—.-— .—

 

 

 

EXPERIMEWAL ANALYSIS OF THE
VIBRATION 0F VARIABLE THICKNfl-S PLATEE

Thesis for the Degree of M. 5.
MICHEGAN STATE UMVERSITY
MyrI W. Thompson
1962

   

LIBRAT’V

Michigan State
University

 
     

This is to certify that the

thesis entitled

EXPERIMENTAL ANALYSIS OF THE
VIBRATION OF VARIABLE THICKNESS PLATES

presented by

Myrl W. Thompson

has been accepted towards fulfillment
of the requirements for

M.S. degree in Apolied Mechanics

O x" .-- -»-r-f’4f____--..
L1" 1% [1? Z’ 24;"

Major professor

Date July 26, 1962

 

 

._, ._.-q——'-_-“

 

 

 

 

EXPERIMENTAL ANALYSIS OF THE

VIBRATION OF VARIABLE THICKNESS PLATES

by

Myrl W. Thompson

AN ABSTRACT
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Applied Mechanics

1962 ‘

Approved by (I 42 Eta—Z2

ABSTRAC

Previous work in the Applied Mechanics Department at
Michigan State University has resulted in the theoretical
analysis of the various mode shapes of vibrating, variable
thickness plates. The objectives of this thesis are to
develop an experimental method of measuring the relative
amplitudes of vibration of plates and to make readings of
these amplitudes for variable thickness plates such that a
comparison can be made between the experimental and theos
retical results that are available.

The development of equipment proved to be a major part
of the work that was done in this investigation. The
capacitive transducer that was used as a measuring device
was a variable-gap, parallel plate capacitor having a shielded
probe as one of its plates and the vibrating plate itself
as its second parallel capacitor plate. Readings were
taken at each of 225 data points by the probe as it was
carried over the plate by a traverse mechanism that was
developed for the purpose.

The experimental results that were obtained indicated
that further refinements of the capacitive probe were
needed to insure accurate measurement of the more complicated
mode shapes. The fair to good agreement between the theo-
retical and eXperimental results for simpler mode shapes
and the reliability exhibited by the capacitive probe in

static tests warrant its further development.

EXPERIMENTAL ANALXSIS OF THE VIBRATION
-0F VARIABLE THICKNESS PLATES

BY

Myrl W. Thompson

A THESIS

Submitted to
Michigan State University _
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Applied Mechanics
1962

j, A kn w e ements
The author wishes to take this opportunity to *

acknowledge the guidance and assistance given by Dr.
Clement A. Tatro in this investigation. The method
of displacement measurement was, in fact, the result
of a suggestion made by Dr. Tatro. .The author slso
wishes to express gratitude to Dr. William A. Bradley
for his assistance in the work and to Hr. Carroll
Lenning of Oldsmobile Division of General Motors
Corporation for the financial assistance that was
made available.

Appreciation is also expressed to the author's
sister. Judy, for her long hours of typing and proof-

reading.

.1. 1

TABLE F CONTENTS

 

cmprsa PAGE
assrnscr
scxnostDssnsnrs . . . . . . . . . . . . . 11
LIST or rxsuass . . . . . . . . . . . . . iv
LIST OFTABLES . . . . . . . . . . . . . . v
LIST or snares . . . . . . . . . . . . . . vi
I INTRODUCTION . . . . . . . . . . . . . . . 1

II OBJECTIVES OF EXPERIMENTAL ANALXSIS . . . 2
Summary of Previous work . . . . . . 2
Experimental Analysis . . . . . . . . 12

III EXPERIMENTAL EQUIPMENT . . . . . . . . . . 15
Transducer Design and Application . . 15
Static Calibration . . . . . .,. . . 20

Dynamic Measurements . . . . . . . . 28
Sources of Error . . . . . . . . . . 32

IV. PRESENTATION OF RESULTS . . . . . . . . . S7
ModeShapes . . . . . . . . . . . . . as
Capacitive Probe Output Signal . . . 53

V CONCLUSIONS. . . . . . . . . . . ... . . . 57
'Summary . . . . . . . . . . . . . . . 57
Recommendations . . . . . . . . . . . 57

BIBLIOGRAPHY . . . . . . . . . . . . . . . 59
APPENDIX . . . . . . . . . . . . . . . . . so

111

FIGURE

tomqoscnuacntot-a

ta F‘ r4 us h‘ F4
(n .a at a) r4 <3

LIST 9}; FIGURES

The Variable Thickness Plate . . . . . .

Capacitance vs.

Displacement Curves . .

Schematic Diagram of Circuit. . .

The Shielded Capacitive Transducer . .

Static Calibration Apparatus.

The Traverse Mechanism.

The Setup in Final Form .

Capacitance vs.

Pictorial View
Pictorial View
Pictorial View
Pictorial View
Pictorial View
Pictorial View

Displacement Linearity.
of First Mode Shape. . . .
of Second Mode Shape .
of Third Mode Shape
of Fourth Mode Shape .
of Fifth Mode Shape
of Sixth Mode Shape

The Signals Read from the Probe . .

iv

RAGE

. IS

17
17
21

. 26

29
29

. 35
. 41
. 43

45

. 47

49

. 51

55

'L;§1.QE TABLES

TABLE PAGE
1 First Mode-Finite Difference Results . . . . 6
2 Second Mode-Finite Difference Results 6
3 Third Mode-Finite Difference Results . . . 7
4 Fourth Mode-Finite Difference Results . 8
5 Fifth Mode-Finite Difference Results . 9
6 Sixth Mode-Finite Difference Results . . . . lO
7 First Mode-Experimental Data . . . . . . . . 61
8 Second Mode—Experimental Data . . . . . . . 62
9 Third Mode-Experimental Data . . . . . . . . 63
10 Fourth Mode-Experimental Data . . . . . . . 64
11 Fifth Mode-Experimental Data . . . . . . . . 65
12 Sixth Mode-Experimental Data . . . . . . . . 66

LIST 9}; amass

GRAPH PAGE

1 Comparison of Capacitive Probe, Dial Indicator,
and Finite Difference Results - Static Test. . 24

2 First Mode-Experimental Data . . . . . . . . . 42
3 Second Mode-Experimental Data . . . . . . . . . 44
4 Third Mode-Experimental Data . . . . . . . . . 46
5 Fourth Mode-Experimental Data . . . . . . . . . 48
6 Fifth Mode-Experimental Data . . . . . . . . . 50
7 Sixth Mode—Experimental Data . . . . . . . . . 52

vi

I. Introduction

Background

Several individuals in the past generations of master's
and doctoral candidates have worked on variable thickness
plates in the Applied Mechanics Department at Michigan State
University. The experimental work that has been done has
resulted in equipment that is being used in a continuing
investigation of the behavior of plates in vibration and
under static loading.

Pugpose

The purpose of this investigation is to experimentally
verify the finite difference approximation of mode shapes
for vibrating plates made by someone else. These calculations
predict the mode shapes of a variable thickness, aluminum
plate and provide approximations of the relative amplitudes
of vibration to be expected.

An important part of the work done in fulfilling the
objectives of this investigation has been the development
and proving of measuring instruments and equipment.
digg 2; Reporting

The presentation of the work done in this investigation
will be given according to the following order of reporting.

1) Objectives and Procedure of Experimental Analysis

2) Experimental Equipment

3) Presentation of the Results

II. Objectives 2; Experimental Analysis

The objectives of this thesis stem from a recommendation
made by a previous investigator of variable thickness plates.
This recommendation...

"...(to develop)... a technique to measure
the amplitudes in the vibrating plate.“

came about after numerical calculations had been made to
approximate the relative amplitudes of the various modes of
vibration.1 The following sections present a summary of the
work previously done with these plates and outlines the work
done in this thesis to extend the investigation.

Summary 2; Previous Work

In as much as the purpose of this thesis is to experi-
mentally verify the results of finite difference calculations,
it is appropriate at this time to present a brief summary of
the numerical method and the results that it produced.

The Finite Difference Method The theoretical analysis
of plate deflections, either under static loading or in
vibration,requires the solution of differential equations
which describe the behavior of the plate. The finite differ-
ence method offers a means of approximating the solution of
these differential equations whose exact solution is very
difficult or perhaps even impossible in some cases. In
applying the finite difference method to plates, the plate

y

1 Raju, B. Basava, Bending and Vibration is Plates o_r_ Variable
Thickness. Michigan State University Ph.D. Thesis, Applied
Ffichanics, 1961.

is first divided with a grid such as that shown in Figure 1
page 13 . Note in the Figure the designation of the inter-
section points of the grid. It is at these points that the
differential equation is approximated. The approximations

used in a plate analysis may be stated as...

 

6:0: fl --”f1 (1)
6x 211

02: _ r. - 2r.+ r. 2
6320— h2 ___ ( )

where in the case of plate deflection, f is a function of
the deflections, where x is the coordinate in the direction
of the deflection, and where h is the size of the grid over
which the approximation is made. Note that in the equations
(1) and (2) above that the subscripts l and 2 refer to points
just to either side of some point 0 at which the differential
equation is being approximated. In actual practice, the
subscripts are replaced with the subscripts e,w,n, and s to
indicate the function (f) evaluated at the points that are
respectively east, west, north, or south of the point 0.
Forpurpossi:of this thesis it is sufficient to say that this
method requires the application of boundary conditions such
that a series of simultaneous algebraic equations are obtained
which may be solved. The solution of these simultaneous
equations using a matrix method produces the desired approxi-
mate solution of the differential equations of the plate.
This then is the basic method that was used in the

theoretical analysis of variable thickness plates. The

method, although rather simple in concept, is tedious in
application to an involved problem such as that presented
by the variable thickness plates. The actual results that
are to be experimentally verified in this thesis were
obtained using the digital computer which greatly reduced
the effort and time involved in solving the differential
equations that were encountered.

Predicted Mode Shapes The results that were obtained
by a previous investigator using the finite difference
method described above are approximations of the relative
amplitudes of vibration of variable thickness plates. The
results for the one plate that was experimentally measured
in this thesis are presented on the following pages.5
through 10 in Tables 1 through 6. This one plate, that
designated plate as plate "A" in the reference cited at
the bottom of page 2, was chosen because the clamped edge
promised less boundary condition problems than the other
harder-to-obtain simply supported edges. It was felt that
in as much as the technique of measurement to be used was
essentially untried, it would be best to not introduce
unnecessary complications.

The finite difference results are presented for a
grid which is one-half that used in the experimental work.
The theoretical mode shapes are shown in pictorial views in
a later chapter for each of the mode shapes in Figures 7

through 12 on pages 41, 43, 45, 47, 49, and 51, respectively.

 

 

 

.018 .064 .110 .129 .110 .064 .018
.064 .217 .370 .433 .370 .217 .064
.110 .570 .665 .794 .665 .370 .110
.129 .433 .794 1.00 .794 .433 .129
.110 .570 .665 .794 .665 .370 .110
.064 .217 .370 .433 .370 .217 .064
.018 .064 .110 .129 .110 .064 .018
_,--b
1 2 3 4 5 6 7

The data shown on this page for the first
mode and on the following pages for modes
2 through 6 is that obtained by B. Basava

Raju in his Bending and Vibration $3
of Variable Thickness.

EBttom of page 2 .

Table 1

First Mode-Finite Difference Results

Plates

( See reference at

 

 

 

 

.067 .166 .150 .000 —.150 -.166 .067
.200 .506 .468 .000 -.468 -.506 .200
.323 .826 .841 .000 -.841 —.826 .323
.372 .958 1.00 .000 —1.00 -.958 .372
.323 .826 .841 .000 -.841 -.826 .323
.200 .506 .468 .000 —.468 -.506 .200
.067 .166 .150 .000 -.150 -.166 .067
l 2 3 4 5 6 7

Table 2

Second Mode-Finite Difference Results

 

 

 

 

.191 .421 .365 .000 -.365 -.421 -.191
p.421 1.00 .896 .000 -.896 -1.00 -.421
.365 .896 .929 .000 -.929 -.896 -.365
.000 .000 .000 .000 .000 .000 .000
-.365 -.896 -.929 .000 .929 .896 .365
-.421 -1.00 -.896 .000 .896 1.00 .421
-.191 -.421 -.365 .000 .365 .421 .191
1 2 3 4 5 6 7

Table 3

Third Mode-Finite Difference Results

 

 

 

 

.000 .099 .365 .497 .365 .099 .000
-.099 .000 .626 1.00 .626 .000 -.099
-.356 -.626 .000 .589 .000 -.626 -.356
—.497 ~1.00 -.589 .000 -.589 ~1.00 -.497
-.356 -.626 .000 .589 .000 -.626 -.356
-.099 .000 .626 1.00 .626 .000 -.099

.000 .099 .356 .497 .356 .099 .000

1 2 3 4 5 6 7

Table 4

Fourth Mode-Finite Difference Results

 

 

 

 

.081 .173 .173 .146 .173 .173 .081
.173 .363 .282 .163 .282 .363 .173
.173 .282 -.053 -.387 -.053 .282 .173
.146 .163 -.387 —1.00 -.387 .163 .146
.173 .282 —.053 -.387 -.053 .282 .173
.173 .363 .282 .163 .282 .363 .173
.081 .173 .173 .146 .173 .173 .081
1 2 3 4 5 6 7

Table 5

Fifth Mode-Finite Difference Results

 

10

 

 

 

.255 .490 .375 .000 -.375 ~.490 -.255
.380 .774 .554 .000 —.554 -.774 -.380
.065 -.008 -.243 .000 .243 .008 -.065
-.186 -.647 -1.00 .000 1.00 .647 .186
.065 -.008 -.243 .000 .243 ‘ .008 -.065
.380 .774 .554 .000 -.554 —.774 —.380
.255 .490 .375 .000 -.375 —.490 -.255
1 2 3 4 5 6 7

Table 6

Sixth Mode-Finite Difference Results

 

ll

Qualitative Measurements Work has been done that has
qualitatively verified the mode shapes predicted by the
finite difference method. This previous work amounted to
using fine sand or glass beads on the vibrating plate to
outline the nodes of vibration such that each mode shape
could be observed and photographed? The results of the
qualitative work that was done show that good clear mode
shapes were obtainable by the previous investigator. The
symmetry exhibited by these photographs of the glass beads
outlining the nodal lines appears to be good. This same
method of using glass beads on the vibrating plate was
used in the present work to adjust the frequency and ampli-
tude of the vibration such that the clearest and finest
mode shapes could be obtained. .It should also be noted
at this point that although good outlines of the modes
were obtained in this investigation, the degree of symmetry
in the second, third, and fourth modes was not as sharply
defined as those presented in the reference above. Instead,
it was found that certain nodal lines tended to blur as if
there was not a good node existing. Once the mode shape
had been obtained using the glass beads, the quantitative
measurements to be presented later were taken with the equip-

ment that will be described later in this thesis.

 

2 Raju, B. Basava, Bending and Vibration in Plates 2; Variable
Thickness. Michigan State University Ph. D. Thesis, Applied
Mechanics, 1961. See Figure 30, page 59.

12

Experimental Analysis

The experimental work that was done made use of a plate
that was used in the qualitative measurements taken by a
previous investigator. This plate was the same as one used
in the theoretical analysis described in the ggggggy g:
Previous Work section previously presented.

The Plate The plate used in this work was a square
aluminum plate of variable thickness. This plate was held
rigidly in a heavy frame such that its edges were effectively
clamped. The variable thickness of the plate was obtained
by machining a .125 of an inch thick plate such that the
plate was symmetrical about 4 axes. The thickness variation
was from .125 of an inch at its edges to .0625 of an inch
at its center as shown in Figure l , page 13.

To facilitate the measurement of the displacement at
each point of the plate, the flat surface was marked with
a grid. This grid was drawn such that it was approximately
twice as fine as that used in the theoretical analysis
using the finite difference method. The intersection points
of the grid were then taken as the measuring pointsduring
the tests. The 15 space by 15 space grid resulted in 225
points of measurement for each of the six mode shapes
investigated.

Quantitative Measurements In experimentally verifying

the results of the finite difference analysis, there are

13

I

O
5
2
1..

 

The Variable Thickness Blate

Figure l

14

certain specific areas of interest. These areas are listed

below...

1) To establish the accuracy and reliability of
the measuring technique.

2) To obtain the relative amplitudes of vibration
and to check their agreement with predicted
results.

5) To analyze the actual symmetry of the data
obtained in the light of the symmetry expected
with the plate that was used.

The procedure used and the equipment involved in the
investigation of the above areas of interest are presented

in the following chapter.

I I I. .Elxaerissat .41 .Es__u 121119111

A great deal of the apparatus and equipment that was
used in this investigation was available as the result of
previous work. The work that was required to adapt the
existing equipment to this investigation proved to be,
however, a large part of the work that was done. Specifi-
cally, the development and calibration of the capacitive
transducer required extensive work to establish its accuracy
and reliability in the measurement of the deflection of
plates. This transducer as well as the circuitry it required,
and the two degree of movement traverse mechanism that was

designed are presented in the following section.

Transducer Desigg and Application

The choice of the transducer to measure the dynamic
displacement of a vibrating plate was based on the following
criteria:

1) The transducer must have a sensitivity that
will enable it to read changes in deflection
of .0001 inch.

2) The transducer must not change the behavior
of the plate in its application...i.e. it
must not touch the plate or restrain its
deflection.

3) It must have sufficient stability to enable
a set of readings to be taken over a period

of time.

15

16

On the basis of these criteria, the capacitive trans-
ducer was chosen for the deflection measurements. Not only
did it theoretically satisfy each of the requirements, but
it also offered an ease of application which further prompted
its choice.5

Principle 2; Operation The principle stating the
relationship between the capacitance of a two plate capacitor

and its dimensions is...

0 =.225 _€_.i_t, (1)
d

where €h,is the dielectric constant of the material between
the two plates of the capacitor, A is the overlapping area
of the two plates, and d is the distance between the two
plates. Note in equation (1) above the inverse (hyperbolic)
relationship of capacitance (C) to the distance between the
plates (d). It is this hyperbolic relationship, referred
to later as the Calibration Curve, upon which the conversion
of data from relative capacitance to amplitude of deflection
will be made. A typical calibration curve is shown in
Figure 2 , page 17.

By using the plate itself as one-half of a variable

capacitor, it is Possible to construct a circuit in which

 

7—

5 To avoid confusion it should be noted at this point that
the capacitive transducer is often referred to as the
capacitive probe or pickup in the remainder of this text.
This nomenclature is the result of common usage.

17

Curves are for Probes of
different diameter as indicated.
Note increased range of
approximate linearity as diameter
of probe increases.

   
   
 

 

 
 

 

 

 

 

C v

a

P

a L.

c

i

t -

a

n .750 Dia.
c

e

.500 Dis.
.250 Die.
I l I V I I I I +
Displacement

Figure 2 Capacitance vs. Displacement Curves

 

 

T----— 19—4

Shielded Bridge Circuit

  

 

 

 

3h181d\
\\ _. /// D> means to type “Q“
Plate ‘ plug—in unit of the
oscilloscOpe.

   

   

Figure 3 Schematic Diagram of Circuit

 

18

the capacitance varies with the amplitude of displacement
of the plate. The variable capacitor and the circuit that
it requires are shown schematically in Figure 3 , page 17.
Note the amplifying bridge circuit in Figure 3 . This
amplification was necessary so that the signal could be
observed on the Tektronix Oscilloscope.

Sensitivity Because of the hyperbolic relationship
between capacitance and displacement, the capacitive trans-
ducer has a theoretically infinite sensitivity. In other,
words, as the distance between the plates of the capacitor
becomes smaller, the change in capacitance per unit change
in displacement becomes greater. In actual practice, there
is a limit to the sensitivity caused by the voltage break-
down limit of the gap between the plates of the capacitor.4

The sensitivity of the capacitive probe can be brought
to within .0001 of an inch without having the probe touch
the plate by choosing the proper reference distance between
the plates of the capacitor and the corresponding probe
diameter. The probe finally chosen for measurement of the
plate deflections was a .750 of an inch diameter probe with
an initial gap of .0300 of an inch between the plates of

the capacitor. These dimensions were the result of a

compromise between sensitivity, amplitude of oscilloscope
response, and of the range of reasonable capacitance vs.

displacement linearity.

 

4 In umentation in Scientific Research, Kurt S. Lion,

str
MoGraw Hill Book-Company, Inc., page 67.

19

Plate Restraint The second criterion that the trans=

 

ducer have no effect on the behavior of the plate is well
satisfied by the capacitive probe. Not only is there no
physical contact between the probe and the plate, but the
forces.exerted by the plates of the capacitor upon each
other are found to be small. The principle expressing this
force is...

F ___ 8.85 x 10"12 E2 A (2)
2 <12

 

where E is the voltage applied across the plates of the
capacitor, A is the area of the overlapping plates, and d

is the distance between the plates. If we use the dimensions
of the capacitive probe used in this work in the above
equation (2) and make the calculation, we find that the forces
involved are of the order of a few dynes...where 105 dynes s

1 pound of force.

The vibrating plate will also displace air from between
the capacitive probe and the plate, and this too could very
well be a source of force exerted on the plate. Consider~
ations given this problem in this work are presented in
the Appendix of this Thesis on page 67.

Stability The criterion of stability is the most diffi-
cult to satisfy in the case of the capacitive probe. Not
only is capacitance a function of temperature, pressure, and
humidity, but it also is affected by stray signals, noise
in the highly amplifying circuitry, and the surrounding back-
ground vibration. This problem is best approached with

20

extensive shielding of all leads and the capacitive probe
itself, and by such precautions as taping all leads down
to prevent movement and noise in the circuit, by using
rubber insulation where possible, and by using very heavy
supports for the plate and probe carriers so that surrounding
vibration from running machinery and operator movement could
be damped out. As a result of these precautions, all problems
of stability were sufficiently reduced except for an annoying
internal drift in capacitance caused by extreme drafts or
temperature changes.5

The setup in its final form is shown in Figure 4 ,
page 21 . The cross section shown is a detailed drawing of

the capacitive transducer shown in the photograph.

Static Calibration

This part of the development of the capacitive probe
was necessary to establish the accuracy and reliability of
the capacitive pick~up. The preliminary calibration attempts
amounted to a comparison of the deflection measured with a
dial indicator with those measured with the probe. In the
final calibration tests, the same plate that was to be used
in the dynamic vibrations tests was uniformly loaded for the
static tests. Measurements of this deflected plate were
then taken again with both the dial indicator and the capaciw

tive probe that by this time had been fullv developed,

 

5 This problem was resolved by eliminating the source of the
drafts and temperature changes. Data was taken late at
night when traffic through the area was nonexistent;,

Figure 4

The Shielded Capacitive Transducer
(See Next Page)

21

 

Micrometer

Ground
Wire

®

 

 

 

 

 

 

 

Insulator

Shield

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ground
Wire

23

Procedure 3; Calibration The procedure used in
determining the deflection of the statically loaded plate
was to first run a calibration curve of the probe. This
was nothing more than Just an accurate, very detailed
capacitance vs. distanceabetween-plates curve. This curve
was then used to convert the relative voltage changes read
from the oscilloscope to actual amplitudes of deflection
of the uniformly loaded static plate. This conversion not
only resulted in quantitative amplitudes of deflection,
but also corrected for the slight nonwlinearity of the
capacitance vs. displacement relationship that existed in
the range in which the probe was used.

Once the calibration curve was determined, the probe
was carefully gapped at .0500 of an inch from the center
of the plate at zero load. From this point a complete set
of readings was taken of the 225 grid points of the plate
in its undeflected condition. This gave a picture of the
plate surface before loading. This step was necessary
since the plate was not perfectly flat. After taking
readings of the loaded plate, the total deflection of the
plate from unloaded to loaded condition then was the differ-
ence between initial and final readings at each point of
the plate. Readings were taken at each point of the plate
by moving the probe with the traverse mechanism that had
been designed for the purpose.

The measurements that were obtained using the probe

and the dial indicator were then compared to the results of

OdeDHom

mdzosoomH'UmH-U

24

 

 

 

 

 

 

 

  

 

 

1.00
,'K 2: Finite Difference
\ Results
’/
P
/ l
l
.7’ ’ i
.5
g
.2;
Legend
Dial Indicator
I
----- Capacitive Probe

 

 

2 4 ' é a £0 " 12 14
Centerline Grid Positions

Graph 1 Comparison of Capacitive Probe, Dial Indicator,.
and Finite.Difference Results- Static Test.

25

a finite difference analysis which were available. The
results of this comparison are shown in Graph 1 , page 24.
From this graph it can be seen that there is good agreement
between the three sets of data plotted. A capacitive probe
will, then, give an accurate representation of a deflected
plate under static loading. There remains now only the
application of this probe to the dynamic vibration of the
plate.

Equipment pg Calibratiog The equipment used in the

 

 

calibrations tests was the same as that used later in the
vibrations work, except for the method of loading. In the
calibration work, the load was applied with a head of
water through a water bag placed in a closed chamber below
the plate. The water pressure standpipe and the apparatus
used for calibration are shown in Figure 5 , page 27 .
The water sac was made of plastic material bonded together
with Goodyear Pliobond Cement.

In obtaining the calibration curve, which was required
only for the static tests ..... a large diameter plate was
mounted on a micrometer head which enabled data to be taken
for the plotting of the calibration curve. The data for
the curve was taken as the deflection of the trace on the
oscilloscope screen. The oscilIOSCOpe was rebalanced after
each full scale deflection of the trace. The data taken
showed excellent repeatibilitv. It should be mentioned,
however, that any geometric change of the probe or its

components requires that a new calibration curve be run.

Figure 5

Static Calibration Ap
(See Next Page

paratus

26

4

4532:?

 

 

 

 

 

 

 

 

n—Pressure

Stand Pipe '

Static Loading Setup
Traverse Mechanism
Measuring
Device
Plate
Pin-u —\ *

rJ—_—L.
‘F—‘v

      

Water Sac

 

 

 

 

 

 

 

 

28

Dynamic Measurements

The actual taking of dynamic data represented the final
realization of the objective of the investigation. The equipe
ment used in these measurements and the procedure that was
followed are presented in the following section.

Equipment In the experimental work that was done, it
was necessary to vibrate the plate at both varying frequency
and amplitude. This required range of vibration was obtained
with a siren and a high volume air compressor that provided
a pulsating air blast which was used to vibrate the plate.
Measurements of the amplitude of vibration of the plate
were taken at each of the 225 points on the plate by moving
the capacitive probe across the plate with the traverse
mechanism. In as much as the compressor and siren were
available for use from previous works, only a detailed
description of the traverse mechanism will be presented
here.

Measurement of the amplitude of vibration of the plate
required that the traverse mechanism have two degrees of
motion that could maintain an accurate vertical position
of the capacitive probe. For this reason, the traverse
was built heavy enough so that there would be no problem
in keeping the probe in position with respect to the plate.

The traverse mechanism, shown in Figure 6 , page 30.

 

6 For detailed description of the equipment and the compres~
sor, see RaJu, B. Basava, Bending and Vibration in Plates
2; Variable Thickness. Michigan State University‘Ph.D.
Thesis, Applied Mechanics, 1961.

Figure 6

The Traverse Mechanism
(Next Page)

Figure 7

The Setup in Final Form
(Next Page)

29

THE TRAVERSE MECHANISM

THE SETUP IN FINAL NRM

 

31

consists primarily of two high quality dovetails. As shown
in'Figure 6 , the larger screw driven slide supports the
small dovetail and the rack and pinion driven slide. It is
this smaller cross slide that supports the capacitive probe.
The final setup is shown in Figure 7 , page 30.

Procedure Dynamic measurements of the amplitude of
vibration of the vibrating plate were made in a manner
similar to that used in the static calibration work. Readings
of the change in capacitance were read from the oscilloscope
as the peak to peak height of a signal displayed on the
screen., This signal was that read from the circuit containing
the variable capacitor consisting of the capacitive probe
and the vibrating plate. Readings were taken in the dynamic
case at the same points used in the static work...i.e. at
the 225 intersection points of the grid drawn on the plate.

In processing the data resulting from the dynamic readings,

the capacitance vs. displacement relationship was considered

to be linear. This assumption is permissible since the part

of the hyperbolic curve along which readings were taken was
small. A discussion of the errors introduced by this assump-
tion is presented in the section Sources 3; Egggg. At this
time it is sufficient to say that the assumption of linearity
greatly reduced the problems of data taking and data processing
without introducing appreciable error.

The problem of instability that is inherent in the

capacitive probe made it advisable to incorporate some type

52

of reference check in the data taking procedure. This was
done by taking the data in a spiral pattern and by checking
at a point previously read at every two or three rounds in

the 45 to 60 minute data taking period.

Sources 2: £332;
The sources of error can be broken into three general

areas...

1) Errors in Capacitive Probe

2) Errors from Traverse Mechanism

6) Errors Resulting from Method of Handling Data
Each of these sources of error and the probable amount of
error that they contribute are explored in detail in the
following sections.

Errors from the Capacitivg Probe As mentioned previously

 

in the discussion of the stability of the capacitive probe,
the capacitance is affected by the pressure, temperature, and
humidity, as well as, it seems, by drafts and air movements.
Examining first the effects of changes in pressure on the
capacitive probe measurements, it is known that for air at
66°F, a change in pressure of from 1 atmosphere to 100 atmos~
pheres causes a change in the dielectric constant of from
1.0006 to 1.0548. This represents a deviation of 5.42% in
the dielectric constant with an extreme change of 100 atmoss
pheres of pressure. In order to get some idea of the pressure
variations existing between the vibrating plate and the

capacitive probe, consider the most extreme case possible

33

which would be if the varying capacitor is thought of as a
closed cylinder of changing volume. In such a case it is
found that the pressure changes would be of the order of 2
atmospheres. From this we can conclude that the errors
caused by pressure changes can be neglected in the light of
the 5.42% error with a change in pressure of 100 atmospheres.
The fringe effect or stray capacitance at the edges of
the plate introduce some error in the simple expression for
capacitance shown in equation (1 ) on page l6. According

to Lion in Instrumentation for Sgientifig Researghs the

 

fringe effect introduces an error of about 6% if the ratio
of the radius to distance between the plates is 50. In the
case where this ratio can be kept over 200, the error is
less than 1%. In the case of the probe used in this work
with its .750 diamenter and .931{»0f an inch gap, the ratio
of radius to gap is 125. In the light of this ratio and
since the probe used was shielded, the error can be consida
ered to be well below 5%.

The errors caused by temperature and humidity changes
may be considered to have negligible effect on the measure»
ments that were made since these variables may be considered
to be constant in the time that it took for a set of readings
to be taken.

Errors from Traverse Mechanism The traverse mechanism

 

supported the capacitive probe on a cantilever and could

34

conceivably contribute error to the readingsthat were taken.
For this reason, the traverse mechanism was designed to
have very rigid components. Checks made of the deflection
of the cantilever showed that a very heavy load would be
required to have an appreciable deflection at even the most
extreme cantilever position. In the light of the ruggedly
built mechanism, and the care that was used in moving the
probe, this source of error can be neglected since the .75fi‘
load of the probe caused an insignificant deflection in the
l" X 4” steel cantilever even at its maximum extension of
18 inches.

Ema getting free Elsie. lialsaiiaa An important
source of possible error is that of the assumptions made and
the technique used in data handling and evaluation.

The one assumption made in converting the dynamic capac=
itive reading to relative amplitudes of vibration was that
the capacitance vs. displacement relationship was linear.
This assunption introduced little error as shown in Figure 8
page 55. As seen in the figure, the range in which the change
in deflection occurred about the .0600 inch initial gap at
the center of the plate amounted to a $20 units of oscillo-
scope deflection. This portion of the curve is a small and
essentially linear portion of the total hyperbolic calibration
curve. Even considering the fact that the plate was not flat,
which would vary the .0500 inch initial gap, we find that the
:;.0015 inch variation in the undeflected plate surface

35

 

 

GODDd‘l-J'OW’UQO

    

   
 

".M‘“ A __L s _ i . 1 A
v v r v

ent

  

Displace

-.._|..A9 121.18 l C

 
   
  

20
-10

-10
-20

 

Maximum Range of Use

Figure 8 Capacitance vs Displacement Linearity

 

would exyand the range of measurement to that indicated on
the curve in Figure 8. The percent of error introduced
by the nonlinearity of the calibration curve, then, is 3 %.
Another source of error resulted from readings of the
scope trace deflection. Thesignalthat resulted, especially
in the high er modes, v ried somewhat in the absolute peak-
tompeak height. This variation was ”averaged outW by setting
the scope each that a wide .a d was displayed on the screen
instead of the Mar single sweep signal. The reading taken
as data then was the height sf the brightest part of the
band. Stray fringes of the hand were considered to be the
variation cf the signal. in as much as the bright band that
was measured was stable, the primary siurce of error here
ould be the reading of the height. The screen was gridded
with 40 units, and considering that readings could be esti:
mated with an accuracy of 1:1 unit, this amounts to an error
of approximately i 2%.

Sumr is v 3f grrors A summary of the sources of errors

and the amount that each is considered to have contributed to
the total error of the results that were obtained is presented
in the table below.

Probe Errors
Pressure Ghanges . . . . . . . o . . Negligible
Fringe Effect . . . . . . . . . . 3;0%
Humidity, Temperature . . . . . . . Negligible

Traverse Mechanism
Cantilever Deflection . . . . . . . Negligible

Data Evaluation
Linearity Assumption . . . . . . . . {5%
Data Reading . . . . . . . . . . . . ;: 2.0%

IV. Presentation 9; Results

 

The quantity of experimental data to be presented and
compared with the finite difference results presents a
problem. This problem, one of keeping the data presentation
compact and to the point while still presenting enough
detail to enable the reader to get a feel for the results
of the thesis work, was handled in the following manner.

For each mode shape a pictorial view of the theoretical mode
shape has been developed. The pietorial mode shapes are
shown as absolute values, although positive and negative
areas have been marked where their inclusion aids in visual-
izing the actual physical shape of the plate in vibration.
Following the pictorial view of each mode shape, the experi-
mental data is plotted on a set of graphs which present the
relative deflection of the plate along each of the cross
sections formed by the 15 by 15 space grid which was drawn
on the plate. The sets of graphs are keyed to the data
sheets in that the column of graphs marked N S is that data
taken from the appropriate data sheet as the rows marked 1
through 15, while the column of graphs marked W E is that
data appearing as columns 1 through 15 on the data sheets.

These same graphs have superimposed upon them the finite
difference results in red which were available. The sets
of graphs, then, not only indicate the degree of symmetry

and the experimental mode shapes that were found to exist,

37

58

but also provide an excellent comparison between experimental

and finite difference data.

Mode Shapes
First Mode A pictorial view of the absolute value shape

of the first mode shape is presented in Figure 9, page 41.
The experimental data that was obtained for the first mode is
shown plotted in the set of graphs shown in Graph 2, page 42.
The agreement of the experimental data with the finite differ-
ence results for this simple mode is good. The degree of
symmetry is also good. *Note that the finite difference
results in this case are neither consistently above nor
below the actual experimental values obtained.

Second Mode A pictorial view of the absolute value
shape of the second mode is shown in Figure 10, page 43.
It should be emphasized here that the actual physical shape
of the vibrating plate is that implied by the positive and
negative signs shown in the pictorial drawing. The data
plotted for each of the cross sections of the 15 by 15
space grid is shown in Graph 3, page 44. It can be seen
that there is fair to good agreement between the experi—
mental data and the finite difference results shown in red
on these graphs. Symmetry is not as good as that obtained
with the first mode, but is fair. The lack of symmetry
may be attributed to the fact that the plate in its second
mode of vibration must by necessary be driven from a point

off center.

39

Thigg'gggg The pictorial view of the third mode
absolute value shape is presented in Figure 11, page 45.
The mode shape is marked with positive and negative areas
to indicate its actual physical shape. The plots of the
data for each of the cross sections is shown in Graph 4,
page 46.. There is a marked lack of symmetry in the.
experimental data, although the mode shape is apparent.
Note also that for this more complex mode shape, the .750
diameter probe has been unable to detect the detailed shape
of the mode. The lack of good agreement between the experi-
mental and finite difference data might again be attrib-
uted to the fact that the plate had to be driven off center
in this mode of vibration.

Fourth Egg; The pictorial view of the fourth mode is
presented in Figure 12, page4fl’. The positive and negative
signs are again shown to indicate the physical shape of
the vibrating plate. The agreement between the finite
difference results and the experimental data shown in
Graph 5, page 48, is only fair at best. The large diamenter
probe that was used was unable to pick out the fine detail
of the complex intersection of the four surfaces ofthe
fourth mode shape. The probe only read an average of the
true surface in which much detail was lost. Symmetry in
this case is not good. This is attributable to the fact
that the plate was by necessity driven at one corner
since in this mode of vibration a node passed thru the

center.

40

Fifth £993 The pictorial view of the fifth mode
shape is shown in Figure 13, page 49. The agreement of
the experimental data to the finite difference results
as shown in Graph 6, page 50, was fair to good. The‘
mode shape is clearly evident in the graphs of the experi-
mental data and the symmetry is seen to be fair to good.
Note that in this case that the finite difference results
were consistently less than the experimental data.

glxth,§gdg The pictorial view of the sixth mode of
vibration is shown in Figure 14, page 51. The mode shape
is discernible from the experimental data shown plotted
in Graph 7, page 52. The symmetry, however, is not good.
As in the other complex mode shapes, namely the third and
fourth modes, the capacitive probe has rounded off the
data to such an extent that detail of the mode shape has
been lost. Here, again, the plate was driven off center.

General Comparison 9; Egg; Shapes ‘In general, the
mode shapes that were simplerLin.form and which could be
obtainedby driving the plate from the center provided
good agreement between the finite difference and experi-
mental results. The lack of good symmetry in some of the
experimental data and the lack of good agreement between
relative amplitudes of the experimental and finite differ-
ence methods may be attributed to the following...

1) The capacitive probe that was developed and

proven in static test was too large to read the

41

 

   
 

Figure 9oPictorial View of First Mode Shape

 

42

 

 

 

 

 

 

 

 

 

 

 

 

Figure _10 Pictorial View of Second Mode Shape

 

w:
JJJ

 

 

 

 

‘L-
I
L

L.--

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LL-

 

 

 

 

VI

10

 

 

 

I

L.

 

 

cf

 

 

F-
_ld

 

 

 

 

F"

3

Al J!

_JLJ

 

 

 

 

F ,

 

 

 

 

 

F I

 

 

 

46

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- 1 A ”Ll—fl “-2- fl-“n—ii—Anfi-n1 “n+n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

._ i i Y Y a: ‘ r , fl 7" A far 1‘1
LL _1_ v I. Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1“ A ‘
"V-‘V‘ ‘

 

 

.- 1.. r- an L1. ‘r-:- 13.... _R_-_L-‘I “-5.

49

 

 

 

 

Shape

of Fifth Mode

W
e
i
V
l
a
.1
r
O
t
C
i
P
"0
1

Figure

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T V
i '
V‘v v v ff?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

51

 

Figure 14. Pictorial View of Sixth Mode Shape

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

53

detail of complicated intersection of some of
the surfaces of the vibrating plate.

2) The plate was by necessity driven off center for
certain modes. This caused a marked lack of
symmetry as the plate was forced to vibrate at
larger amplitudes at the points of driving.

5) The probe reads only absolute values and loses
detail in the area of the nodes.

4) The finite difference method does not produce
exact results. Better results would require a
finer grid, and considerations which would make
the theoretical model more representative of the
real structure.

Capacitive Probe Output Signal

The output from the capacitive probe was presented on
an oscilloscope. The absolute value of the signal, being
proportional to the deflection of the vibrating plate, was
read as the data upon which the previously presented graphs
are based. For each mode shape a single signal was displayed
on the screen of the oscilloscope and photographed. These
photographs are shown in Figure 15, page 56. Note that for
the first mode shape both large and small deflection signals
are presented. The small deflection signals for the first mode
is essentially a sine wave, while the large deflection
signal is distorted as the plate is drive beyond its nat-
ural amplitude of vibration. In all but the first mode,

54

a great deal of noise was found to be superimposed upon
the signal obtained. The source of this noise, it is
felt, is a combination of a non-linear behavior of the
plate at more complicated mode shapes and of a non—
sinusoidal signal produced by the siren used to vibrate
the plate.

It was found during the investigatidn that the
output signal from the capacitive probe was stable in
form only for the more simple mode of vibration such as
the first, fifth, and to some extent the second. The
more complicated mode shapes, the third, fourth, and ‘
sixth, were less repeatible. The photographs of these
more complicated mode shapes shown in parts 3, 4, and 6
of Figure 15, page 56, ahould be considered only to be
representative of the type of signal obtained, rather

than the actual signal that could be obtained at a'givenn.

run .

Figure 15

(Next Page)

First Mode
First Mode
Second lode
Third Mode
Fourth Rode
Fifth Mode
Sixth Mode

(Small
(Large
(Small
(Small
(Small
(Small
(Small

The Signals Read from the Probe

Deflections)
Deflections)
Deflections)
Deflections)
Deflections)
Deflections)
Deflections)

55

 

U1

/_
C)

Signals from Capacitive Probe

V. Conclusions

Summary
The capacitive probe that has been developed has been

shown to have satisfactory sensitivity, accuracy, repeati-
bility, and ease in application in static tests that were
made of the measuring device. Actual measurements of a
statically deflected plate produced experimental results
that agreed with a dial indicator measurement to within a
10% deviation.

The results of applying the capacitive probe to dynamic
measurements of the amplitude of vibration of a plate in
general are not as good as those obtained in static test runs.
There is good or only Just fair agreement between the experi-
mental and finite difference data for only the simpler mode
shapes. The more complicated mode shapes lack symmetry
because of the necessity of driving the plate at a point
not on center. The capacitive probe was unable to read the
finer detail of the more complicated mode shapes since it
read only absolute values and could not pick out surface
intersections.

Recommendations

Even though it has been shown that the specific capaci-
tive probe that was developed and used in this investigation
was unable to provide good readings of all mode shapes, it
should not be interpreted that this technique is not a good

57

58

one. On the basis of the good qualities that the capacitive

probe technique of dynamic measurement offers, and in the

light of the results of static calibration tests, it is felt

that this method warrants further work. Specifically,recom-

mendations are....

1)

2)

a)

Develope a probe of smaller diameter so that
increased detail can be read by the probe. This
will involve an increase in the non-linearity of

the probe to displacement, a decrease in the range
of measurement that may be covered, and perhaps

will require increased amplification of the signal
which will further amplify the problems of stability
and shielding that will be encountered.

Investigate the large deflection of statically
loaded variable thickness plates using the equip—
ment that is now available from the calibration

work that was done with the capacitive probe. A
dial indicator was found to produce satisfactory
results for static measurements, although the
capacitive probe provides a more sensitive, although
harder to use method of measurement.

Investigate the forces existing between the probe
and the plate.

Investigate higher modes of vibration of plates.

BIBLIOGRAPHY

RaJu, B. Basava, ”Bending and Vibration of Variable
Thickness Plates”. A Ph.D. Thesis submitted to the
Michigan State University, 1961.

Lion, Kurt S., "Instrumentation in Scientific Research”.
McGraw Hill Book Company, Inc.

Jaeger, J.C., Elasticity, Fracturg, 52%,ELEE» John Wiley
and Sons, New York, N.Y.

59

APPENDIX

60

(OCDQO‘JUIhOllOI—J

h’ r4 a4 F‘ +4 rd
0: w» ca to a» C)

61

 

 

.05
.05
.05
.05
.05
.06
.06
.08
.08
.06
.06
.06

.03,

.05
.05

.05
.06
.08
.11
.14
.17
.20
.22
.22
.20
.17
.14
.11
.06
.05

.05
.08
.11
.17
.22
.28
.55
.56
.55
.51
.28
.22
.17
.08

.06
.08
.14
.20
.28
.55
.59
.45
.45
.59
.55
.25
.22
.14

.06
.11
.20
.25
.59
.50
.62
.67
.64
.58
.50
.55
.28
.17

.06 .06 .08

.06
.14
.22
.51
.47
.64
.78
.89
.85
.75
.62
.42
.55
.19

.06
.14
.22
.55
.50
.70
.89
.95
.92
.81
.64
.42

.56—

.19

.08
.14
.22
.55
.55
.75
.89
1.0
.95
.85
.67
.55
.55
.19

.08 .08 .11

.06
.14
.22
.51
.47
.67
.85
.92
.86

.78

.62
.59
.28
.17
.08

.06
.11
.20
.25
.42
.56
.70
.75
.72

.67
.50

.55
.22
.14
.08

.05
.08
.14
.22
.55
.45
.50
.58
.55
.50
.59
.28
.19
.11
.08

.05
.08
.11
.14
.22
.28
.55
.55
.55
.51
.28
.20
.14
.08
.06

. 03
.os
.08
. 11
.14
. 17
. 20
.22
. 20
.20
.14
.11
.08
.06
.06

.05
.06
.06
.08
.08
.11
.11
.14
.14
.11
.11
.08
.06
.06
.05

.05
.05
.05
.06
.06
.06
.06
.06
.06
.06
.06
.05
.05
.05
.05

 

1

2

5

Table 7

4

First Mode—EXperimental Data

5

6

7

8

9 10 11 12 15

14 15

 

¢> CD ‘0 05 (n pl on to IH

P' h’ ha ha F‘ +4
01 as cs to ta 0

62

 

 

.14
.14
.14
.17
.25
.25
.29
.29
.54
.29
.29
.25
.14
.09
.06

.14
.09
.14
.20
.29
.29
.51
.57
.46
.54
.54
.51
.17
.11
.11

.14
.14
.20
.51
.46
.57
.68
.77

.71
.68
.57
.57
.20

.14
.17
.29
.46
.57
.71
.85
1.0

.80 1.0

.91
.86
.74
.57
.25

.17
.14
.29
.45
.57
.71
.86
.94
.97
.91
.80
.77
.57
.51

.ll

.14 .17

.14
.14
.25
.54
.45
.57
.74
.91
.97
.94
.71
.80
.48
.25
.20

.14
.ll
.17
.25
.25
.51
.57
.51
.57
.66
.57
.54
.57
.17
.17

.14 .14
.11 .14
.17 .29
.20 .57
.25 .40
.25 .45
.17 .57
.09..57
.20 .48
.29 .57
.29 .51
.54 .29
.26 .29
.17 .14
.14 .14

.14
.20
.54
.40
.57
.57
.77
.77
.74
.60
.48
.45
.51
.17
.14

.17
.14
.54
.45
.54
.74
.71
.85
.85
.71
.60
.51
.40
.25
.14

.14
.17
.51
.45
.48
.65
.65
.80
.77
.71
.54
.48
.54
.20
.17

.14
.17
.20
.29
.45
.46
.57
.65
.54
.57
.46
.54
.29
.17
.11

.14
.11
.14
.20
.29
.29
.54
.57
.57
.54
.29
.26
.20
.14
.11

.11
.09

.11

.14

.17

.20
.25
.26
.25
.20
.17
.14
.11
.09
.09

 

l

2

Table 8

5

4 5

6

7

8 9 10 11 12 15

Second Mode-Experimental Data

14 15

 

(OOQOSOIF-GNH

10
ll
12
15
14
15

65

 

 

 

.11 .11 .ll .11 .15 .11 .11 .08 .11 .15 . .11 .ll .11 .08
.ll .15 .16 .16 .25 .22 .19 .16 .19 .22 . .19 .16 .16 .15
.ll .19 .42 .44 .42 .55 .22 .16 .28 .55 . .55 .28 .19 .15
.15 .16 .50 .47 .61 .55 .56 .22 .59 .44 . .42 .59 .25 .15
.11 .19 .55 .58 .75 .61 .42 .25 .56 .44 . .44 .42 .25 .16
.11 .16 .55 .55 .72 .55 .42 .22 .28 .44 . .42 .55 .28 .16
.11 .15 .28 .42 .47 .42 .28 .28 .28 .55 . .55 .28 .16 .15
.08 .11 .15 .19 .19 .16 .25 .28 .28 .25 . .25 .16 .15 .08
.08 .15 .55 .47 .47 .42 .55 .22 .28 .55 .56 .50 .15 .08
.11 .28 .69 .69 .78 .58 .42 .19 .42 .55 . .55 .42 .28 .11
.15 .28 .85 .85 .85 .69 .55 .19 .42 .64 . .55 .55 .50 .11
.11 .28 .85 1.0 .94 .86 .55 .25 .55 .58 . .55 .55 .50 .11
.ll .25 .78 .97 .89 .64 .42 .15 .56 .55 . .64 .47 .28 .11
.ll .19 .55 .50 .28 .28 .16 .ll .16 .22 . .55 .25 .16 .ll
.15 .15 .11 .15 .15 .15 .ll .08 .11 .11 .15 .15 .ll .11 .08
1 2 5 4 5 6 7 8 9 10 ll 12 15 14 15
Table 9 Third Mode—Experimental Data

 

(O a) '0 O) (n oh 0: to Id

#4 t4 F‘ r4 F' r4
OI es cs to F‘ C)

64

 

 

.16
.16
.16
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20
.20

.16
.16
.20
.20
.20
.20
.20
.20
.20
.16
.20
.20
.20
.20
.20

.20
.20
.24

.24
.52
.42
.60
.80
.64
.56
.56
.44
.40
.24
.20

.20
.20
.24
.28
.44
.42
.76
1.0
.80
.64
.44
.28
.40
.28
.20

.20
.20
.28
.52
.24
.44
.60
.88
.80
.44
.28
.40
.40
.52
.20

.20
.20
.40
.44
.56
.28
.56
.56
.44
.28
.44
.60
.56
.40
.20

.20
.20
.40
.60
°56
.56
.44
.56
.44
.44
.56
.84
.60
.40
.24

.20 .20 .20

.20
.48
.60
.60
.56
.60
.56
.56
.64
.60
°6O
.68
.40
.24

.20
.40
.56
.60
.42
.56
.60
.56
.64
.60
.88
.64
.40
.24

.20
.56
.48
.40
.40
.56
.68
.64
.42
.56
.64
.56
.56
.24

.20
.20
.24
.52
.52
.56
.60
.68
.64
.56
.44
.56
.56
.56
.20

.16
.16
.20
.52
.28
.56
.60
.68
.68
.56
.40
.56
.40
.24
.20

.16
.16
.20
.28
.52
.40
.56
.60
.60
.56
.40
.52
.52
.20
.20

.12
.12
.16
.20
.20
.20
.28
.52
.24
.28
.20
.20
.20
.16
.16

.12

.12
.12
.16
.16
.16
.20
.20
.20
.20
.20
.20
.16
.16
.16

 

Table 10

8

9

10

ll 12 15 14 15

Fourth Mods-Experimental Data

 

K) (D -q 0: cm »- on to IA

F4 r4 F‘ r4 I4 F‘
U! as C» to F‘ C)

65

 

 

.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17
.17

.17
.26
.26
.52
.52
.52
.52
.52
.54
.54

.57

.54
.29
.25
.14

.17
.26
.52
.54
.57
.57
.40
.45
.45
.49

.52

.52
°40
.54
.17

.17
.29
.54
.40
.45
.40
.45
.40
.46
.49
.54
.49
.45
.52
.25

.17
.26
.54
.45
.57
.52
.29
.29
.29
.57
.45
.45
.46
.54
.20

.17
.26
.57
.57
.29
.29
.45
.49
.45
.29
.54
.54
.45
.52
.17

.20 .17
.26
.29

.29

.26
.54
.54
.29
.45

.29
.46
.65 .77
.77 1.0
.66 .72
.45 .54
.26

.26

.29
.26
.54 .57
.29 .26
.17 .17

.14
.26
.29
.29
.29
.40
.66
.86
.72
.49
.25
.29
.57
.29
.14

.14
.26
.54
.52
.26
.29
.45
.49
.45
.29
.29
.54
.57
.54
.20

.17
.26
.54

.54

.29
.26
.26
.25
.25
.26
.54
.40
054

.29

.17

.17
.26
.52
.57
.54
.52
.29
.29
.29
.54
.40
.40
.57
°29
.17

.14
.25
.29
.52
.54
.52
.52
.29
.54
.54
.54
.54
.52
.26
.17

.14
.20
.25

.26

.29
.29
.26
.26
.29
.29
.29
.29
.29

.20

.17

.17
.17
.17
.17
.17
.17
.17
.17
.17

.17

.17
017
.17

.17

.17

 

Table 11

10 11

12

Fifth ModeaExperimental Data

15 14 15

 

to (D 'Q o: (n a» on to I“

+4 r4 F‘ #4 r4 F'
0‘ we Cd to ta 0

66

 

 

.18

.25
.28
.25
.25
.18

.18

.25
.28

.50

.25

.25
.25
.18
.15

.25
.25
.57
.40
.25
.25
.50
.57
.45
.45
.45
.65
.58
.45
.20

.25
.57
.45
.50
.55
.50
.50
.68
.65
.50
.50
.68
.75
.65
.20

.25
.57
.40
.52
.45
.55
.68
.95
.88
.58
.68
.95
1.0
.75
.25

.20
.50
.45
°50
.45
.57
.68
.90
.80
.50
.58
.90
1.0
.70
.25

.18
.50
.57
.57
.58
.55
.58
.78
.75
.45
.50
.68
.75
.50

.15

.26

.25
.25
.28
.25
.45
.50
.45
.55
.58
.45
.55
.58

.15

.15

.18
.20
.20
.20
.20
.25
.20
.25
.50
.25
.50
.25

.55 .55 .15

.18
.18
.20
°2O
.25
.25

.35
.38 .

.55
.28
.25
.58
.55
.25
.15

.18
.25
.28
.55
.25
.55
.45

.50
.55
.55
.50
.50
.45

.18
.25
.50
.40
.55
.55
.50
.65
.55
.55
.40
.55
.65
°50

.20
.50
.25
.40
.50
.50
.40
.50
.50
.55
.45
°50
.65
.45

.25 .57 .57

.18
.25
.25
.40
.50
.25
.58
.45
.45
.50
.45
°50
.45
.45
.50

.18
.20
.25
.25
.25
.25
.25
.25
.28
.50
.50
.50
.57
.50
.25

.15
.15
.15
.18
.15

.15

.15
.18
.18
.20
.20
.50
.50
.25
.18

 

Table 12

5

Sixth ModemExperimental Data

6

7

8

9 10 11 12

15

14 15

 

67

Air Displacement Force Consideration

The complete analysis of the problem dealing with the
force acting on the plate that is caused by the flow of air
from between the plate and the probe is quite involved. The
problem, one of forcing air out from between a disk and a

‘large plate, is schematically presented in the drawing below.

,_Although a complete and detailed analysis is beyond the scope

of this investigation, the consideration that was made is

presented below.

 

 

 

The solution of a similar problem is available in a

reference7 and through this it is possible to at least obtain
some idea of the order of magnitude of the force of the air
on the plate. The case for which the solution is known is
shown schematically in the drawing above. From the reference
in the footnote on this page,‘it is found that the force (P)
is expressed in the form of

P = envo 13/h‘5

 

7 Jaeger, J.C., Elasticity, Fracture, a d Flow

n , John Wiley
and Sons, New York, N. Y. (See page 140 — 142)

 

68

where (V0) is the velocity of the plates with respect to one
another, (1) is the width of the strips, and (h) is the
distance between the strips. 7} is viscosity of fluid between
plates.

Making the calculation indicated in equation (5), using
1 - .750 inches, h = .0500 inches, Vo=15 inches per second,
we find that the force is small, less than .Ol#,but of a
magnitude that could have an effect. Since the case for
which the solution is known is definitely a worse condition
than that encountered in this investigation, we can say
that the effect is small, although not necessarily negligible.
There definitely is need for further work in this area, as

indicated in the recommendations at the end of this thesis.

HllllllHlllii

||
||
ullll
I
ll

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