AN ATTEMPT TO APPLY
ELEMENTARY VIBRATION THEORY
TO A STACK OF PACKAGES '

Thesis for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
THOMAS JOSEPH KUSZA

- 1973

ABSTRACT

AN ATTEMPT TO APPLY ELEMENTARY VIBRATION
THEORY TO A STACK OF PACKAGES

BY

Thomas Joseph Kusza

This thesis is a determination of the degree of
applicability of elementary vibration theory to a stack of
packages. If such an application could be used to explain
the behavior of a stack of packages, a simple and very
powerful tool would be available to better design a
functional package.

From the research it can be concluded that there
is little correlation of elementary vibration theory to
the laboratory data. The difference between the laboratory
value and a corresponding theoretical value, increases as

the number of containers in the stack increases.

AN ATTEMPT TO APPLY ELEMENTARY VIBRATION

THEORY TO A STACK OF PACKAGES

BY

Thomas Joseph Kusza

A THESIS

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

MASTER OF SCIENCE

School of Packaging

1973

To Nita Jo, my wife

ii

ACKNOWLEDGMENTS

I would like to thank Dr. James W. Goff for his
help during my program. Without his guidance and support
this thesis could not have been written. I would also like
to thank Dr. William Sharpe, Jr. whose patience and
direction were tremendous aids in getting me to the level
of understanding which was necessary to perform the

research in this paper.

iii

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . vi
LIST OF FIGURES. . . . . . . . . . . . . Vii
INTRODUCTION. . . . . . . . . . . . . . l
ELEMENTARY SINGLE MASS VIBRATION THEORY. . . . . 4
EXPERIMENTAL INVESTIGATION OF CUSHIONING MATERIALS . 8
Static Spring Constants . . . . . . . . . 8
Single Mass Natural Frequency as a Function
of Static Loading. . . . . . . . . . . 9
Single Mass Natural Frequency as a Function
of Input Acceleration . . . . . . . . . 13
DISCUSSION OF RESULTS FROM CHAPTER III . . . . . 17
TWO MASS SYSTEM. . . . . . . . . . . . . 20
Elementary Theory . . . . . . . . . . . 20
Experimental Procedure. . . . . . . . . . 22
Results. 0 O O O O O O O O O O O O O 22
Discussion. . . . . . . . . . . . . . 22
FOUR MASS SYSTEM . . . . . . . . . . . . 25
Elementary Theory . . . . . . . . . . . 25
Experimental Procedure. . . . . . . . . . 25
Results. I O O O O O O O O O O O O O 27
Discussion. . . . . . . . . . . . . . 27
CONCLUSIONS . . . . . . . . . . . . . . 29
BIBLIOGRAPHY. . . . . . . . . . . . . . 30

iv

Page

APPENDICES

Appendix A-l . . . . . . . . . . . . . . 31
Appendix A-2 . . . . . . . . . . . . . . 32
Appendix B. . . . . . . . . . . . . . . 33

LIST OF TABLES

Page

Laboratory Spring Constants and Theoretical
Natural Frequencies . . . . . . . . . 11
Natural Frequency of a Single Mass. . . . . 15
Input Acceleration vs. Natural Frequency. . . 16
Data From the Two Mass System . . . . . . 23
Data From the Four Mass System . . . . . . 28

vi

LIST OF FIGURES

Figure Page
1. Single Degree of Freedom System . . . . . 5
2. Sample of a Static Load Deflection Curve . . 10
3. Plate Illustrating the One Mass Model

Mounted on the Table . . . . . . . . l3
4. Acceleration Time Plot From Brush Recorder . 14
5. Two Degree of Freedom System . . . . . . 21
6. Four Degree of Freedom System. . . . . . 26

vii

INTRODUCTION

In transit packages are stacked in various multi-
ples to efficiently use space. Vibrations are transmitted
through the floor of the truck or rail car and are experi-
enced by the units because they are conveyed through the
stack. Each container's response to vibration input will
differ due to the influence of the other stack members.
Since the stacked configuration has a marked influence,
it is important to study its effects and determine if they
are predictable.

Little effort has been spent in trying to apply
elementary vibration theory to stacked packages. The
application, if successfully made, would provide the
designer an invaluable tool.

The purpose of this paper is to determine if a
simple linear undamped model could be used to describe
stacked packages. A more complicated model with damping
and nonlinear springs would be expected to be more
accurate, but the object is to see how useful a simple

model could be.

There are two reasons why simple vibration theory
might not work. First, the package stack might be too
complicated to treat as a discrete mass system with only a
few elements and second, the typical packaging material
may not behave as a linear undamped spring. This investi-
gation will attempt to isolate the second premise and avoid
the first by using a simple model made up of steel plates
to simulate package mass.

Dow Ethafoam* 220 and 275 (mullen test in lbs)
c-flute corrugated fiberboard were the materials proposed
for the simple model. They were tested individually, while
bonded together, and under various loadings, to simulate
real conditions. The high and low loadings for the
polyethylene cushion were taken from a static-stress-peak—
deceleration curve1 and a resonant frequency curve2
respectively. The high loading gives optimal cushion
performance and the low gives a measure of predictability
to the natural frequency.

The bulk of the research involves the determi-
nation of the error introduced by treating the materials
as linear undamped systems. Linearity allows ready
prediction of performance and this allows for great

savings in time and money.

 

*Registered Trademark, the Dow Chemical Company.

The body of the thesis is broken up into six parts.
The first discusses single mass vibration theory. The
second is composed of three sections: the first section
deals with measuring static-spring constants; the second
contains an investigation of single mass resonant frequency
as a function of static loading; the third section has a
study of resonant frequency as a function of input
-acceleration. The third part is a discussion of the
results of the second. The fourth part is an analysis of
a stack of two packages. The fifth is an analysis of a

stack of four packages. The last contains the conclusions.

ELEMENTARY SINGLE MASS VIBRATION THEORY

The subject of vibration deals with the oscillatory
motion of dynamic systems. A dynamic system is a
combination of matter which possess mass and whose
parts are capable of relative motion. All bodies
possessing mass and elasticity are capable of vibration.
The mass is inherent in the body, and the elasticity
is due to the relative motion of the parts of the
body.

The objective of the designer is to control or
minimize the vibration when it is objectionable and to
utilize and enhance the vibration when it is desirable.3

The elements that constitute a vibratory system

are illustrated in Figure 1. The mass is assumed to be a
rigid body. The spring element is elastic and is assumed

to be of negligible mass. A linear spring is one that

obeys Hooke's law, that is, the spring force is proportional
to the spring deformation. The constant of proportionality,
measured in force per unit deformation, is called the

spring constant (K). The damping element has neither mass
nor elasticity. Damping force is proportional to the
relative velocity between the two ends of the damper.

The work energy or input into a damper is non-

conservative. Energy enters a system through the appli-

cation of an excitation force to the system.

 

MASS

 

 

 

SPRING CONSTANT (K)

 

 

 

It::JI DAMPER

 

C

EXCITATION FORCE

F(t)

 

\\\\\\\\\\\

Figure 1. Single degree of freedom system.

The focal values are the mass and the spring
constant. The excitation force is discussed later and the
damping factor is neglected completely. Damping does exist
in all physical systems to some extent, but the purpose is
to compare linear undamped systems with actual stacks of
packages.

Natural frequency is a property of a dynamic
system. The natural frequency is determined by the mass

and spring constant. This value is calculated from the

f=_l_vé;:
n 2n mass

The natural frequency will be in cycles per second (Hz).

following equation:

The mass is determined by dividing the weight in pounds by
386, the acceleration of gravity in inches/second. The
spring constant is determined by taking the slope of the
static load deflection curve for that material. If the
graph is curvilinear, then the spring constant is not a
constant and varies as a function of the load on the
cushion. The system is no longer simple and prediction
from the simple theory is impossible.

"When the frequency of the exciting force coincides
with one of the natural frequencies of the system, a
condition of resonance is encountered, and dangerously

large amplitudes may result."4

If the natural frequency of the system is known,
the resonance condition maybe avoided through design.
This is why natural frequency is so important. Tremendous
savings could be realized if the simple equation, previ-
ously mentioned, could be used in place of tedious

laboratory testing.

EXPERIMENTAL INVESTIGATION OF

CUSHIONING MATERIALS

Static Spring Constants

In the preceding chapter the formula for natural
frequency is presented. The spring constant (one of the
independent variables, the other being the mass) is
important since each product's mass is virtually fixed.

By treating the mass as a constant, the natural frequency
varies as a function of the spring constant. This section
is an experimental determination of the spring constant,
with an application of the formula to demonstrate its
effects on natural frequency.

The specimens were cut to proper dimensions,

1.5" x 1.5" x l" and 3" x 3" x l", to yield 2.0 and 0.5 psi
loading respectively for an 18 pound mass. The corrugated
fiberboard was only tested at the 0.5 psi loading.

An Instron laboratory testing maching, model TT—B,
was used to test the samples. The test procedure used is
similar to ASTM D 1225-66 (flat crush of corrugated board).
The corrugated fiberboard was tested two sheets at a time
with the flutes running perpendicular to simulate their

arrangement in the flaps of a box.

The testing procedure produces load deflection
plots such as the one in Figure 2. Since the masses weighed
18 pounds and 4 cushions would be used for each mass, each
cushion would bear 4.5 pounds. From a tangent drawn at
this point, the spring constant can be calculated by
taking the slope of the tangent.

With the natural frequency equation, a theoretical
value may be determined by substituting for the mass and
spring constant. The results of the tests and calculations
are presented in Table 1.

Single Mass Natural Frequency_as a
Function of’Static Loading

This part deals with the laboratory determination
of the natural frequency of a system. The product, while
mounted on the cushions, is subjected to a vibration input
through a range of 0-50 Hz. The acceleration of the mass
is monitored to locate the resonance point, which is
defined to be the point at which the frequency of the
vibrating table is equal to natural frequency of the
package.

The cushions were tested at four loadings. Along
with the polyethylene cushion and corrugated fiberboard,

a combination of the two was tested to simulate actual

package conditions.

10

Load (lbs)

I Spring Constant =

Load

Deflection = lb/in

 

__._._. 4.5
Tangent
Point

I
|
I
I
lLoad

--.-u-—---A

Deflection

 

 

Deflection (inches)

Figure 2. Sample of a Static Load Deflection Curve.
Material: Ethafoama 220; Size: 1.5" x 1.5" x
l".

aRegistered Trademark, The Dow Company.

11

TABLE 1.--Laboratory spring constants and theoretical natural

frequencies.

 

Ethafoama 0.5 psi

Ethafoama 2.0 psi

 

 

 

 

 

 

 

Crosshead
Test No. Speed Spring Natural Spring Natural
(in/min) Constant Frequency Constant Frequency
(lb/in) (Hz) (lb/in) (Hz)
1 0.5 240.1 22.9 133.3 17
2 0.5 201.6 20.9 123.1 16.4
3 0.5 246.5 23.2 123.3 16.4
1 0.05 237.5 22.7 114.6 15.8
2 0.05 194.4 20.6 . .b . .
3 0.05 250 23.3 . .b . .
1 0.5 271 24.3 126.5 16.6
2 0.5 255 23.6 98 14.6
3 0.5 225 22.1 131.8 16.9
1 5 233 22.5 130.8 16.9
2 5 305 25.8 137 17.3
3 5 277 24.5 145 17.8
Crosshead Corrugated Fiberboard 0.5 psi
Test Speed . .
(in/min) Spring Natural
Constant Frequency Load Range
(lb/in) (Hz) (lbs)
1 0.1 176.5 19.6 200
1 0.1 205 21.1 100
1 0.1 152.5 18.2 50
l 0.1 196 20.6 50

 

bTest was not repeated due to length.

aRegistered Trademark, The Dow Company.

12

A hydraulic shaker was used to vibrate the model
through the frequency range. The cushions were placed
under the corners of the mass as seen in Figure 3. A small
amount of adhesive was used to prevent slippage and
possible damage. The equipment was then calibrated and
set up (see Appendix A-l).

Figure 4 is a plot which was used to calculate
natural frequency (see Appendix A-l). The sample calcu-
lations show how the natural frequency was determined.
Each loading for each material (polyethylene foam,
corrugated fiberboard, combination of first two) was
tested at least twice and the results are presented in
Table 2.

Single Mass Natural Frequency as a
Function of Input Acceleration

In trying to design a test that was repeatable,
the possibility of keeping displacement constant was
investigated. It was found that natural frequency varied
from run to run. By keeping the input excitation force
constant the results became very consistent.

To investigate what might actually happen if the
input changed, the input was varied and its effects on
natural frequency were observed. All three materials were
tested but at only one loading, 0.5 psi. The results are

presented in Table 3.

l3

 

 

 

Figure 3. Plate illustrating the one mass model mounted
on the table.

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15

TABLE 2.--Natura1 frequency of a single mass.

 

 

 

 

Material
Loading Ethafoama Ethafoama
Corrugated Corrugated
Fiberboard Fiberboard
0.5 psi 39.4 H 42.2 H 19.4 H
z z z
0.5 psi 39.0 H 38.5 H 26.6 H
z z z
1.0 psi 32.3 29.8 25.6 Hz
1.0 psi 32.1 32.4 23.0 Hz
1.5 psi 30.1 29.9 24.9 Hz
1.5 psi 30.6 26.4 22.1 Hz
2.0 psi 25.1 20.8 20.4 Hz
2.0 psi 25.2 18.5 20.1 Hz
aRegistered Trademark, The Dow Company.

16

 

 

 

 

 

TABLE 3.--Input acceleration vs. natural frequency.
Corrugated
Ethafoama Fiberboard Combination
Input (9.5) Natural Natural Natural
Frequency Frequency Frequency
(Hz) (Hz) (Hz)
0.071 38.5 17.5 47.5
0.143 37.5 17.5 47.5
0.214 36.5 17.5 45
0.286 34 17.5 43.5
0.357 32.5 17.5 42.5
0.428 31 17.5 40
0.50 30 17.5 36
0.571 31 16 36

 

aRegistered Trademark, The Dow Company.

DISCUSSION OF RESULTS FROM CHAPTER III

In comparing the results of the theoretical natural
frequency calculated from the spring constants and the
laboratory results for the polyethylene cushion loaded at
0.5 psi, there is no agreement. The lab value is almost
twice that of the calculated natural frequency. When the
cushion is loaded to 2.0 psi the natural frequency values
are much closer. The difference is approximately 28 per-
cent. This does not allow for a reasonable amount of
predictability, but it does lean closer to the theory.

It can be concluded that as you go to the higher loading,
towards the flat part of the resonant frequency curves and
away from the optimal range on the deceleration curves,
the laboratory values agree more with the theoretical
natural frequency.

On the other hand, the corrugated fiberboard
material appeared to be very predictable at all loadings
tested. Laboratory values when compared to the theoretical
results, appeared to show little difference. The greatest
difference was when the cushion was loaded to 0.5 psi.

The corrugated fiberboard, for the loadings tested,

17

18

appears to be a linear spring, which should allow for a
great deal of predictability.

In analyzing the spring constants alone, the
polyethylene foam loaded to 0.5 psi had the largest
values. The corrugated pads followed next with the
polyethelene cushions loaded at 2.0 psi being last. In
looking at the effects of changing the crosshead speed,
the differences were negligible.

The corrugated board acts like a linear spring
making the natural frequency very predictable. For the
loadings which were tested, the amount of loading on the
corrugated pads has no effect on the natural frequency.
For the foam alone, there is a shift of 15 Hz from the low
loading with a high natural frequency of 39.0 Hz to the
higher loading with the low natural frequency of 25.0 Hz.
In the combination the polyethylene foam dominates at the
lower loading, 0.5 and 1.0 psi, while the corrugated
dominates at the 1.5 and 2.0 psi loadings. By "dominating"
it is meant that the natural frequency of the combination
is similar to the natural frequency of the material which
dominates it.

The final area of discussion is that of input
acceleration. Tests run on the corrugated material show
the natural frequency was not affected by input acceler-
ation. The polyethylene foam loaded to 0.5 psi (as was

the corrugated fiberboard), showed that by changing input

l9

acceleration the natural frequency could be moved. By
moving from a low of 0.071 g. to a high of 0.571 g. the
natural frequency could be lowered by 7.5 Hz' Thus, the
input acceleration is very critical in determining natural
frequency for non-linear materials.

While this simple model is not adequate, obviously
a model could be developed, but this work indicates that
it would have to include damping, non-linear effects or
other considerations that would make it very complicated,

and it is this complexity which has caused restricted use.

TWO MASS SYSTEM

Elementary_Theo£y

 

The degrees of freedom of a system are equal to
the number of independent coordinates necessary to
describe the motion of the system. A system with n
degrees of freedom will have n natural frequencies
which characterize the behavior of the system.5

Any system which requires two independent co-
ordinates to describe its motion is a two-degrees-of—
freedom system, as illustrated in Figure 5 The spring
constant (K) is the value determined in Chapter III,
multiplied times 4 (4 cushions). The spring constant (K)
will be half the value of spring constant (K1) because each
cushion is 2" thick. The double thickness results in a
softer cushion. This configuration was used to simulate
the arrangement of one package on top of another.

This system will have two natural frequencies and
when it oscillates at one of them it is said to be
vibrating at a principal mode.

The system does not really appear to be much more
involved but its complexity can be viewed in the equations.

The following equation is used to find the angular natural

frequencies (K2 for this system is zero).

20

21

 

MASS

 

 

 

SPRING CONSTANT (K)

 

MASS

 

 

 

SPRING CONSTANT (Kl)

 

\\\\\\\\\\\\

Table

Figure 5. Two degree of freedom system.

22

 

Experimental Procedure

 

Another accelerometer was used so that both masses
could be monitored. Previous procedures were followed

with the additional equipment (see Appendix A—l).‘

Results
The phasing and amplitude ratios were determined
from pictures (see Appendix A-2). The theoretical phasing
and amplitude ratios were determined from a computer
program (see Appendix B). The data along with its

theoretical counterpart is presented in Table 4.

Discussion

 

There is very little agreement between laboratory
data and the theoretical values. In comparing the natural
frequencies, the lower frequencies agree with their
theoretical counterparts while only the second frequency
of the polyethylene cushion loaded to 0.5 psi agrees with
its theoretical value. The first natural frequency could
be predicted for all three cases and the generalization
that all first natural frequencies may be calculated for
all loadings of the foam and corrugated fiberboard could
be made.

The amplitude ratios show no agreement.

 

23

 

 

 

 

 

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24

In checking the phasing for the first natural
frequency, there is total agreement. For the second
natural frequency, only the polyethylene foam loaded to

0.5 psi agrees.

FOUR MASS SYSTEM

Elementary Theory

 

There is no real way to sufficiently present
vibration theory for multiple-degree of freedom systems
in a few lines. The step from one mass to two masses is
complicated, but the step from two masses to four masses
is infinitely more so.

Figure 6 is a four mass system. Two masses and
springs have been added on the same manner as in Chapter V.
The values of K2 - K4 are the same as the spring constant
in Chapter V. The solution for this system involves a
series of matrices, which are solved though a computer
program. This program is on file at Michigan State Uni-
versity (see Appendix B).

Like the two mass system, there are three areas of

comparison: natural frequency, amplitude ratio, and

phasing.

Experimental Procedure
The equipment used was the same as in Chapter V.
Since only two accelerometers were used, their positions

were shifted to get the reaction of all four masses.

25

26

 

MASS

 

 

 

 

 

 

 

 

 

 

 

 

4
:E%: SPRING CONSTANT (K4)
MASS 3
$ SPRING CONSTANT (K3)
MASS 2
::%:: SPRING CONSTANT (K2)
MASS 1

 

 

 

$ SPRING CONSTANT (Kl)

\\\\\\\K\\\\

Table

Figure 6. Four degree of freedom system.

27

Results
The data is collected in the same manner as was
previously stated. The results of the tests along with

its theoretical counterpart is presented in Table 5.

Discussion

 

There were four natural frequencies found, which
shows some agreement with theory, but the agreement ends
there. The frequencies do increase as do the theoretical
ones, but actual values show no agreement.

The amplitude ratios show no agreement.

The phasing shows occasional agreement but there

is not enough consistency to allow for any predictions.

28

TABLE 5.--Data from the four mass system.

 

3"x3"xl" Ethafoama 0.5 psi

 

 

 

 

 

Natural Natural Natural Natural
Frequency 1 Frequency 2 Frequency 3 Frequency 4
b c
L.R. T.R. L.R. T.R. L.R. T.R. L.R. T.R.

 

5.6 H 16.6 H 14.4 H 19.6 H 24 H 29.4 H 48 H 83.6 H
2 Z Z Z Z Z Z Z

 

 

Amplitude Ratios

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mass 1 Mass 2 Mass 3 Mass 4
L.R. T.R. L.R. T.R. L.R. T.R. L.R. T.R.
Mass 1 . . . . 3.86 1.766 .274 1.18 1.50 .199
Mass 2 .599 .351 . . . . .336 5.029 2.58 .234
Mass 3 .599 .234 1.39 5.029 . . . . 1.28 .351
Mass 4 .399 .199 .819 1.180 .150 1.766 . . . .
Phasing
Mass 1 Mass 2 Mass 3 Mass 4
L.R. T.R. L.R. T.R. L.R. T.R. L.R. T.R.
Mass 1 . . . . IN IN OUT OUT OUT OUT
Mass 2 OUT IN . . . . IN OUT IN IN
Mass 3 OUT IN OUT IN . . . . IN OUT
Mass 4 OUT IN OUT OUT OUT OUT . . . .

a .
Registered Trademark, The Dow Company.

b
L.R. Laboratory results.

c .
T.R. Theoretical results.

CONCLUSIONS

The data in general does not allow for any predi-
cability from the theoretical calculations. The differences
are wide and too frequent to put complete faith in the
theory. Yet, the few instances of agreement are not
chance occurrences. They are real and give a certain
amount of credibility to the analysis. It was found that
as the investigation became more complex, the disparity
between values increased till they possessed no similarity.

Although there is little proof, the theoretical
approach is still a valid plan of attack. What has been
shown is that both materials defy analysis through ele-
mentary theory. If the effective spring constant is
treated as a non-linear function and damping is also
introduced, there could be an aligning of the laboratory

and theoretical values.

29

BIBLIOGRAPHY

1.

BIBLIOGRAPHY

Howard C. Blake, III. Acceleration-Stress Properties
of Selected Cushioning_Materials. School of
Packaging, Micfiigan State University,

January 1, 1962, Fig. 5.

 

Stephen R. Pierce and John Wambaugh. Vibration
Transmissibilityjof Resilient Package Cushion-
inngaterials. Sohool of Packaging, Michigan
State University, January 15, 1973, p. 8.

 

Francis S. Tse, Ivan E. Morse, and Rolland T. Hinkle.
Mechanical Vibrations. Boston: Allyn and
Bacon, Inc., August, 1964, p. 3.

 

William T. Thompson. Vibration Theory and Applications.
Englewood Cliffs, N.J.: Prentice-Hall, Inc.
1965, p. 2.

Ibid., p. 159.

30

APPENDICES

APPENDIX A-l--A-2

APPENDIX A-l

An Endevco accelerometer no. 2265-20 was mounted
to the steel plate to measure the acceleration of the
mass. The accelerometer output was fed into an Endevco
signal conditioner, no. 4470 and calibrated to 625 mv/g.

A Krohn-Hite filter no. 3750 was used to filter the output
of the signal conditioner on a low pass of 60 Hz' The
output of the filter was fed onto a Gould 280 brush
recorder which was set at a speed of 2 mm/sec and a
sensitivity of 50 mv/line. These charts are then used to
calculate natural frequency (see Figure 4).

To keep the input constant, a Kistler accelero-
meter no. 818, calibrated for 10.4 mv/g., was mounted on
the table. The accelerometer was monitored on a Bruel and
Kjaer RMS meter, no. 2417 and kept at a constant input of
0.0015 v RMS (0.21 g.). The input was kept constant to
allow the easy comparison of results.

To aid in the accurate location of the natural
frequency a Quan-Tech Wave Analyzer no. 304T was used to
sweep through the frequency range of 0-50 Hz in either 50

or 500 seconds (both time intervals produced comparable

31

32

results). By starting the analyzer and simultaneously
marking the start on the brush recorder, the point of

resonance was very easily determined.

APPENDIX A-2

Along with the additional equipment for the two
mass tests, a Tektronix no. 502 oscilloscope and camera
were used to determine phasing and amplitude ratios.

Pictures were used to provide permanent records for

future use.

APPENDIX B

APPENDIX B

The mass and stiffness matrices were computed for
the two and four mass models, through a computer program.
The program name is EIGEN, NROOT and is on file at
Michigan State University. This program calculated the
theoretical values for the natural frequencies, mode

shapes (phasing), and amplitude ratios.

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