ings during long-term storage carried out by Pickett and Lemcoe [29] .

During this study, they found that microscopic residual stresses are left

in vinyl phonograph records by the molding process. The relaxation of

these stresses is both time and temperature dependent, and results in a

warping of these records due to differential shrinkage of different sec¬

tions or sides of the records.

The presence of these residual stresses in the record samples under

current investigation is exhibited by the photograph shown in Fig. 20,

The five records shown in this figure were subjected to various amounts

of heat, and the results are rather astonishing. Record A was supported

at its center by a ring while it was in a water bath at 65°C for one

minute. Record C was similarly supported, but was in a water bath at

46-51¾ for about an hour. Seme of the defonnation of both records is

undoubtedly due to a general sagging of the record rim, although the

various small warps cannot be attributed to that cause. Record B was cut

into quarters and the sections were hung vertically for 10 seconds in a

water bath at approximately 78¾. Gravitational forces thus had little

influence on the defonnations which occurred Record D, which was placed

on a flat plate in an oven at 60°C for about 30 minutes, was also not

significantly influenced by gravitational forces. In both B and D,

therefore, the deformations must be entirely due to the relaxation oí'

29. A. G. Pickett and M. M. Lemcoe, Preservation and Storage of Sound

Recordings . (Library of Congress, Washington, 1959).

TM 1+9 -69-

residual stresses. It is interesting to note that although the radial

edges of the four quarters of record B were not constrained while the

sections were in the heated bath, the deformations along these edges,

where the record was cut, line up extremely well. An extreme case of

defonnation is shown by record E. In this case, a ring again supported

the center of the record, and it was placed in an oven at 140¾ for about

10 seconds. The rim of the record sagged considerably, but left the

seven flutes in the original plane of the record.

For purposes of comparison, the five records in Fig. 20 are each

oriented by means of a small mark molded into the surface of each disk,

and are positioned to correspond to the diagram in Fig. 15, which for

convenience has also been included in Fig. 20. Records A, B, and C show

the same face as the diagram, while records D and E show the reverse

face. A rather sharp warp in each of the records A, B, and C falls into

Sector A, although enough other warps are also present to make any pre¬

cise correlation with the measured values of somewhat difficult.

Since we see that residual stresses are present in our record

samples, it seems reasonable to expect that these residual stresses

would, under certain conditions, react with the indenter and thus produce

some effect on the measured value of Kr, and that the extreme effect on

fCr would be localized over certain portions of the record. One mechanism,

other than the effect of general heating revealed in Fig. 20, which could

release these stresses is plastic deformation. Thus a different value of

Kr might be expected to arise before and. after plastic deformation, and a

transition region would lie in between. The gradually rising values of

kr" shown in Fig. l8 at a bias load of approximately I50 mg, like the

knee in the k^/ curve at the same bias load, referred to earlier, suggest

TM 1+9 -66-

that subsurface yielding probably begins at a bias load near I50 mg.

Below 150 mg, therefore, the measured values of include all the

effects of the residual stresses, whatever they might be. Above some

other bias load, which according to Fig. l8 might be 1+00 mg, the stresses

produced as the material slides under the indenter increase fast enough

and are large enough to release and swamp out the effect of the residual

stress. Why the value of k ' should be higher for one sector of the r

record than for another after the residual stresses have been so swamped

out is not yet clear. The application of stress by the indenter is

almost impulsive in all cases. Thus, in the region referred to above as

the transition region, we might expect that the incompleteness of plastic

yielding would allow a time relaxation effect to produce some anomaly

which would reveal itself in the imaginary component, i.e., in kr". The

associated variations in k ' and k " should provide a clue to the under- r r

standing of this anomaly, but at present we do not know how to interpret

this clue. Further investigation is obviously needed to reveal the true

cause of these anomalies.

We had originally Intended to express the data obtained for the

effective stiffness K of the record material in terms of a complex elas- r

tic modulus E. The only formula currently available to relate these two

quantities, ^owever, is obtained from the Hertz theory of the contact

between elastic solids [8]. This theory assumes that the deformations

are elastic and the load static, two conditions which are not satisfied

by the experimental conditions of the present investigation. The theory

of linear iscoelastic behavior, which might better describe these exper¬

imental conditions, has not yet advanced to the state where this particu¬

lar problem of a hard, spherical indenter sliding on a smooth, plane

TM 1+9 -67-

surface has been solved. Lee and Radok [j50] discuss the Hertz contact

problem on the assumption of ein incompressible, viscoelastic material

(Poisson's ratio v « 1/2). However, they still treat the problem as a

static one and do not allow for the complication of moving boundary con¬

ditions. An approach which could be taken toward solving this problem,

and the difficulties which might be encountered, are indicated in two

additional papers by Lee ¡2, Jl].

Some indication of the validity of the Hertz theory in describing

our experimental results can still be obtained, however. For the special

case in which one of the solids has a plane surface and the ether solid

is perfectly rigid, the Hertz formula, as given by Timoshenko [52, p. J7|,

reduces to

a? - 9F2/l6REy2 , (5-1)

where a represents the depth of penetration of the indenter into the

plane surface of the material, F represents the force load on the inden¬

ter, R represents the indenter radius, Ey = E/(l-y^) is a "constrained"

Young's modulus, and y is Poisson's ratio. If Kr is represented by the

ratio of the variational force to the variational displacement, and if the

variational force is small compared with the bias force, we.then obtain

Kr - dF/da = (6gR)1/5 Ev2/5 . ( 5.2)

The force has now been expressed in mass units and is designated

JO. E. H. Lee and J. R, M. Radok, Trans. Am. Soc. Mech. Engre. -J. Appl. Mech. 27, Series E, I+38-I+U+ (I960).

31. E. H. Lee, Quart. Appl. Math. 1¿, I83-I9O (I955).

32. S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill

Book Co., Inc., New York, 1951), 2nd ed.

TM U9 -68-

If E-y is a constant of the material and not a function of a

graph of Kr as a function of Mg, with logarithmic acales, should give

rise to a straight line, the slope oí' which should have a value equal to

the exponent relating the two quantities, i,e. , a value of 0, 7i3. Refer¬

ring to Fig. 14a, we noted that the slopes of the two straight-line

portions of the curve of k^' were 1.30 and 0.23. The values of k^' are

small enough, relative to the values of k^' over this range of bias load,

that a graph of the magnitude of Kr would not have a slope significantly

different from the slope of the kr' curve. To aid comparison we have

included in this figure a Hertz-formula straight line with slope 0.33,

based on the values E = 2.6xl0xu dynes/cm and ‘V » 0.1+4 which were

obtained by Pickett and Lamcoe [29]. The only conclusion we can draw

from this comparison is that either E-^ is not constant but is a function

of the bias load Mg, or the Hertz relation is not valid for describing

our expreimental results. An investigation of Fig. 16 leads one to the

same conclusion, since its curves are not significantly different from

the curves in Fig. 14a.

If we assume that *> Ey(F), the expression for K , is altered to

the form

K r

(6RE 2F) ( 5-3)

If, furthermore, E-y is a function of some power of F, which would be the

most reasonable assumption of functional dependence to mek; for a limited

range of F, we can set E.y = E^F" and Eq, (5-3) becomes

K = ( 6RE-y* ¿:)1/5 r( 1+ 2x)/ 3

1- X (5-M

TM 1+9 -69-

Frcan this equation, it can be seen that no power of F can provide a

log K - log F elope greater than one for positive values of K . On the r r

other hand, a slope of 0. 25 is given by a value of x equal to -0,12.

This ’.'alue implies that decreases with increasing load, which is a

reasonable effect to expect from subsurface plastic yielding. We find,

therefore, that the Hertz relation does not adequately describe our

experimental results for small bias loads, but does describe them sur¬

prisingly well for medium bias loads if we assume a strain-dependent

elastic modulus.

One possible effect on the experimental results, which was indi¬

cated in Chapter I but has not been mentioned since that time, is the

effect of a skin on the surface of the record. It is not unlikely that the

molding process produces such a skin, and one would expect its properties

to differ from the properties of the material in the interior of the

record. Under deformation, the effect of the skin would be superimposed

on the effects which would otherwise be observed, and thus could cause

some apparently anomalous results. Whether this effect could explain the

results observed above, however, is uncertain. In any case, a more

nearly complete theoretical understanding of the actual experimental con¬

ditions involved is necessary before a better description of the results

can be given.

Considerable care was taken during the measurements to ensure that

the indenter was always working on virgin material. The effect on K of r

the indenter reworking previously stressed material is shown in Fig. 21,

where the values of k ' and k " at several positions on the record are

plotted versus time as the turntable comes to a stop. The actual time at

which the record stopped moving, as near as could be determined, is indi-

TM 1+9 -70-

ca ted by t «0. The Indenter was vibrating at ^00 c/s, and had a bias

load of 1.01+ pa. The increase in the values of k^' and k^" for the sta¬

tionary record can be seen to be as much as 35$ over the corresponding

values obtained from fresh material. An intermediate stage cetween these

two extremes would be an interesting area for further investigation; that

Is, the case where a particular volume of material undergoes several

stress cycles as it passes under the indenter.

It should be emphasized that the results and conclusions presented

above were obtained for a 1-mil radius indenter vibrating at 5OO c/s

while indenting a vinyl chloride-acetate copolymer phonograph record

rotating at 1 rpm. The measurements were all made at room temperature

(76°F). If any of these quantities is altered, it is quite likely that

the experimental results will be affected. The most immediate area for

further research, therefore, lies in the direction of making changes in

these quantities and comparing the results with those obtained above. We

must not forget, however, that since the material properties vary from

point to point on the record, and since no particular volume is ever

worked twice, we are never roally certain what the measured result from a

particular volume would have been under different measurement conditions.

Fortunately, the material properties do not appear to vary too rapidly

from point to point, so that as long as the two comparison measurements

are made relatively close to one another, no difficulty should be encoun¬

tered. Otherwise, comparison results can only be significant if they

differ by more than the above-mentioned point-to-point variations.

Averaging over an area of the record would, of course, decrease the rango

of uncertainty, but would no longer provide a point-to-point measure of

the comparison.

EF

FE

CT

IVE

ST

IFF

NE

SS

Kr

= kr

'♦ j

kr"

, D

YN

ES

/cm

.

Fip. 21. Real and imaginary parts of the effective stiffness of a vinyl

copolymer phonograph record at five different locations, plotted as a

function of time immediately after the turntable stopped rotating.

TM 1+9 -71-

As we pointed out at the beginning of this chapter, the above

results were intended merely to illustrate the kind of information which

this type of measurement can provide. The questions raised in the first

chapter of this report for the most part remain unanswered; in fact,

additional questions have been raised by the foregoing results. For

example, present indications seem to point to the absence of a size

effect of the kind proposed by Hunt, although this conclusion can not be

confirmed until measurements are performed using an indenter with a dif¬

ferent radius. On the other hand, there does appear to be a minor

enhancement of the hardness of the material over parts of the record

tested, and this enhancement may be due to the residual stresses left in

the material by the molding process.

In conclusion, even though we have raised more questions than we

have answered, the way is now open to approach these questions systema¬

tically through use of the measuring device which has been developed.

Furthermore, if the actual stress system produced by this device can be

solved theoretically, the device can have widespread application as an

essentially nondestructive method for obtaining the elastic moduli of the

materials being tested.

ACKNOWLEDGMENTS

I wish to thank Professor F. V. Hunt for suggesting this work to me

and for giving me much encouragement and many helpful suggestions through¬

out the course of the investigation. The assistance of the other members

of the Staff of the Acoustics Research Laboratory has also been deeply

appreciated.

TM k9

Append! j: A

REDUCTION OF THE EQUATION OF MOTION

FOR A LOADED, FREE-FREE VIBRATING BAR

We start with the following Eqs. (2-6), which were obtained on

page 20,

u(x) o (cosh kx + cos kx) + ( ainh kx + sin kx)

+ -^P (cosh \x - cos \x) 0£x5i/2, 2k ’

(2-6)

u(x) = (cosh \x + cos kx) + (sinh kx + sin \x)

+ ^ (cosh kx - cos kx) - K'b[sinh \(x- |) - sin k(x- |)]

l/2$x$2 .

To the second of the above equations, which is valid from 2/2$xSl, we

apply the two boundary conditions at x« i. These conditions are the same

as the boundary conditions at x* 0, which are u"»D and u'" »0. After

performing the indicated operations, we then obtain from (2-6)

2

D . (coah Kl . coe kl) + (sinh Ki . sln Kl)

+ § (cosh kl + cos kt) - K'b\2(sinh™ + sin™) ,

5 2 (A_l) o = (8inh kt + Bin kt) + n'iojA (C08h kl _ C0B ki)

+ ^ (sinh ki - sin kl) - K'b\5( coshy + cosy) .

Using the theory of determinants to solve for u(0) and u'(0), we obtain

-73-

Au(o) (cosh \l - COB kt)

Au'iO)

“ V[f"l^c08h *** + 008 + K,t^2(sinh^r + sin^f)

+ sinh kt - sin kt) - K'b\5(coshY + cosy-) ( sinh kt - sin kt),

(A-2)

2 , 5r

y(sinh \l - sin \l) - K'b\^( coshy + cosy)

kt kt, Dyicosh kt + cos kt) + K'b\ (sinhy + ainy)

(cosh kt - cos kt)

(sinh kl+ sin kt).

By some manipulation, we reduce these expressions to

p 2ûu(0)/\ ■ D(coeh kt - cos kt - sinh kt sin kt)

+ 2K'b*.2 (coehy siny - sinhy cosy)(coshy + cosy) ,

(A-3)

-2Au'(0)/X.^ » d[(1-C08 \i)sinh kt + (1-cosh \Z)sin

+ ac'b\2(coshy siny - sinhy cosy)(sinhy - siny) .

The symbol A represents the deteminant of the coefficients of

(A-l), and is given by the expression

A - I \\l- cosh kt cos kt) . (A-h)

xhe factor in parentheses can be recogaized as the characteristic

equation for an unloaded, free-free bar. The roots of this equation give

the resonance frequencies of the unloaded bar, and occur for values of

kt equal to U.73O, 7.853, 10.996, 1^.137, and (2n+l)*/2, n - 5, 6, 7, ... .

We consider now only the region t/2gx£t, and substitute Eqs. (A-3)

into the second equation of (2-6). After some rearrangement of terms,

we obtain

TM i*9 -75-

where

= (cosh Kl - coa Kl - ßinh Kt sin \/)(cosh + cos Kx)

- Ql-cos Kl) sinh Kt + (l-cosh \Osin kt\ (sinh Kx + sin Kx)

+ (l-cosh Kl cos A.I)(cosh Kx - cos Kx) . (A-^

C2 = (l-cosh Kl cos hf)Qinh K{ x- ) - sin

- (cosh“ sin— - sinhy- cob~) |( cosh“- + coe'y-)(cosh Kx + cos Kx)

- (sinh-- - sin^')(sinh Kx + sin \x)] .

anà Cg are now reduced by repeated application of various trigonometric

and hyperbolic Identities until a fairly concise form results, giving us

Au(x) b BKi ij)oA( x) + K,bX.1+^oB(x) , (A-6)

in which we have defined

2B(x)

<f>o * cosh^- sin'll— sinh^- cos^- ,

A(x) « sii'cosh \(x-|) - sinhTf- cos \(x-|) ,

sln^ sinh K( l - x) - sinlr— sin K(l - x) + cosy- cosh \( ! - x)

+ coshy cos \(/- x) + cosh \(x- |) + cos \(x-|) .

A further look at A, given by Eq. (A-k), reveals that it too can be

broken down into several factors, and when this is done it becomes

A = \\coshy siny - sinhy cosy) ( coshy sin™- + slnhy cosy ) ,

A = è ToTe

which is rewritten as

TM U9 -76-

It was stated that the roots of A give the resonance frequencies

of the unloaded, free-free bar. If it is recognized that the first root

of A occurs for \l = 0, the second, fourth, and all subsequent even

roots are ßiven b.y the roots of and the third, fifth, and all subse¬

quent oad roots are given by the roots of ¿ .. ihus ^ and |>^ determine

theevenand odd modes of vibration, respectively, and since the right-

ntiiiu side of Eq. (A-6) contains ^ as a common factor, it cancels out and

11 the odd modes disappear. ihis fact is not surprising when it is

remembered that the boundary conditions were given symmetrically; that

is, u"(0) * D and u"(l) « D. Hence the antisymmetric or odd modes are

not excited, and can occur only as a result of nonlinearities in the

system, which have not been taken into account here.

Equation (A-b) thus reduces to

u(x) . K'. (A-7) \ 'e ye

We can now set x * i/2 and solve for b = u(i/2), obtaining

_D_ A( i/2) . 2 $ - K ' B ( i /¿T *

(A-e)

where A(i/2) » sin(\i/2) - sinh(\i/2) and B(i/2) = cosh(\i/2)cos(\i/2) f 1.

If we define

C - (|>e - K'B( i/2) (A-9)

and insert this and (A-8) into (A-7), we obtain the final solution

u(x) . ^Lx) - K' L Te -I

TM J+9 -77-

which can be further reduced to

u(-) = d[a(x) -K'B'(x)]A2C . (A-10)

In Eq. (A 10) we have defined B'(x) by the expression

2B'(x) = sin^- sinh \(x-'|) + sinli^f sin \(x -^) + cos^f cosh \(x-^)

ht t - cosh— COS \( X - 7>) + cosh \( / - x) - cos \( / - x) .

It must be remembered that the above solution is valid only for

//2¾ xí: i. The other half of the solution, valid over the region Osx 4//2,

is obtained from Eq. (A-5) by simply eliminating the term

(1 - cosh \/ cos \t) [sinh \(x -1) - sin \( x - |)J

from the coefficient of 2K'b\2. Without too much difficulty, the result

reduces to the equation given above with x replaced by / - x, which could

have been anticipated in view of the symmetry of the problem.

C, as given in Eq. (A-9), turns out to be the characteristic

equation for the loaded bar, and several interesting results can be

observed. If K' = 0, then C » which is the symmetric part of the

characteristic equation for the free-free unloaded bar. However, the

bar does not need to be unloaded for K1 to vanish. K' was defined as

K' * (a+j£ +^)/2k3, which for a lossless mass-spring load becomes

K' 1 -u)gM+ k

2\5 ESK2

2 This vanishes when w M » k, that is, when the mass-spring load has its

own resonance. If this resonance coincides with a resonance of the

TM 1+9 -78-

unloaded bar, the resonance will remain undisturbed by the load; hence

in theory the bar can be loaded (or supported) without disturbing one

of its natural modes, and if the load is a multiresonant system, sereral

natural modes can be left undisturbed. Naturally, any losses introduced

will tend to invalidate this conclusion.

On the other hand, if ve let K' approach infinity, Eq. (A-9)

reduces to

u(x) n X2

The characteristic equation is n;w simply B(l/2) = cosh(X.l/2)cos(\i/2) + l,

which is precisely the characteristic equation of a ciamped-free bar of

length i/2. The reason for this is that a clamped condition can be con¬

sidered equivalent to an infinite mass or spring load. The resonances

of the free-free loaded bar are thus determined by both the resonances

of the unloaded bar and the resonances of the clamped-free half bar,

modified by the "softness" of the clamping.

TM h9

Appendix B

AMLYSIS OF THE CLAMPED-LOADED VIBRATING BAR

This analysis proceeds much more simply than the corresponding

analysis of the free-free loaded bar, since the constraints now occur

only at the ends of the bar and thus can be handled as boundary con¬

ditions. The equation of motion is the same as that given in Eq. (2-1),

which is

(B-l)

If, as before, y(x,t) * u(x)e^u,t, we obtain

d^u/dx1* - Kku = 0 , (B-2)

4 2,2 where again \ = w p/Ex . The general solution of (B-2) is given by

u(x) *» A cosh \x + B sinh \x + C cos + D sin \x , (B-3)

where A, B, C, and D are constants whose values are to be detemined from

the boundary conditions.

The boundary conditions at the clamped end x* 0 are u= u' o 0, while

at the free end x» i there is no bending moment (u" = 0), but a shearing

force is present due to the constraints produced by the load. This

shearing force is given by

dx^

and must equal the sum of all the external forces produced by the con¬

straints. Hence, in general,

TM 49 -80-

where M, r, and k represent maes, velocity-dependent resistance, and

spring loads, respectively, and f is a sinusoidal driving force. From

this we obtain the final boundary condition, which is

u,m . u --½. (a+f+J)u - f/ESK.2.

EStc ESk

Inserting the boundary conditions at the clamped end, we find that

0 = A + C , 0 = B + D ,

and consequently

u(x) = A(cosh kx - cos kx) + B(sinh kx - sin kx) . (B-4)

Applying the boundary conditions at the loaded end x=i, we have

u"(¿) - \2A(cosh kl + cos kt) + k^isinh kt + sin \l) - 0 ,

u'»'(i) - \5A(sinh kt - sin kt) + A(cosh kt + cos kt)

» (a+$+r)[A(cosh " 008 *•*) + B(sinh kt - sin \i)] - f/ESK. ,

which immediately reduce to

A(cosh kt + cos kt) + B(sinh kt + sin kt) = 0

A [(sinh kt - sin kt) - 2K'(cosh kl - cos kt)]

+ B[(cosh kt + cos kt) - acisirh kt - sin kl)] = - f .

As before, K' - (crffî+y)and we have defined

f Uf AH

Solving for A and B, we obtain

A - f'(sinh kt + sin kl)/à , (B-5)

B = -f'icosh kt + cos kt) /ù ,

TM 1+9 -8l-

where A is the coefficient determinant given by the expression

1 A = (cosh \t cos + 1) + 2i'(coBh kt sin \i - sinh \f cos kt) . 2 (B-6)

Substitution of (B-5) into (B-1+) then yields

u(x) = f'Qsinh kt + sin \/)(cosh kx - cos kx)

- (cosh kt + cos \l)(sinh kx - sin \x)]/a , (B-7)

which gives for the motion at the end x = /

u(/) = 2f'(co8h kt sin kt - sinh kt cos kt) /& . (B-8)

The resonances are given by the values of kt that make A vanish.

The first term in the expression for A can be recognized as the charac¬

teristic equation for a clamped-free bar, while the second term shows

the perturbation of the zeros of the unloaded bar by the load K'.

Equation (B-8) can be put in very interesting form if it is recalled

that

ir. K M anA r< Klf K ’ "sT T ^ " "27 ’

where K is the effective stiffness of the applied load. A slight

rearrangement of terms gives the result,

2 f ui M^(cosh kt cos kt + 1)

u(t) “ ^ + \|(cosh kt sin kt - sinh kt cos kt) * ^ ^

Equation (B-9) represents the stiffness of the loaded bar at the end

x=i, and since the first term is the stiffness of the applied load, the

second term gives the effective stiffness of the bar itself at the loaded

end. For kt small, i.e., at very low frequencies, this term reduces to

TM k'î -82-

- 5ESx2//5 , (B-IG)

which 1b the effective stiffness of the bar under static deflection.

TM 49

Appendix C

THE DRIVING-POINT IMPEDANCE OF A PIVOTED, LOADED VIBRATING BAR

The eagle support bar represents the third type of bar > j be con¬

sidered. This bar is driven at the end x=0, is loaded at the end x=/,

and is constrained by a pivot located a distance ai from the driven end,

as shown in Fig. 22. The axis of the pivot is normal to the length of

the bar and to the plane of vibration. The technique for solving this

problem is to consider separately the two segments of the bar on either

side of the pivot, and then obtain the final solution by requiring con¬

tinuity of the boundary conditions at the pivot.

I -3»

Fig. 22. A pivoted, loaded vibrating bar.

The general solution of the equation of motion of a laterally

vibrating bar was given by Eq. (B-3) and is

u(x) = A cosh \x + B sinh \x + C cos \x + D sin \x , (C-l)

where we have assumed sinusoidal variation of the displacement u with

time. A, B, C, and D are constants whose values are to be detemined

-83-

h 2 2 i'rom the boundary conditione, and \ => w p/Ek . The remaining symbole

are defined on pp. 14 ff. The boundary conditions at the loaded end x=i

can be taken directxy from Appendix B, and are u"(i) =0 and

u'‘'(jf) » (-w2M + >r+k) u(l)/ESx2 = (a+ß + u(/) * 2k5K'u(í) ,

where M, r, and k represent the mass, resistance, and stiffness compo¬

nents, respectively, of the load attached at that end. The boundary

conditions at the driven end x»0 can also be taken from Appendix B

with one slight modification; since the di'ving force in that situation

was applied at the opposite end of the bar x= i, the algebraic sign of

the force must be reversed. The reason for this reversal can be seen

most readily by considering Fig. 22 and noting that a positive deflec¬

tion of the bar under a positive force produces a positive curvature

(u">0) in the vicinity of the driven end, except where the curvature

must vanish at the end itself. Thus when a force is applied at the end

x»0, the curvature is increasing as x increases and u','(0)>0, but when

a force is applied at the end x« i, the curvature is decreasing as x

increases and u'''(/)< 0. For this reason, the desired boundary condi-

tlons at x*0 become u"(0) « 0 and u'1'(0) * +f/ESK” ■ where f is

the magnitude of the driving force and f = f/ESxk . Finally, the

boundary conditions at the pivot are u(ai) » 0, i.e., the displacement i

zero, and the slope and curvature are continuous as x Increases from ai

to ai*. The continuity of slope states that the bar cannot have an infl

nitely sharp bend at the pivot, and the continuity of curvature states

that the pivot cannot apply a bending moment. Gathering all these

boundary conditions together and designating the 0-to-ai and ai-to-i

TM 1+9 -85-

segments 01' the bar by the subscripts 1 and 2, respectively, we have

u1"(0) = 0 u2"(/) = 0

u1"'(0) = \*r u2’"(i) = 2\5K'u2(/)

u^ia/) = 0 = Up(al)

Uj^'iai) = Ug' (ai)

u]L"(ai) = Ug” (a/) ,

(C-2)

which are applied to Eq. (C-l) with the proper subscripts added. When

this is done, we are left with the following eight independent equations

in the eight unknowns A^, 1^, C^, D^, Ag, Bg, Cg, and Dgi

0 = A1+ 0 — C x t 0

f = 0 + B1 + 0 - D:

u ^ A^^cosh a\i + B^sinh a\/ + C^cos a\/ + D^sin aki

0 - A^sinh a\/ + B^cosh aki - C^sin aki + D^cos aki

- AgSinh aki - Bgcosh aki + CgSin aki - DgCos aki

0 = AjCOsh aki i B^inh aki - C^cos aki - D^in aki

- AgCosh aki - BgSinh aki + CgCos aki + DgSin aki

0 = AgCosh aki + BgSinh a\X + CgCos a\/ + DgSin a\l

0 » AgCosh ki + BgSinh ki - CgCos ki - DgSin ki

0 = Ag( 3i’ coshX.X - einh ki) + BgiSK'sinh ki - cosh ki)

+ Cg(2K'co8 ki - sin ki) + DgiSK'sin ki + cos ki)

(C-5)

In the last equation of (C-5), the expression for Ug(/) obtained from

Eq. (C-l) has been substituted into the equation for the boundary

condition.

The complete solution involves solving Eqs. (C-5) for the eight

unknowns. However, we are interested only in a particular solution,

TM Ut -86-

namely the ratio oi' the driving Torce to the displacement at the driven

end. This means we need find only the two quantities appearing in the

equation

u^(0) = + . (C-U)

Because of the nature of the first two equations of (C-3), the three

required 8x8 determinants quickly reduce to five 6x6 determinants, and

we have

D , „ + D, , - D,™, - 1212 ;12lU ' ^1223 ' 123b

2 D 1213

(C-5)

where D^212 represents the coefficient determinant of (C-3) with the

first and second rows and first and second columns deleted, Dj_2lU the

first and second rows and first and fourth columns deleted, and so forth.

The evaluation of these five determinants is straightforward, and the

solution obtained can be put in the form

at'A -B (c_6)

ÿôT uk'c-d ’

where

A = 2?^ - p2q2 B « p^g + P2q1

c • ¡ 2q? + P5Í2 D = P2q2 -

and

p^ a 1 + cosh a\t cos akt

P2 = cosh akt sin akt — sinh akt cos akt

p^ ■ sinh akt sin akt

q^ = 1 + cosh(l-a)\i cos(l-a)\l

q2 = cosh( 1 - a)\/ sin( 1 -a)kt - sinh(l -a)kt cos(l -a)\i

q^ sinh(l-a)k/ sin( 1 -e )kt .

TM 1*9 -87-

Recalling from Appendix B that 2K' = KXl/co^M^ and f = f\i/w M^ , where K

is the effective stiffness of the attached load and Mb is the mass of the

bar, we can put Eq. (C-6) in the form

f KA -ooSfbA/

u(°) ” - D K sb

(C-7)

which defines the effective stiffness of the loaded bar as seen at the

driving end, and is thus designated K8t.

We would also like to find the change in due to a change in K,

which we shall call AK , . Thus, from Eq. (C-7) ven the

expression

AK (K + AK)A KA - oj^B/U

8b * 2(K +AKÍCkí/^Mh-D æCU/^-D (C-8)

If we place everything over a common denominator, we obtain

AK sb - AKAD + 2AKBC

( 2A KCki/u^Mh + E)E (C-9)

where E is the denominator of the right-hand term in Eq. (C-8). The

numerator of Eq. (C-9) can be reduced still further, and we finally obtain

a AK( sinh a\i t sin akl) 2 Talnht l-a)\i 4 sln(l-a)\l]2 (C-io)

Bb (RAKCkf/w^ + E) E

We must now particularize Eq. (C-10) to the physical situation

which prompted this investigation. Since the pivot on the eagle support

bar is 2/3 of the distance from the eagle to the counterweight, we have

a - 2/3. The load on the support bar consists solely of the counterweight

mass, and the change in this load is the change in the counterweight mass

TM U9 -88-

which is made in order to produce the eagle bias force. Since the eagle

2 is statically balanced when it is calibrated, K = -<*> M.cw, where Mcw

represents the counterweight mass necessary to achieve this balance, and

the K . term has been absorbed into the calibration constant K in sb a

2 Eq, (U-2). The change in load AK is represented by -w AM, but to

express this in terms of the bias force Mg we must remember that AM is

negative for positive Mg and is twice the value of Mg due to the l-to-2

static lever-arm ratio. Hence, we find that AK * aij^Mg. Finally,

AKeb, which is the effect of A K seen by the eagle, is given by ¿oj Mg,

where Ô has been defined so that its value is 1/2 at very low frequen¬

cies when the bar is rigid. Making the above substitutions in Eq. (C-10),

we find Immediately that

2(sinh^-4 sln^)2 (sinh^-i sln^ )2

(h Mg C U/Mb + E)E (C-ll)

where E - - 2 Mcw

Mi Ckl -D .

Equation (C-ll) is a very complicated function of frequency (the

frequency is contained in \), so this equation was put into the fonn

s

and programmed for an IBM 7^90 computer. Values for ^ and (5g were

then obtained for kl in the range 0.1 - 5.0, assuming values of 1.0k for

and 5.7O for Mcw. A few representative values of and §2 are

shown in Table C-I. Two conclusions can be drawn from these values. The

first is that for small \f, and hence for very low frequencies, 6 = 1/2;

this means that the bar is essentially rigid. This value of 6 is

TM 1+9 -89-

TABLE II

Representative values of and as e function of \l,

with the corresponding frequencies of the eagle support bar.

Kl

0.1

1.0

2.0

2.5

2. 765

5.0

1+.0

5.0

freq., c/s

6

65

262

409

500

589

1040

1640

-TO I90.

-8.91

. 210

.O525

.0156

.OO92

,,0027

.0108

-140 58O.

-11.15

2.056

2. 555

2. 674

2. 742

2. 872

2.945

independent of Mg, which can range in value only as high as Mcw/2, since

Mg is produced by the removal of mass from M^. The second conclusion

concerns the value of <5 at the operating frequency of 500 c/s, The

largest Mg obtained by removing counterweight mass is 2 gm; any larger

blal loads are obtained by adding mass at the stylus. Under this

restraint, the value of 6 ranges from 0. OO58 to O.O25. At a bias load

of 2 gm, therefore, iw^dg = 0. 59 x 10^, while the measured value of k^1

is 24x_0^, and the error in neglecting the dw^g term never exceeds

about 1.6# . This is less than the best estimated relative error in kr'

of 2.5#. For smaller bias loads, kr' decreases only slightly with Mg

until Mg reaches about I50 mg, while ¿to^Mg decreases linearly. Thus

both the correction term and the error become considerably smaller as

decreases. We can conclude that á •» 0, within experimental error, under

the conditions imposed on Mg. A new eagle support bar is being developed

to eliminate this error entirely.

ïM Uy

REFERENCES

1. J. D. Ferry, "Experimental Techniques for Rheological Measurements

on Viscoelastic Bodies," Chap. 11 in Rheology, Theory and Applica¬

tions, Vol. 2, Frederick R. Eirich, Editor (Academic Press Inc.,

New York, 19?8).

2. E. H. Lee, "Stress Analysis for Viscoelastic Bodies," Chap. 1 in

Viscoelasticity, J. T. Bergen, Editor (Academic Press, New York,

I960), p. h; see also Chap. 2 in this book.

3. M. J. DiToro, "Distortion in the Reproduction of Hill-and-Dale

Recording," J. Soc. Motion Picture Engrs. 2£, h93-5°9 (1937).

h. J. A. Pierce and F. V. Hunt, "Distortion in Sound Reproduction from

Phonograph Records," J. Soc. Motion Picture Engrs. ¿1, 157-186 (1938).

5. W. D. Lewis and F. V. Hunt, "A Theory of Tracing Distortion in Sound

Reproduction from Phonograph Records," J. Acoust. Soc. Am. 12,

3L8-365 (19M).

6. 0. Kornei, "On the Playback Loss in the Reproduction of Phonograph

Records," J. Soc. Motion Picture Engrs. ¿2, 569-590 (19^1).

7. F. G. Miller, Doctoral Dissertation, Harvard University, I95O.

Results summarized in F. V. Hunt, "Stylus-Groove Relations in the

Phonograph Playback Process," Acústica U, 33-35 ( 195*0.

8. H. Hertz, "Ueber die Berührung fester elastischer Kbrper," Journal

ilirdie reine und angewandte Mathematik §2, 156-171 (1882). For

English translation see H. Hertz, "On the Contact of Elastic Solids,"

Chap. 5 in Miscellaneous Papers, trans. D. E. Jones and G. A. Schott

(Macmillan and Co., Ltd., London, I896).

9. F. V. Hunt, "On Stylus Wear and Surface Noise in Phonograph Playback

Systems," J. Audio Eng. Soc. 2, 1-17 (1955). See also F. V. Hunt,

"Elastic-Plastic Instability Caused by the Size Effect and Its

Influence on Rubbing Wear," J. Appl. Phys. 26, 85O-856 (1955).

-90-

TM 49 -91-

10. A. M. Max, "Record Stylus Pressure," J, Audio Eng. Soc. 66-69

(1955).

11. D. A. Barlow, "Comments on the Paper, 'On Stylus Wear and Surface

Noise in Phonograph Playback Systems,'" J. Audio Eng. Soc. 4,

116-119 (1956).

12. D. A. Barlow, "Limiting Factors in Gramophone Reproduction,"

J. Audio Eng. Soc. IO9-II7 (1957).

IJ. D. A. Barlow, "The Limiting Tracking Weight of Gramophone Pickups

for Negligible Groove Damage," J. Audio Eng. Soc. 6, 216-219 (195Ô).

14. D. G. Flom and C. M. Huggins, "The Dc : rmation of Plastics with Hard,

Spherical Indentors," J. Audio Eng, Soc,, 2> i. t - ~ '1959).

15. J. Walton, "Gramophone Record Deformation," Wireless World

555-557 (1961).

16. J. Walton, "Versatile Stereophonic Pickup," Wireless World 67,

4C7-413 (1961).

17. R, M. Davies, "The Determination of Static and Dynamic Yield

Stresses Using a Steel Ball," Proc. Roy. Soc„ (London) A197. 416-452

(1949).

1Ô. Frederick V. Hunt, Electroacoustics (Harvard University Press,

Cambridge, Massachusetts, and John Wiley and Sons, Inc , New York,

19510.

19. J. P. Den Hartog, Mechanical Vibrations (McGraw-Hill Book Company,

Inc., New York, 1934).

20. Lord Rayleigh, The Theory of Sound (Dover Publications, New York,

1945), 2nd ed.

21. Horace Lamb, The Dynamical Theory of Sound (Dover Publications,

Inc., New York, I960), 2nd ed.

22. Harry F. Olson, Elements of Acoustical Engineering (D. Van Nostrand

Company, Inc., New York, 1947), 2nd ed.

23. Philip M. Morse, Vibration and Sound (McGraw-Hill Book Company,

Inc., New York, 1948), 2nd ed.

TM U9 -92-

21*. Warren P. Mason, Electromechanical Transducers and Wave Filters (D. Van Nostrand Compani', Inc., New York, 191*8), 2nd ed.

25. Kin N. Tong, Theory of Mechanical Vibration (John Wiley and Sons, Inc., Hew York, I960).

26. N. W. McLachlan, Theory of Vibrations (Dover Publications, Inc.,

New York, 1951)•

27. Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York 195: ).

28. H. A. Schenck, Internal Memorandum, Acoustics Research Laboratory, Harvard University, April,1962.

29. A. G. Pickett and M. M. Lemcoe, Preservation end Storage of Sound Recordings (Library of Congress, Washington, 1959).

50. E. H. Lee and J. R. M. Radok, "The Contact Problem for Viscoelastic Bodies," Trans. Am. Soc. Mech. Engrs. - J. Appl. Mech. Series E, U38-UUU (i960).

31. E. H. Lee, "Stress Analysis in Visco-Elastic Bodies," Quart. Appl.

Math. IJ, 183-190 (1955).

32. S. Timoshenko and J. N. Goodler, Theory of Elasticity (McGraw-Hill

Book Conqjany, Inc., Hew York, 1951)» ^

I

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