1H1; UAhflHNKJ \AE’P‘MH l VI wwluuvvo nu..—

PAPERBOARD AS A FUNCTION OF
STRESS AND FREQUENCY

 

Thesis {or the Degree of DH. D.
MICHEGM STATE UNIVERSITY
James. Winthmp Cmff
1957

' W/Ifif/W/Yflfili/Mi/7/fW]HEWWEIR/WM

3 1293 105 4 1639

fig, .

This is to certify that the

thesis entitled

The Damping Capacity of Corrugated
Paperboard as a Function of Stress and
Frequency

presented by

James W. Goff

has been accepted towards fulfillment
of the requirements for

_Eh._D.._degree in WQOd Technology

Ears/W

Majg professor

Date May 10, 1957

 

0-169

 

 

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THE DAMPIWJ CAPACITY OF CORRUGATED PAPERBOARD
AS A FUNCTION OF STRESS AND FREQUENCY

By
JAMES WINTHROP GOFF

AN AESTRACT

Submitted to the School of Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Forest Products

1957

Approved: flag; / flM

ABSTRACT 1

This study was undertaken to determine the damp-
ing capacity of corrugated paperboard and to ascertain
the effect of stress amplitude, static load and fre-
quency upon that property of the material.

Two methods, utilizing circular Specimens similar
to those used for the standard flat-crush tests of
corrugated board, were developed. The first method,
which produced load deflection diagrams for complete
cycles at rates up £5 one-quarter cycle per second,
made use of a Baldwin FGT-SR-# universal testing mach-
ine equipped with a low range load cell and cycling
controls. The ranges of loading used were 5 to 35
pounds, 35 to 65 pounds, and 5 to 65 pounds on a Spec-
imen which was two and one-half inches in diameter.

A Special cyclic loading fixture was designed for
use in the second method which involved loading in the
same ranges, but at frequencies of five and seven and
one-half cycles per second.

.Thetnaterial investigated was.A-flute corrugated
board, manufactured with two #2 pound kraft liners and
a semi-chemical corrugating medium. All board used was
supplied by the.American Boxboard Company of Grand
Rapids, Michigan. Specimens were cut, conditioned at
73 degrees Fahrenheit and 50 percent relative humidity

for one week, selected for thickness, and samples of

ABSTRACT 2
five were drawn at random. Each sample of five was
tested at a single frequency and within a single load-
ing range.

The total energy and energy dissipation per cycle
were measured for each load deflection curve using a
polar planimeter. The damping capacity and actual
energy loss per cycle were calculated from these data.

Test results indicated that the damping capacity
of the board under investigation lay in the range of
0.009 to 0.026 as determined at the 100th cycle of
loading. The damping capacity was found to decrease
with increased duration of cyclic loading.

The damping capacity and the actual energy loss
per cycle appeared to be somewhat dependent on the fre-
quency of vibration but no definite conclusions were
drawn because of the lack of continuity of data in the

overall frequency range studied.

THE DAMPING CAPACITY OF CORRUGATED PAPERBOARD
AS A FUNCTION OF STRESS AND FREQUENCY

By
' JAMES WINTHROP GOFF

.A THESIS

Submitted to the School for Advanced Graduate Studies of
Michigan State University of Agriculture and
‘ Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Forest Products

1957

LIBRARY

Michigan State
University

 

ii

ACKNOWLEDGEMENTS‘

The writer wishes to express his sincere ap-
preciation for the help and guidance given in the
pursuit of this investigation by Dr. A..I. Panshin,
Head of the Department of Forest Products and Dr.
Charles 0. Harris, Head of the Department of Applied
Mechanics at Michigan State University.

' The generosity of the American Boxboard Company
of Grand Rapids in providing the materials used in the
eXperimental phase of the work is also genuinely ap-

preciated.

111

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . ii
LISTOFTABLES................. iv

LIST OF FIGURES . . . . . . . . . . . . . . . . v
I. INTRODUCTION . . . . . . . . . 1
II. PREVIOUS WORK . . . . . . . . . . . . . . 3

III. EXPERIMENTAL PROCEDURE . . . . . 6
Iv. THEORETICAL ANALYSIS OE THE PROBLEM . . . 17
v. ANALYSISOFTESTDATA 21_
VI. CONCLUSIONS . . . .-. . . . . . . . . . . -32

LIST OF REFERENCES . . . . . . . . . . . . . . . 3n

.APPENDIX.A. NOTES ON EQUIPMENT AND INSTRUMENTA-

TION . . . .... . . . . . . . . 36

APPENDIX B. OPERATIONAL TECHNIQUES . . . . . . 47

Table

I.
II.
III.
IV.

5‘

VII.

Test Data
Test Data
Test Data
Test Data

Test Data

LIST OF TABLES

Method A
Method A
Method A
Method B
Method B

Mean Energy Loss and

for Selected Cycles . . . .

-_l/8 cps — 100th cycle
1/6 cps - 100th cycle

- l/fl cps - 100th cycle
— 5 cps - 100th cycle
- 7.5 cps - 100th cycle

Mean Damping Capacity

Effect of Extended Number of Leading Cycles

at 7.5 cps and 5-65 Pound Range . . .

iv

Page

22

23
24

25

26

28

29

LIST OF FIGURES

Figure

1.
20

9.
10.

Dead Weight Board Micrometer .

Baldwin, 50,000 Pound, FGT-SR—Q, Universal
Testing Machine Equipped with an Auxiliary
500 Pound Load Cell, Load Cycling Controls,
and a Differential Transformer Type.Auto-
graphic Recorder as Used in Test Method.A.

Arrangement of Apparatus on Testing Machine

Platen for Test Method.A .

Loading Mechanism, Load Cell, and Deflect-

ometer Arrangement Used in Test Method B

Instrumentation Used in Method B .

Schematic Cross-Section,.AqFlute Corru-

gated Board . . . . . .

Free Body Diagram of a Flute Quadrant
Load Deflection Curves for a Single Specimen
with 3 Load Range of 5-75 Pounds and a De-

flection of 0.0088 inches at a Frequency of

705 cps o o o o o o o 0
Loading Platen . . . . . .

Baldwin Model PD-lM Deflectometer as Modified

for Use in Test Method A

?

IO

11

13
l6

17
18

30
37

38

vi

Figure Page
11. Deflectometer Showing the Modifying Parts

Detached . . . . . . . . . . . . . . . . . . . 39
12. Loading Device for Test Method B Showing

V—Belt Drive from the Variable Speed

Motor on the Baldwin Testing Machine . . . . . 41
13. Resistance Wire Strain Gage Deflecto-

meter . . . . . . . . . . . . . . . . . . . . 42
1%. Load Deflection Curves for the Method B

Deflectometer without a Specimen . . . . . . . #3
15. Schematic Diagrams for Phase Shifter and

Switching System . . . . . . . . . . . . . . . 44
16. Load Deflection Curves and Corresponding

Phase Shift Traces . . . . . . . . . . . . . . #5

I. INTRODUCTION

The damping capacity of a material is defined as
the ratio of the energy loss or dissipation per loading
cycle to the total vibrational energy. Damping capacity
is an important dynamic property of materials used for
shock and vibration isolation in packaging and other
areas. In packaging materials damping serves not only
the function of reducing the amplitude of motion of a
cushioned article at near-resonant frequencies, but is
also important in that it may affect the magnitude of
shock transmitted to the cushioned article as a result
of instantaneous changes in velocity. For isolating
near-resonant frequencies of vibration it would be de-
sireable to have a maximum amount of damping, but from
the standpoint of shock isolation too much damping can
greatly increase the peak acceleration imposed on the
article (12).

Corrugated paperboard in its several forms has
long been used as a package cushioning material and it
seems strange that only occasional mention is made of
its damping characteristics (13).

This investigation was concerned with the de-
ve10pment of methods for measuring and the measurement

of the damping capacity of corrugated paperboard, with

the variation in damping as a function of frequency,
and with the variation in damping as a function of
stress. Test frequencies studied were purposely se-
lected to represent those possible of attainment on
available cyclic testing equipment (one-eighth, one-
sixth, and one-quarter cycle per second) and those
‘which represent the continuous vibration Spectrum im-
posed on packaged goods in rail-freight movement (five
and seven and one-half cycles per second) (5). Three
loading ranges were used as follows: five to sixty-
five pounds, five to thirty-five pounds, and thirty-
five to sixty-five pounds. The maximum stress in any
range was well below that which would cause the first
flat crush failure, consisting of buckling in the corru-
gating medium adjacent to the single-face liner, to
occur. .In.genera1, no measurable permanent deflection
of the board was anticipated, and the loads were con-
fined to those which might reasonably be expected to
occur in use under normal handling.

The material used for'the investigation was A-
flute board manufactured with two 42 pound kraft paper-
board liners and an aSpen semi-chemical corrugating med-
ium. The board was fabricated by the American Boxboard
Company in its Grand Rapids, Michigan plant.

II. PREVIOUS WORK

Methods 2;.measurigg dampigg. - The most direct
method of measuring the damping capacity of a material
involves the measurement of the total energy and the
energy loss for a given cycle. This method has been
described by von Heydekampf (18) who explained its use
in connection with measurements of damping in metals at
a maximum cycling rate of five cycles per minute.
Bishop (2) treated the method mathematically for the
hysteretic damping of an anchored Spring but did not
verify his treatment eXperimentally. No original re-
port of actual damping measurements conducted by this
method could be found. Apparently the difficulty in
obtaining accurate load and deflection measurements at
the rates of cycling involved had precluded the use of
this procedure.

Damping capacity is approximately equal to twice
the logarithmic decrement of decay of a damped free
vibration when the logarthmic decrement is small, as
shown by Zener (21). Kimball (7) describes the mathe-
matical procedure involved in the determination of the
logarithmic decrement, and Fusfeld (h), Pattison (1“)
and Smith (16) have deve10ped rapid and accurate elec-

tronic instruments for determining this quantity.

Another method which has been widely used in-
volves the measurement of the magnification factor Q
'which can be obtained from the band width of the fre-
quency-amplitude curve between the half-resonant-ampli-
tude values on either side of point of resonance.

Zener (21), Wegel and Walther (l9), Kruger and Rohloff
(9), Horio and Onogi (6), and Save and Perrin (15) have
made use of this method with various materials.

A fourth method proposed and verified experiment-
ally for a single material by Weissman, Fee, and Marin
(20) involves the prediction of damping from static
creep data for the material in question.

Dampigg measurements 9_r_1_ 322$ _a_r_1g_ m.- Kruger
and Rohloff (9) using the method involving the measure-
ment of the quantity Q found damping capacities for
'wood in bending parallel to the grain varying from
0.050 to 0.07u depending on species. Damping capacities
perpendicular to the grain for the same group of Species
varied from 0.1%0 to 0.23. These investigators also
found that damping capacity was dependent on the ampli-
tude of vibration and independent of the frequency of
vibration when measured in flexure between 10 and 700
cycles per second. Barducci and Pasqualini (l) have
also investigated the damping capacity of wood and
their results agree substantially with those of Kruger
and Rohloff.

Horio and Onogi (6) have conducted damping meas-
urements on wrapping paper using the resonance method
involving the measurement of Q.and report a value for
the tangent of the mechanical loss angle of approximat-
ely 0.047. Zener (21) defines the mechanical loss angle
as the angle by which deformation lags behind applied
force in a periodic oscillation and Meredith (11) gives
the proportionality factor between the tangent of this
angle and the damping capacity as 6.28 or er , result-
ing in a damping capacity for the wrapping paper mention-
ed above of 0.30. These investigators also found that
the damping capacity was essentially independent of the

frequency of vibration.

III. EXPERIMENTAL PROCEDURE

Sampligg and conditionigg.- Specimens were cut

from two sheets of board representing the same corru-
gator run and with identical liners and corrugating med-
ium. The specimen cutting tool, which produced two and
one-half inch diameter circular sections of board, was
made according to specifications furnished by Hinde and
Dauch, Inc. of Sandusky, Ohio.

The Specimens were coded so that the location of
the sample in the board from which it was cut could be
later determined if necessary for prOper analysis of the
test results. A series of four numbers was used: (1)
the first digit represented the board sampled, (2) the
second digit represented the a length of sixteen inches
in the direction of the flutes, (3) the third number,
one through eleven, represented a length of five inches
in the machine direction, (k) the fourth digit repre-
sented the Specimen number within each five by sixteen
inch piece of board. The extensive coding system was
thought desirable in order to reduce the board to
sections small enough to handle in the actual specimen
cutting Operation.

The Specimens were conditioned for at least one

'week following the cutting at seventy-three degrees

Fahrenheit and fifty per-cent relative humidity. Select-
ed samples from the group were subsequently oven-dried
and found to be at an average moisture content of eight
and two-tenths per-cent on the basis of the dry weight
of the board.

Moisture content of the Specimens was maintained
throughout the testing program within one-tenth of one
per-cent through the use of a moisture vapor proof bag
and a moisture barrier surrounding the platens for pro-
tecting the specimens en-route to and in the testing
machine. A number of preliminary tests conducted during
a time when the atmOSphere surrounding the machine was
extremely dry indicated that this procedure was highly
effective in maintaining a uniform moisture content in
the material under test.

Following the conditioning period all Specimens
were checked for overall caliper (thickness) by means of
a deadaweight board micrometer as shown in Figure 1.
Random samples of five Specimens each.were selected from
those measuring 0.217 and 0.218 inches. These two thick-
ness groups were the most highly pOpulated in the thick-
ness distribution and the mean thickness lay in the in-
‘terval between the two values. A total of twenty such
samples were drawn for the subsequent tests.

Testigg procedures.- Each Specimen was tested in-
dividually in a manner similar to that employed for the

Fig. 1.- Dead weight board micrometer.

 

standard flat crush test of corrugated board. Specially
designed loading platens were used to restrict lateral
movement of the liners and thus reduce the possibility.
of leaning-flute failures inherent in the test method.

A drawing of one of the platens will be found in appen-
dixzh. Two different test procedures were deveIOped and.
used as required to gain the desired information.

T_e_s_t_ method A--' This method involved the use of a
Baldwin, 50,000 pound capacity, FCT-SR-u, universal
testing machine equipped with an auxiliary 500 pound
load cell, load cycling controls, and a differential
transformer type autographic recorder for load and de-
flection. The load was applied through the regular
machine drive and controlled by the cycling system.with-
in the desired limits. The one-hundred pound load range
provided by the load cell was used on all tests.

Deflection of the Specimen was measured by means
of a Baldwin model PD-lM deflectometer modified to pro-
vide a 400 to l magnification when used with the record-
er set on its high range. Details of the modification
are Shown in appendix.d. Figure 2 Shows the machine as
used for the tests, while Figure 3 provides a close-up
view of the arrangement of apparatus on the machine
platen.

The switching and acceleration limitations of

the load mechanism and cycling controls as well as the

10

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12

response limitations of both the load and deflection
recording systems restricted the test frequencies avail-
able with this test procedure to a maximum of one-
quarter cycle per second. In the case of the maximum
loading range used the greatest frequency obtainable
was one-eighth cycle per second.

2§§§_gethod,§.- The frequency limitations of the
first method made it necessary to seek another way of
loading the specimens and of recording their behavior
at higher frequencies. This second procedure utilized
the Baldwin testing machine as a loading frame, as a
load indicator, and as a variable Speed drive for the
loading system. The same platens were used as was the
500 pound load cell.

The lower platen was attached to a reciprocating
mechanism rigidly constrained to move in a vertical di-
rection only. This mechanism provided an adjustable
stroke through the use of a four-jaw lathe chuck which
held the crank pin driving the platen support. The
loading system is shown in Figure h and is more com-
pletely described in Appendix A.

An oscillographic recording system was used in
‘this series of tests to provide the necessary reSponse
<3haracteristics. The output of the load cell was con-
rlected to a Tectronix Type 122 Low Level Preamplifier
13hrough a phase shifting device to the horizontal axis

13

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of a.DuMont Type BOMA cathode—ray oscillograph. De-
flection was measured by a specially built cantilever
beam fitted with resistance strain gages. The output of
this deflection sensing beam was connected to an Ellis
Type BA-Z BridgeaAmplifier and then to the vertical axis
of the oscillograph. A DuMont Type 302 Oscillographic
Record Camera provided a means of recording the patterns
appearing on the face of the instrument.

The phase shifting device mentioned in the pre-
ceding paragraph.was designed to overcome the difficulty
caused by the variable phase shift inherent in both of
the external amplifiers used. In using the phase shift-
er, the signal from the load cell at the desired test
frequency was applied to the input of both amplifiers
through the use of a specially designed switching box.
Since the internal amplifiers provided in the oscillo-
graph were of the direct coupled variety, the resulting
pattern on the oscillograph screen represented the phase
difference due to the external amplifiers. This phase
difference was adjusted to zero by the shifting circuit.
The load signal was again connected to the horizontal
channel only, and the deflection signal was restored to
the vertical axis. The system was then ready to use.

The test frequencies used in this procedure
Were too high to determine by the simple expedient of

tlining a single cycle. For this reason the frequencies

15

were adjusted by the use of an electronic counter. The
output of the load cell as amplified by the preamplifier
was connected to the input of a Hewlett-Packard Model
521A Electronic Counter equipped with a crystal controll-
ed time base. The counter was manually gated for periods
of ten seconds while the motor speed control was adjusted
to give the desired frequency. The Speed dial settings
were noted and recorded for use in the subsequent tests.

All instrumentation is shown in Figure 5 and is
further described in detail in Appendix.A.

gperatiogg; Details.- The actual operating tech-
nique for both test procedures and a description of the
difficulties and shortcomings of the apparatus used are

given in Appendix B.

 

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IV. THEORETICAL ANALYSIS OF THE PROBLEM

A theoretical consideration of the question of

damping in corrugated board can best be approached

,Iasvl

 

 

 

 

 

 

 

Fig. 6. Schematic cross-section, A-flute

corrugated board .

byen analysis of the behavior of the material when it
is subjected to a compressive load. Figure 6 above is

a Iline representation of the cross-section of the.A-flute
board used in this investigation. The dimensions shown
are based on the assumption of 36 flutes per foot, a
°°Prugator flute depth of 0.181 inches and a corrugating
me’<'1:lum.thickness of 0.009 inches.

 

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18

The deflection of such a material under load will
cause either a sidewise collapse of the flutes in a
horizontal direction or a bending of the individual
flutes adjacent to the liners to a sharper radius. The
former cannot occur because each flute tip is fixed to
the board liner by means of an adhesive. As a result
deflection of the board is primarily a function of the
bending of the corrugating medium immediately adjacent
to its point of adhesion to the liners. This-bending
can be better analyzed by taking a single quadrant of
the flute from one of the points A on the mid-axis of
the board to point C where the flute is rigidly bonded
to the liner. A consideration of the moments act‘ing
about point A resulting from a load P applied to the
faces of the section of board in Figure 6.will bring
one to a realization that the bending moment in the
cornlgating medium at this point is zero. As a result

the loads acting on a quadrant are as shown in Figure 7

below.
p

e: 17"
Fig. 7. Free body
diagram of a
flute quadrant. “X— 083‘
p
.——— Sing-fl
geese

   

19

The corrugating medium, adhered to the liner at point
6, can be considered as fixed at this point.

Initially, prior to any appreciable deflection
of the board, the bending moment at point C is given by

the expression

a .2... .. _._....__._P
(1) M (0.083) 2 (0.095)2 0086 sine

As deflection proceeds, the quadrant, fixed at point C
and restrained from any horizontal movement at point A
by the geometry of the situation, rotates about point A.
This action results in a change in the angle 6 and the

bending moment at point C becomes

.. P P -
(2) M - (0.083)—§-— — (0.095)m sin(9 A9)

icesulting in an increase in the bending moment at point
(3. Rotation of the medium about point A also effects a
reduction in the direct compressive stress exerted on
tile portion of the quadrant which becomes more vertical
arui an increase in the direct stress on the portion
which approaches the horizontal. Similarly, the rota-
tidan affects the shear stresses acting in the web.

From the above analysis it can be seen that small
deflections of corrugated board sections bring into play
Pather complex combinations of direct stress, bending
S1:I‘esses, and shear. The damping inherent in the

ma"serial, by definition related to the angle by which

20

the strain in the material differs from the stress app-
lied, can be expected to be a highly complex function.
It would seem, however, that the strains caused by the
bending would be of major importance and for this reason
the damping capacity should approach that of paper ma-

terials in flexure.

21

V. ANALYSIS OF TEST DATA

Areas representing the total energy and energy
loss per cycle were taken by means of a polar plani-
meter from each of the load deflection curves in test
method A. The photOgraphs of the load deflection curves
obtained in test method B were photographically enlarg-
ed and the total energy and energy loss data were ob-
tained in a similar way- The damping capacity was com-
puted directly from the areas involved in each case
While the actual energy loss was calculated from the
hysteresis 100p area using a scale factor determined
from the known loading range and known deflection of
the specimen.

Damping capacities and the actual energy losses
during the 100th loading cycle are shown in Tables I
through V for both methods of testing and for all fre-
quencies and loading ranges. The actual energy losses
Shown are in inch-pounds per two and one-half inch dia-
meter specimen.

Comparison of actual energy loss and damping
capacity with the loading range shows that both are
directly dependent on the amplitude of motion at all

f 1' equenci es .

TABLE I

22

TEST DATA - METHOD A — 1/8 CPS - lOOTH CYCLE

 

 

 

 

 

 

 

 

 

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TEST DATA - METHOD A - l/lb CPS - lOOTH CYCLE

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25

TABLE IV

TEST DATA - METHOD B - 5 CPS - lOOTH CYCLE

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27

The effect of the static load level is shown by
a comparison of the damping capacities and the actual
energy losses in the 5 to 35 pound range with those in
the 35 to 65 pound loading range. In all except one
case the actual energy loss decreases with an increase
in static load, and in every case the damping capacity
decreases with an increase in static load.

Both damping capacity and energy loss per cycle
appear to be somewhat dependent on the frequency of the
dynamic load, although the lack of information on these
quantities in the frequency range ‘1‘ to 5 cycles per sec-
ond precludes any statement regarding this dependence.

Table VI is a compilation of Tables I through III
for the 100th cycle with the addition of similar data
for the 10th cycle of loading with the same specimen.
The damping capacity and the energy loss per cycle are
Shown to be inversely related to the number of cycles of
loading to which they are subjected. Table VII shows an
identical but less pronounced trend for loading at 5 and
7-5 cycles per second.

Figure 8 shows, left to right, load deflection
cuIli‘ves at 1,800, 3,600, 5,400, and 10,800 cycles of
1(Wading for a single specimen with a loading range of
5 to 75 pounds and a deflection of 0.0088 inches at a

Jfallln

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TABLE VII

29

EFFECT OF ...'~"XTEZEDED NUMBER OF LOADII‘IG CYCLES AT

7.5 CPS AND 5—65 POUND RANGE

 

 

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4,500 th
Specimen Pr0perty’ Cycle Cycle
Energy Loss I:
in.lb/spec. .0440 .0379
2315
amping
Capacity .26 .22
Energy Loss
in.lb/Spec.1 .0396 .0363
23102
Damping
Capacity .21 .19
yifiergy Loss
in.lb/spec. .0408 .0408
2414 ‘—
Damping
Capacity .20 .21
Evne rgy Loss
in.lb/spec. .0408 .0354
2225 ‘ *‘
Damping
Capacity .22 .20
line rgy Loss W
in.lb/spec.’ .0374a .0354
2344 *
Damping a
CaPBCity 019 019

 

8These values represent the 1,500th cycle.
Camera difficulties prevented the photographing
of the 75th and 750th cycles.

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31

frequency of 7.5 cycles per second. Little difference
can be discerned in energy loss per cycle between the
first and the last trace, but careful examination of the
coordinates will show that the area under the loading
curve is increasing with each successive trace resulting
in a decrease in damping capacity.

Direct examination of the individual photographs
and of the testing machine records revealed the pres-
ence of creep during the dynamic tests and pointed up
a basic difference between the two test methods. Since
the cycling controls on the testing machine are con-
trolled by the motion of the load indicator, load was
the controlled function. The machine maintained the
load between the prescribed limits regardless of defor-
mation of the Specimen. This resulted in a very slow
continuous motion of the strain recorder, amounting to
as much as 0.0005 inch actual deflection during the
course Of a 100 cycle run.

Test method B, on the other hand, made use of a
lOading device in which the deformation of the specimen
Was fixed by means of an eccentric and a constant de-
formation was applied to the specimen regardless of the
load. The resulting photographs for some Specimens
Show a slight relaxation of the load during the course

of a 500 cycle run of as much as two to three pounds.

 

L'M‘
{‘1

32

VI . COI‘ICLUSI OHS

It can be concluded that damping in corrugated
paperboard is dependent on stress amplitude and is
inversely dependent on the level of static stress.

The exact nature Of the dependence was not determinable
on the basis of the experimental data.

NO statement regarding the relationship between
damping and the frequency of the disturbing force seems

warranted in view of the discontinuity of the data in
terms of the frequency Spectrum and in view of the fact
that different methods were used to obtain the data in
the two frequency ranges studied.

Values of damping capacity were found to range
between 0.13 and 0.20. These results compare favorably
with the value Of 0.30 as found by Horio and Onogi (6)
for wrapping paper, and with those for wood as found by
Kruger and Rohloff (9) when it is considered that the
fibers in paper are oriented in a much more random man-
ner than those in wood.

The investigation shows that for the range of
frequencies studied the damping capacity tends to de-
cSlf’ease with an increase in duration of cyclic loading,

particularly at the lower frequencies. To the extent

33

that the degradation may not be reversible with removal
of the load the reuse of corrugated board and contain-
ers would seem to merit further consideration.

Practical use of the damping capacity values as
determined in this investigation can be made in the,
design of container systems. Crede (3) and Mindlin
(12) have treated this area extensively.

l.

9.

10.

11a

34

LIST OF REFERENCES

Barducci, J. and.G. Pasqualini. Measurements of
internal friction and elastic constants of wood.
Nuovo Cimento, 5:416, 1945.

Bishop, R. E. D. The treatment of damping forces
in vibration theory. Jour. Royal Aero. Soc.
59 (11): 783-742, Nov. 1955.

Grade, C. E. Vibration and Shock Isolation. John
Wiley and Sons, New York, N.Y., 1951.

Fusfeld, H. T. Rapid measurement Of internal
friction. Review of Scientific Instruments 21:
612, 1950.

Guins, S. G. and J. A. Kell. Vibrations in rail-
road freight cars. A. S. T. M. Bulletin (173)
:46 April 1951.

Horio M. and S. Onogi. Dynamic measurement of
physical properties of pulp and paper by audio
frequency sound. Journal of Applied Mechanics
22:971. 1951.

Kimball, A. L. 'Vibra tion prOblems, part IV, fric-
tion and damping in vibration. Journal of App~
lied MechanicsB :A-37, 1941

w. Vibration problems, part V, friction and
damping in vibration. Journal of Applied Mech-
anics :A-l35, 1941.

Kruger, F. and E. Rohloff. Uber die innere Reibung
von Holz. Zeitschrift Physik 110:58, 1938.

Iazan, B. J., J. Brown, .A.. Gansett, P. Kirmer, and
J. Klumpp. Dynamic testing of materials and
structures with a new resonance vibration excit-
er and controller. Proceedings A. S. T. M. 52:
858-876) 1952 e

Meredith, R. (ed.) Mechanical Preperties of Wood
and Paper. Interscience Publishers, Inc., New
York, N.Y., 1953.

12.

13.

14.

15.

l6.

17.

18.

21.

35

Mindlin, R. D. Dynamics of package cushioning.
Bell System Technical Journal 24:353-461, 1945.

Mustin, G. S. .An introduction to shock and vib-
ration for package designers. Proceedings of the
Joint Industry Conference on Cushioning in Pack-
ing, Wayne University, 1953.

Pattison, J. R. An apparatus for the accurate mea-
surement of internal friction. Review of Scien-
tific Instruments 25:490, 1954.

Save R. and H. Perrin. Mesure dynamique du module
d'elastique des papiers. A. T. I. P. Bulletin
No. 1, 1954.

Smith,.A. D. H. .An electronic instrument for mea-
suring the damping capacity of materials. Jour.
Scientific Instruments 28:106, 1951.

Strella, S. Vibrating reed tests for plastics.
A. S. T. :1. Bulletin (214):47-50. 1956.

von Heydekampf,.G. S. Damping capacity of materials
31 2

Proceedings.A. S. T. N. 13157. 1930-

Hegel, R. and H. Walther. Internal dissipation in
solgds for small cyclic strains. Physics 6:141,
193 .

Weissman, G. F., V. H. Pao, and J. Marin. Predic-
tion of creep under fluctuating stress and damp-
ing from creep under constant stress. Proceed-
ings Second U. 8. National Congress of Applied
Mechanics, June 1954.

Zener, C. Elasticity and Anelasticity of Metals.
University of Chicago Press, Chicago, Illinois,
19 8.

36

APPENDIX A. NOTES ON EQUIPMENT AND IE‘ISTRUMEI‘ITATIOI‘I

Loading platen§.- The specimens were loaded by
means of identical upper and lower platens as detailed
in Figure 9. After installation on the testing machine
the platens were checked by means of feeler stock for
parallelism and shimmed as necessary.

T§§E.method.§,- The only Special equipment feature
necessary in this test method was the modification of
the Baldwin PD-lM deflectometer to provide a magnifica-
tion of deformation of 400 from specimen to recorder.
Without such modification the hysteresis loops were too
small for area measurements of any degree of accuracy.
The change made in the instrument consisted of provid-
ing an additional pivot bearing on the crank actuating
the differential transformer and an extension bar to
transfer the bearing point on the movable head to this
new location. Figure 10 shows the instrument with the
modifying extension bar attached while Figure 11 is a
view with the bar removed. The modification in no way
affects the normal Operation of the deflectometer.

leg; methg _B_.- This method required the design
and construction of a Special loading device, a resis-

'tance wire strain gage deflectometer, a phase shifting

37

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Fig. 10. Baldwin Model PD-lM deflectometer
as modified for use in test method A.

 

 

39

Fig. 11. Deflectometer showing the
modifying parts detached.

 

40

circuit and a switching system for making quick checks
on phase relationships.

The loading device was described in the body of
the thesis and is shown in Figures 4 and 12. Adjustment
of the lathe chuck which controlled the eccentricity of
the crank pin was accomplished with the aid of a dial
indicator graduated in ten-thousandths of an inch.

The resistance wire Strain gage deflectometer is
shown in Figures # and 13. A load deflection curve for
the instrument alone at the magnification used in the
actual measurements is shown in Figure l#. The latter
also indicates the absence of measureable hysteresis
loss in both the load and deflection measuring systems.

The purpose of the phase shifting device was de-
scribed previously. A schematic circuit diagram for
the phase shifter is shown in Figure 15 along with the
switching system designed to facilitate the use of the
device.

Oscillograph traces for both in-phase and out-of
-phase signals are shown in Figure 16.

A block diagram of the instrumentation used in
test method B’is shown in Figure 17. Dashed lines in-
tiicate the temporary connections made to the frequency
(counter for determining the speed controlled setting
1:0 provide the frequency desired and to the testing

Inachine indicator for'determining load range.

(fl

41

Fig. 12. Loading device for test me-
thod B showing V-belt drive from the variable
speed motor on the Baldwin testing machine.

 

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47

.APPENDIX B. OPERATIONAL TECHNIQUES

2311'; method 5.

'1. Set up testing machine with the 500 pound load
cell and the special loading platens. Adjust the plat-
ens for parallelism using shim stock under the mounting
plates. Set the platen motion stOps. Turn on power to
the machine, the indicator and the recorder. Keep the
recorder motor turned off.

2. Set the indicator range to O-lOO pounds, ad-
Just the indicator to zero, set the recorder to zero
load and fill the pen. Engage the recorder in the half
scale position making certain that the pen does not move
‘when this is done. I

3. Connect the Baldwin PD-lM deflectometer and
adjust it for the range of motion expected. Switch on
the recorder motor and adjust the drum to its starting
position at the proper platen position. (Keep the re-
corder motor turned off except when it is actually in
use.)

4. Insert a trial specimen (not one of the sam-
ple specimens but of the same caliper) between the plat-
ens with the single-face liner up and the corrugations.
perpendicular to the front of the machine. Wrap a

#8

narrow strip-of saran film around the platens.

5. Start the machine using a slow rate of platen
motion. Set the machine to ''cycle" and adjust the cyc-
ling limit controls and the speed control simultaneously
until the desired frequency of motion is obtained within
the loading ange wanted. (The speed and limit control
settings were determined once for each range and fre-
quency. This means that the frequency may vary slightly
depending on the stiffness of the specimen particularly
in the 5-35 pound loading range.) Record the limit
control and speed control settings, st0p the machine and
remove the trial specimen.

6. Insert a sample Specimen in the same manner
as the trial Specimen, set the cycle counter to zero,
run the machine up to the low end of the load range, and
set the machine to “cycle", ( It must be taken out of
the cycle position in order to stop it at the end of step
five.) If either the speed controller or the limit con—
trols have been moved be sure to return them to their
proper settings. Turn on the recorder motor and set the
recorder cylinder to its starting position by hand.
Lower the pen and record the 10th cycle. Raise the pen
and advance the drum by hand. Record the 25th, the 50th,
and the 100th cycles in the same manner, stOp the mach-
ine, remove the specimen and return it to the condition-

ing room.

49

7. Check the final caliper of the specimen and
record it on the Specimen if it differs from the initial
thickness.

8. Repeat the above procedure for each specimen
in the sample.

9. Measure the total area under each curve and
the area within the hysteresis 100p using a polar plan-
imeter. Record the areas directly on the testing mach-
ine charts.
23st method E.

1. Adjust the upper testing machine platen to
accomodate the special loading fixture, the load cell,
and the two platens. Bolt the loading fixture to the
lower platen and raise the platen to approximately the
proper level. Remove the timing belt which drives the
transmission, replace the timing sheave on the motor
shaft with a three inch V pulley, and attach the V belt.
I To assure the safety of the fixture, the loading
limit controls Should be adjusted at this time so that
both of them are on the compression side. This will
permit the motor to run only in the prOper direction
so that the chuck will not turn on the shaft and be
unscrewed.)

2. Attach the resistance wire strain gage de-
flectometer to the platens and adjust the Screw which

contacts the sensitive beam.

50

3. Turn on the power to the machine, the indi-
cator, and all instrumentation.

b. Adjust the eccentricity of the crank pin to
the desired value using a dial indicator. Insert a
trial specimen as in method.A and raise the platen by
means of the hand wheel on the transmission drive until
the prOper load is indicated on the testing machine
load indicator. Check the loading range and readjust
the eccentricity if necessary.

5. Disconnect the load cell from its span cont-
rol box and connect the cell cable to the switching
system. Connect the output of the Tectronix amplifier
to the frequency counter and set the motor speed to
give the desired frequency. The speed control settings
for each frequency to be used should be recorded at
this time.

6. Stop the machine, disconnect the frequency
counter, and connect the Tectronix amplifier to the
phase shifter. Using the switching system, connect
the load cell output to both axes of the oscillograph
through the external amplifiers. Start the machine
and adjust the phase shift to zero for the frequency
to be used. Remember that this adjustment must be
made every time the frequency is changed.

7. Restore the deflection Signal to the X axis

by means of the switching system and check the size

51

and position of the signal on the screen. Adjust the
signal trace, if necessary, and stop the machine. (Turn
the brightness control on the oscilloscope down when
the machine is off.) Remove the trial specimen.

8. Insert a sample specimen in the platens as
before and connect the load cell output to its span
box so that the loads can be read on the machine in-
dicator. .Adjust the load to the lower load limit by
means of the hand wheel and check the loading range by
rotating the shaft slowly by hand. Connect the load
cell cable to the switching system again.

9. Check the setting of the speed control, start
the machine, and record the 100th and the 500th cycles
using the Polaroid Land camera. ( A.st0pwatch was used
to determine the 100th and 500th cycle since the fre-
quency counter produced considerable noise in the os-
cilloscope trace.)

10. Enlarge the record photographs using the Land
camera prints as paper negatives. Trace the enlarged
curves on vellum paper and measure the areas as in test

method A.

 

 

 

 

 

 

 

 

 

 

 

 

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