PREDICTION OF ACCELERATION DURING
IMPACT USING THE
RELATIONSHIP OF STRESS AND DENSITY
by
LARRY JOEL WEINSTEIN
B. S. E. E., UNIVERSITY OF TOLEDO, 1968
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
2002
Ml
_
2002 by Larry Joel Weinstein
All rights reserved.
This thesis for the Master of Science
degree by
Larry Joel Weinstein
has been approved
by
Ronald Rorrer
Weinstein, Larry Joel (M. S., Mechanical Engineering)
Prediction Of Acceleration During Impact Using The Relationship Of Stress And
Density
Thesis directed by Assistant Professor Ron Rorrer
ABSTRACT
The purpose of this work was to develop an understanding of the dynamics of
impact for a fiber filled container. To accomplish this, a theory was developed to
relate fiber density to bulk fiber stress. The theory also included the air entrapped in
the fiber media. The resulting equations were then solved using numerical methods.
The predicted and measured accelerations were tested in a design of experiments
using a central composite design to develop regression coefficients. No significant
difference was found between the two sets of coefficients. This information was
then used to evaluate container designs for competition sponsored by the Institute of
Paper Science and Technology (IPST) and the US Department of Energy (DOE).
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
Ron Rorrer
ACKNOWLEDGEMENTS
I would like to acknowledge the following people for their valuable help in this
project. Jim Weaver was a great lab partner in designing, fabricating and testing the
containers used in competition. Ron Rorrer for his patience even when he had
doubts about the project. My wife. Sherry, for her patience and help in editing this
paper.
I

I
i
CONTENTS
Figures................................................................viii
Tables...................................................................xi
1. Introduction...........................................................1
2. Existing Technology....................................................7
3. Report Overview.......................................................16
4. Stress as a Function of Density.......................................21
5. Effects of Entrapped Air..............................................42
6. Numeric Methods.......................................................48
7. Fiber Data............................................................57
8. Design of Experiments.................................................61
9. Comparison of Predicted and Measured Performance for Axial Fiber......84
10. Peak Acceleration....................................................90
11. Effects of Fiber Lay.................................................96
12. Forces on the Inner Container.......................................103
13. Effects of Altitude.................................................106
14. Recommendations for Future Work
108
15. Conclusions
112
Appendix
A. Project Photographs...............................................113
References...........................................................127
FIGURES
Figure
1.1 Initial Container Design for the Energy Challenge......................2
2.1 StressStrain Relationships of Various Fiber Densities.................9
2.2 Slope of the StressStrain Curve......................................10
2.3 Typical Cushioning Curves Used in Packaging...........................14
3.1 Mass Spring Model For Predicting Acceleration.........................17
4.1 Micro Photograph of Fiberglass........................................22
4.2 Density of Air as a Function of Pressure..............................24
4.3 Relationship of Stress and Density for Hammermilled Newsprint.........25
4.4 Coordinate System.....................................................26
4.5 Energy Relationship of Radial Fiber Density and Work..................29
4.6 Relationship of Stress and Density Assuming Stress is Zero at the
Starting Density......................................................34
4.7 Relationship of Stress and Density Assuming Stress is Zero at the
Fibers Natural Density...............................................35
4.8 Ideal Model of the Fiber to Include Recoverable Energy................37
4.9 Stress and Density for the Model in Fig. 4.8 That Includes Recoverable
Energy................................................................38
4.10 Fiber Model used for the Numeric Model................................39
viii
4.11 Stress as a Function of Density for the Model Used in the
Numerical Model.....................................................40
6.1 Numeric Model Definitions...........................................49
6.2 Segments to Define the Inner Container..............................52
6.3 Effect of Time Step Size on the Predicted Acceleration..............54
6.4 Summary Sheet from the Numerical Simulations........................56
7.1 Stress Density for Radial Fiber.....................................58
7.2 Viscoelasticity for Radial Fiber....................................59
8.1 Design of Experiments Test Points...................................64
8.2 Test Container for the Design of Experiments........................67
8.3 DOE Comer Point (1 (p= 64 kg/mJ), 1 (Zo=51 mm))Block 1 Run 5...72
8.4 DOE Center Point (0(128 kg/m3), 0 (Z0=128 mm))Block 1 Run 1.........73
8.5 DOE Comer Point (+1 (192 kg/m3), +1 (Z0=102 mm)) Block 1 Run 7...73
8.6 Cross Sections of the Response Surfaces Predicted by the Design of
Experiments.........................................................76
8.7 Predicted Acceleration with a Viscous Damping Term Added to the
Model...............................................................78
8.8 Viscous Damping Proportional to Stress Added to the Model...........79
8.9 Force of the Fiber Multiplied by 2.5 to Account for the Viscoelastic
Properties of the Fiber.............................................80
8.10 Predicted Acceleration of the Model with the Fiber Force Offset to
Zero at the Starting Density........................................81
8.11 Accelerometer Data Starting 1.8 ms Ahead of the Apparent Start......82
8.12 The Results Of Shifting the Accelerometer Data 1.5 Ms and Zero Force
Developed by the Fiber at the Time of Impact.....................83
9.1 Axial Fiber Test Container.......................................85
9.2 Test 1 With Axial Fiber At 128 Kg/m3, Zo =76 mm
and a 6 m Drop...................................................86
9.3 Test 2 With Axial Fiber At 128 Kg/m3, Zo =76 mm
and a 6 m Drop...................................................87
9.4 Test 3 With Axial Fiber At 128 kg/m3, Zo =76 mm
and a 6 m Drop...................................................88
10.1 Predicted Accelerations For a 6m (20 ft.) Drop and a 60 mm Azmax.94
11.1 Comparison of Radial and Axial Fiber Stress to Density...........97
11.2 Ratio of Axial to Radial Fiber Stress............................98
11.3 Spring Constant for Radial and Axial Fiber with a 1 kg Payload
Dropped from a 6 m and Maximum Deflection held at 60 mm..........99
11.4 Leakage Paths For Radial Fiber..................................101
11.5 Leakage Paths For Axial Fiber...................................102
12.1 Failure Mode From Impact........................................104
13.1 Predicted Accelerations For a 6 m (20 ft) Drop in Denver Colorado
and Atlanta Georgia.............................................106
x
TABLES
Table
8.1 Summary of the Test Results by Averaging Two Adjacent Points.....68
8.2 Summary of Regression Equations..................................74
1.
Introduction
In the summer of 1997, the United States Department of Energy (DOE) and the
Institute of Paper Science and Technology (ISPT) (1.1) solicited design proposals
for an insulated and impact resistant container constructed of paper for their Energy
Challenge 98. The universities submitting the top 10 proposals received a $2,000
start up grant and competed for a $15,000 first prize. The proposal, written by
University of Colorado at Denver (UCD) students and Dr. Ronald Rorrer, Assistant
Professor, placed in the top 10. Dr. Rorrer challenged his Designing with
Composites class, ME 5114, to participate in the project. Graduate students Jim
Weaver and the author agreed to work on this project.
The challenge was to design and manufacture a container to hold 1050 cm3 (64 in.J)
of 0 C (32 F) water, a raw egg, and survive a 6 m (20 ft) drop to a concrete
surface using only recycled paper and chemicals customarily used in the paper
making process. A design that met the criteria was built and tested by mid semester.
Final judging was a complex scoring system that graded the submissions on
insulating performance, weight, cost of materials, percentage of recycled paper,
leakage, condition of the egg, and the quality of written reports and video.
The basic design is shown in Fig. 1.1. The container is constructed in three parts; an
inner container to hold the water and egg, a lid to seal the system, and an outer
container to both absorb the shock and insulate the water. The original design was
very rugged and could be dropped many times from 6 m (20 ft) without failure. At
4 kg (9 lb), the team participants felt it was over designed and therefore not
competitive against many of the top paper science schools in the United States.
Figure 1.1 Initial Container Design for the Energy Challenge
2
To make the design more competitive, a program of optimization was initiated. The
major focus in the optimization was reducing the overall weight of the container.
The weight affected several areas of scoring and therefore yielded the most
improvement with a reasonable effort. At the same time the container was being
cost reduced, the reliability to withstand the 6 m (20 ft) drop had to be maintained.
This would have required destructive testing of many containers that would have
been very time consuming to construct. Therefore a system to evaluate proposed
changes without a threetoseven day fabrication period was needed. In addition to
evaluating materials there were many questions about the system dynamics during
impact the participants wanted answered.
Questions to be answered:
The following questions needed to be answered in the course of this study:
1) What is the expected peak acceleration? Early estimates placed the average
acceleration at impact to be 100 g. This meant the peak acceleration would be
on the order of 200 g because acceleration is negligible during the initial stages
of impact. Only as the fibers are compressed do they generate a force. The
generated force peaks at the bottom of travel. It is known by compressing the
fiber by hand its stressstrain relationship is highly nonlinear. Therefore
!
3
acceleration would be lower than predicted by a simple linear model during the
initial stages of impact and higher than expected at the end. The question was
how high were these accelerations. If one could calculate the peak acceleration
for an acceptable known design density, then physical dimensions could be
established for other materials and densities.
2) What effect does fiber orientation have on its ability to absorb shock? The
normal way fiber is introduced into a package is with random orientation. The
fiber is then compressed to the desired density. When the fiber is compressed
most of the fibers end up laying in the radial direction, which is perpendicular to
the direction of loading. As a load is applied, the fibers bend easily and very
little force is generated until the fibers are compressed to nearly a solid mass. A
novel approach was proposed midway through the project to orient the fibers in
the axial direction. This would load the fibers in their strongest direction.
Therefore less material could be used to maintain the system stiffness. A second
advantage was the crushing strength of the new system appeared to be more
uniform. It was suspected that with this shock absorbing system the peak
accelerations would be reduced. The question was how would this proposed
system affect the performance of the container?
4
3) What density should be used to absorb the impact and how large should the
crush zone be? Constructing the crush zone out of low density material meant
the initial forces developed would be low. As the material collapsed, the forces
would progressively increase. If the crush zone was too small these forces
would result in unreasonable acceleration thus destroying the inner container.
As the length of the crush zone is increased, so is the container weight. A
method to estimate the optimum combination of density and length of a crush
zone was needed.
4) How will the inner container fail? One possibility is that it would fail because
of the internal pressure. Another possibility is that it would fail due to the
external forces decelerating the inner container. By understanding how the
container would fail, there may be a better way to design the system. The inner
container held 1 kg (2.2 lb) of water and was constructed out of used newsprint
layed up in a paper mache method. The lamination was 2.2 mm (0.09 in.)
thick. The expected peak pressure was about 1370 kPa (200 lb/in.2) at the
bottom of the container. The internal pressure was counteracted by stress
generated by the fiber absorbing the energy of impact.
5) What effect would altitude have on the performance of the system? Would the
system perform better in Atlanta Georgia than in the Denver Colorado area? All
5
testing of the container was done at elevations in excess of 1500 m (5000 ft),
however, the competition was going to be held at nearly sea level. It was known
relatively early in the design stage that air trapped in the fiber played a major
part in how the inner container decelerated.
This report describes the development of the theory to describe the fiber, entrapped
air, and a numeric model to answer these questions and test results used to verify the
calculations.
6
2.
Existing Technology
The most direct approach to answer the questions posed in the Introduction would
be to assume the system was linear. Using classic linear vibration analysis the peak
acceleration of a body dropped from a height h would be:
where con is the systems natural frequency. The constant k is calculated from the
system geometry and the materials equivalent modulus of elasticity E.
The stressstrain curves for three different densities of fiber are shown in Fig. 2.1.
Two things are apparent from the curves. First is the curves are highly nonlinear
and the second is they appear to have little in common with each other making
generalization or normalization difficult. The modulus of elasticity can be defined
in terms of the differential E=da/de. The results of the differentiation on raw data
2.1
2.2
7
are shown in Fig. 2.2. The irregularities in the curves are due to experimental error
being accentuated by the differentiation process.
5 2
The modulus of elasticity for the 240 kg/m (15 lb/ft ) fiber could be approximated
at 2000 kPa with reasonable results. The modulus of elasticity for the 64 kg/m3 (4
lb/ft3) fiber varied from 50 kPa to 10,000 kPa over the range of expected strain
making any simple linear assumptions invalid. It would have been possible to
approximate the data by assuming the strain was linear up to a break point and
linear thereafter with a new slope. This was considered after a different method had
been developed and therefore not pursued.
8
Stress Strain for Hammermilled Fiber
Figure. 2.1 StressStrain Relationships of Various Fiber Densities
9
Figure 2.2 Slope of the Stress Strain Curve
To complicate the analysis even further, the air entrapped in the fiber must be taken
into account. Early estimates put the force generated by the air to be nearly as great
as the fiber. The air pressure would follow the ideal gas law except for any leakage.
Obviously, numerical methods could be used to solve the system dynamics. The
method was strongly considered because of its straight forward approach. The
major drawback to this method was experimentally developing a stressstrain curve
for each density to be evaluated because it was felt there was too large a variance
10
between the densities to accurately interpolate the data. In reviewing methods to
solve the problem, there was a persistent feeling a more basic approach to the
problem was being overlooked.
A literature search was initiated to gain insight into the dynamics of impact. Little
useful information was found. Eshleman in his paper High Performance Shock
Isolation Systems" (2.1) states Modeling of dynamic systems is often given the
least attention in the process of design and analysis. Specific fundamental research
in this area has been rare; however, data and procedures on modeling can be found
scattered over many texts and papers. This author found this to be true. There
were two general categories of work done in this field. The first is in shipping of
packages for the transportation industry. The second series of work was in
transportation of nuclear waste.
Transporting nuclear waste is a sensitive, highly political topic, so significant
amounts of public money have been put into research to determine the safety and
possible modes of failure for these containers. Timpert, et al, in a series of two
papers (2.2) (2.3) looked at transporting nuclear waste by rail in drums. His method
of analysis is based on a finite element mass spring system. The work was limited
to accelerations of only 6 g and no test data was given. Butler (2.4) in another
article used a finite element method in his analysis for drops of up to 55 m. The
11
extreme was work published by Brown (2.5) where he looked at transporting waste
by air. Drop heights as high as 10 km and velocities as high as 200 km/h at impact
were examined. He, also, used a finite element method of analysis. The predicted
acceleration and deflection generally agreed within 10% of the test data. The
general trend in the work reviewed was to use finite element analysis.
Each year there are thousands of disappointed customers when their packages arrive
damaged from shipping, and consequently millions of dollars in insurance claims
are filed. It is to the benefit of manufacturers, warehousers, and shippers to work
together for the safe delivery of packages at the lowest cost. The Institute of
Packaging Professionals (IOPP) is an organization dedicated to that cause. Their
focus is on drop heights of less than 1.2 m (48 in.) and accelerations generally under
75 g. Studies have shown 50% of the packages dropped will fall less than 0.3 m
(12 in.). These typically occur as a package is being transferred from one conveyor
to another, or as the package is manually placed on a truck. Occasionally packages
will be dropped from waist high. It is rare to have a package fall more than 1.2 m
(48 in.).
In reviewing the IOPP publication 'The Best of Transpack (2.5), a compilation of
the best papers presented over the last five years at their symposiums revealed little
in modeling package performance. In two papers there were references to the
12
cushioning manufacturers proprietary software. Based on the work presented it
could be assumed that the software was a finite element system provided by the
cushioning manufacturer with a proprietary nonlinear addin module to match the
material proposed by the cushioning manufacturer. This software would be of little
use for predicting the performance of recycled paper without the needed addin
modules.
A method available to the public for predicting cushion performance is ASTM 1596
(2.6). In the test, weights are dropped on a sample of cushioning material and the
maximum accelerations are plotted as shown in Fig. 2.3 (2.7). This is a very time
consuming test in that a minimum of 25 points are required for each curve. Each
point would require a new cushion to be fabricated. The standard states in 13.1
Differences produced by these factors, together with specimen performance
variation, might exceed 50% of the ideal. However, use of different systems
testing a consistent material by this method should produce data that agree within
10%. These results are with refined and expensive commercial equipment.
Ground newsprint would hardly be considered a consistent sample. In addition to
needing specialized equipment, separate curves would be need for each density and
thickness to be examined resulting in hundreds of tests. The most important
drawback to this test method is that it only works for flat surfaces. The inner
13
containers used in this project were spherical and therefore the cushioning curves
would not apply.
0.01 0.02 0.03 0.03 0.1 0.2 0.3 0.5 1.0 2.0 3.0
STATIC STRESS, PSI
Figure 2.3 Typical Cushioning Curves Used in Packaging (2.7)
Intuition leads one to suspect there is an approach somewhere between the simple
stressstrain curves, the ASTM 1596 test method, and the highpowered finite
14
element analysis. Development of such a method would fill a needed gap, and lead
to a more basic understanding of fiber dynamics.
15
3. Report Overview
The purpose of this section is to give the reader an overview of the procedures
chosen to answer the questions posed in the Introduction. This will introduce the
reader to some of the deviations from classical vibration theory taken by the author
to model the nonlinear response at impact. The work presented is the result of years
of the model evolving and being refined even subsequent to completion of the
competition. Only the final version of the model is presented in that much of the
work was undocumented, would add little to understanding the technique
developed, and quadruple the volume of the report. It is interesting to note after
several years of refinement and testing, the conclusions drawn from modeling the
impact have not changed.
In its simplest form the model can be reduced to a series of springs and a frictional
damper. This is shown in Fig. 3.1. Force on the container during impact will be
generated by both the fiber and the air entrapped in the fiber matrix.
The air acts as a nonlinear spring and will obey the ideal gas law. The pressure
generated by the entrapped air will be discussed in Section 5. The equations
16
developed will account for the nongaseous material in the cylinder and leakage to
the atmosphere.
Figure 3.1 Mass Spring Model for Predicting Acceleration
The model for the fiber is more complex and is addressed in Section 4. Some
characteristics of the fiber are similar to a spring in that when compressed some
energy is stored and will be returned to the payload. Most of the developed force is
due to friction of the fibers rubbing together as they are being compressed. The
effects of the frictional spring for paper fibers are quite small and will be ignored in
the model. The model assumes all energy absorbed by the fiber will dissipated and
not returned to the payload.
MASS
Fiber
Sectloc
Air
Section 5
17
The force developed is unidirectional in that it is significant only when the fiber is
being compressed. When the load on the fiber is removed and it is expanding, the
internal stress quickly goes to zero and normal fibertofiber force is low. Therefore
the frictional forces are also low. In the model presented force will be a function of
position and only valid while fiber is being compressed.
It is important to forewarn the reader that the fiber equation developed will be in
terms of stress and not force. This was done for several reasons. The most
important reason is that stress and fiber density will be shown to form a basic
relationship. A secondary but very important reason to work with stress is that non
uniform payload shapes can be evaluated. Finally, the units of stress are the same as
pressure so they can be summed with the air pressure in the model.
A second item to advise the reader is that density always refers to the bulk density
of the fiber composite and not the density of an individual fiber. In the same way
stress also refers to the stress of the fiber composite and not the stress on an
individual fiber. The model developed is a macroscopic picture and not
microscopic.
The composite fiber characteristics are developed from a simple experiment that
utilized a press to compress the material. Data consisting of stress and density are
18
then fit into a polynomial equation to be used later in the model. The procedure is
described in Section 7.
Another deviation from classical theory is that strain is not directly used. Instead
the equations are related to density. The advantages of using of density will become
apparent in the derivations for air pressure and fiber stress. A second benefit of
working in density is the equations become more linear. Numeric models are
generally more stable with linear systems.
To solve the differential equations at impact Eulers method is used to calculate the
displacement at finite steps in time. From the initial conditions the position of the
payload is known, the volume of the fiber and density can be calculated. Knowing
the fiber density, the fiber stress can be calculated. The position of the payload also
yields the air pressure. From pressure and stress the total force on the payload is
calculated using cylindrical coordinates.
Total force is then used to calculate acceleration. The process is then repeated in an
Excel spread sheet with a finite time step to approximate a new position and
therefore a new fiber density and air pressure. This process is described in Section
6.
19
To verify the model a twofactor design of experiments was run. A somewhat
unusual approach was taken for the experiment in that regression coefficients for the
factors were developed separately for the model data and the experimental data.
These regression coefficients were then compared to each other by looking at their
differences in terms of standard deviations. The design of experiments and
discussion of sources of error are described in Section 8.
Once confidence in the model has been established it can be used to answer the
questions posed in the Introduction. This is done in Sections 10 thru 13.
20
4.
Stress As A Function Of Density
The purpose of this section is to translate the classical approach of using a materials
stressstrain relationship to the concept of stress as a function of density. The
concept will prove useful in that multiple stressstrain curves can be combined into
a single curve that can be created from easy to obtain data. The curve can be
approximated by a linear function if a high degree of accuracy is not needed and
large changes in density are not required.
While this paper is dedicated to solving the dynamic problem of the impact of a
mass on a paper matrix, it should prove useful in many other fields. The problem is
similar to baling of hay, cotton, trash, packaging of fiberglass, or stuffing a down
coat in a pack. In all the above cases, the package size is reduced by means of
mechanical compression, thus raising the bulk density of the material. Package
volume is inversely related to the mechanical force applied to the package. For a
fixed mass, density is also inversely related to package volume. Therefore
mechanical force, or equivalently stress, should have a direct relationship to the
density. For each material there should be a function that relates stress and density
for a fibrous material.
21
A fibrous network has more in common with gases than it does with its solid
counterparts. Both fibrous networks and gases are mostly open space. In a gaseous
system the molecules are held apart by molecular forces. In a fibrous network the
fibers are held apart by the stiffness of the fibers that cross and interfere with each
other. This is illustrated in Fig. 4.1 which is a photograph of a fiberglass wool.
Figure 4.1 Micro Photograph of Fiberglass (4.1)
22
The ideal gas law states:
PV=nRT
4.1
Rearranging the terms to solve for pressure, which is analogous to stress in a solid
system:
P =
4.2
The term n is proportional to the density of the gas p.
V
If the temperature and mass of the system is held constant:
P=Kp 4.3
Thus for a gas, pressure is linearly proportional to the density. Figure 4.2 is a graph
of the relationship of air density and air pressure. The data was extracted from
Crane (4.2). Crane claims an accuracy of 3%. Over a two decade range of density
the relationship of pressure and density is linear.
23
Pressure in atm
Figure 4.2 Density of Air as a Function of Pressure
Test data for two starting densities of hammermilled paper are plotted in Fig. 4.3 to
show the relationship of stress and density. Even though the starting densities of the
two samples differ by a factor of four, they converge into a common curve. This
would lead one to suspect that there is a common function for all densities and
stress.
24
Figure 4.3 Relationship of Stress and Density for Hammermilled Newsprint
Fig. 4.4 shows the coordinate system used in this analysis. The X and Y dimensions
are held constant so strain sx and sy are equal to zero. This would be analogous to
dropping a package in a cardboard box where the sides did not rupture.
25
z
Figure 4.4 Coordinate System
All deflection is assumed to occur in the Z direction. Since the mass of the
packaging material is constant and strain will be in the Z direction, density and
strain will be related by the following equations:
26
p,
M
AZ0 4.5
p is the original density of the segment under investigation,
where:
M = mass of the segment,
A = area of the segment, and
Zo = the original Z dimension.
When the segment is compressed the new density can be expressed as
P
M
A{Z+S:)
where
5z is the change in the Z dimension
Equation 4.6 can be rewritten as:
P =Po
Z0+Sz
4.6
4.7
27
While strain and density are related, in this analysis the principle concern will be
density.
In a fibrous network the material is gathered and then mechanically compressed to
the desired density. The density for a loose pile of hammermilled old newsprint is
32 kg/nT (2 lb/fT). Beyond that density mechanical work must be utilized to
compress the material further. Figure 4.5 is the relationship of the work required to
compress a kilogram of hammermilled fiber. The curve is valid only when the bulk
density of the sample is increasing (5p>0). If the material is allowed to expand,
some of the energy used to compress the material will be recovered.
Also included in Fig. 4.5 is the unrecoverable energy plotted against density. The
unrecoverable energy could also be called the energy of formation. It is the energy
required to compress the material to a given density with all external forces
removed. To develop this information the fiber was compressed to various densities
and the total energy was calculated from force and distance. The fiber was then
allowed to expand to its new density. The energy returned by the expansion was
calculated from product of estimating the average force as one half the peak force
and the distance of the fiber expanded. The returned energy was then subtracted
from the total energy to get the unrecoverable energy shown in Fig. 4.5
28
I
Density (kg/m3)
Figure 4.5 Energy Relationship of Radial Fiber Density and Work
The total energy, Utotai, of the fiber is represented by the following equation:
Ullllal = U,
wire cov erahle
+ u
re cov erahle
4.8
The term Uunrecoverabie is the energy to form the material to a given density when all
force is removed from the material. The term UreCoverabie is the recoverable energy
when the material expands as the force is released.
29
The term Urec0verabie is important in two situations. The first situation is if rebound of
the payload is going to be considered. In this case maximum deflection and
acceleration all occur at the bottom of travel just before the fiber starts to reexpand.
What happens beyond maximum deflection is not considered in this study.
Therefore Urecoverabie can be ignored.
In the second situation the recoverable energy must be considered when the initial
density is greater than its natural density. The recoverable energy must be supplied
to the material before it will start to yield and permanently change density. In this
situation the energy of the payload is about 1200 J/kg (400 ftlbf/lbm) of paper. At
the starting densities that were expected to be used in the final design, the
recoverable energy was only 6.8 J (2.3 ftlbf/lbm ). Since the desired accuracy is
about 10% the recoverable energy can be ignored.
Based on the previous discussion the energy per unit mass can be written:
U(P) UUnrecoverable(p)^"Urecoverable(p) Utotal(p) 4.9
As long as the recoverable energy at po is small and dp >0, Eq. 4.9 can be written in
differential form.
30
4.10
dU
dp
U'(p)
During impact, work is absorbed by the media and used to change its density:
4.11
or
dW A crdz
4.12
The negative sign appears because during compression dz is negative. Equation 4.9
is work per unit mass. To equate 4.10 and 4.12, Eq. 4.12 must be divided by the
mass.
, A adz
dw =
M
4.13
, adz
dw 
Po^o
Applying the first law of thermodynamics to Eqs. 4.10 and 4.14:
4.14
31
4.15
U'(cr)dp = 
adz
PoZ0
Density and z are related by the following equations:
4.16
dp M
dz Az2
4.17
A , 4.18
dz =z~dp
M
Therefore:

PqZq PqZ1
)dp = U\p)dp
4.19
a^(^fU'{P)
7
4.20
Since p = ^>0
7
32
(J = p2u\p)
4.21
Solving Eq. 4.21 :
a=f(p) +C 4.22
The constant C would be defined by the initial conditions of the fiber. One of two
assumptions can be made about the initial conditions: 1) the fiber at its compressed
density before impact has zero stress, or 2) the fiber has zero stress at its natural
uncompressed density. The assumption chosen will have an effect on the maximum
predicted acceleration. The error induced by either assumption will be a function of
the magnitude of the recoverable energy at the starting density.
If one assumes the fiber at its original starting density has no stress:
C = f(p) 4.23
The stress as a function of density for a 128 kg/mJ (8 lb/ft3) radial fiber sample is
shown in Fig. 4.6 using this assumption. The effect of this assumption is to drop the
entire strain density curve by 23 kPa (3.3 psi). This assumption is reasonably valid
33
i
for fibers that are either very soft at the beginning of the compression or the starting
density is very close to the natural uncompressed density of the fiber. In either case,
the force at the beginning of impact will be underestimated. Because acceleration is
underestimated at the beginning of impact more energy will be absorbed during the
last stage of impact. Due to the non linear nature of the fiber, the peak acceleration
is overestimated with this assumption. This assumption proved to be reasonably
valid for fiber layed in the radial direction.
Figure 4.6 Relationship of Stress and Density Assuming Stress is Zero at the
Starting Density
34
The other assumption is that the fiber is at zero stress only at its natural density. If
this assumption is made, the fiber matrix is assumed to be rigid until the fiber
reaches its yield point. The effects of this are shown in Fig. 4.7 for the previous 128
kg/m (8 lb/ft) example. Generally acceleration at the beginning of impact is
overestimated and the final acceleration is underestimated. This assumption is valid
for materials that are quite stiff at their starting densities such as axially layed fibers
or material bonded together. Most of the work done on the project used axial fibers
and therefore this assumption was chosen. For the design of experiments radial
fiber was used and the two methods are compared.
Figure 4.7 Relationship of Stress and Density Assuming Stress is Zero at the
Fibers Natural Density
35
Physically what is happening in the fiber matrix is the fibers are locked together by
friction. The fibers deform axially in bending, but do not slide when small forces
are applied. This is a linear region that any applied energy is returned when the
force is released. As the stress is increased the fibers begin to slip causing a
permanent change in density. This work is unrecoverable and begins the nonlinear
portion of the stress strain relationship. The point the fibers start slipping is a
function of their coefficient of friction and the normal fibertofiber force. The
normal force is a function of both the direction the fibers are laying in respect to the
applied load and the density of the matrix. As the density goes up both the normal
fiber to fiber normal force due to the Poisson ratio and the number of fibers in
contact with each other will increase. All of these factors result in a highly non
linear response in this region. When the force is released the matrix will reexpand
returning some energy. More fibers at the new higher density will be in contact
with each other. This results in shorter unsupported spans, bridges, and columns that
will result in a stiffer matrix and a new equivalent spring constant for the
recoverable energy portion of the fiber model.
The preceding example can be modeled as shown in Fig. 4.8. The spring represents
the linear portion of the example where the fibers are bending and there is no
slippage. As the applied force is increased and exceeds the frictional forces of the
fiber, energy is dissipated, the density is permanently changed and a new normal
36
fibertofiber force is established. In this model a smooth transition is made
between the two assumptions discussed earlier.
Figure 4.8 Ideal Model of the Fiber to Include Recoverable Energy
The relationship between stress and density for this model is illustrated in Fig. 4.9
with a load applied and then removed. Also on the graph are both assumptions to
show how the addition of a spring makes the transition between them. This would
be the most accurate model of the fiber.
37
200
Density (kg/m3)
Figure 4.9 Stress and Density for the Model in Fig. 4.8 That Includes
Recoverable Energy
While the model presented is the most accurate, it was not pursued because the
additional work required to determine the spring constant as a function of density
was not warranted at this stage of the project. Because the starting densities were
low, errors induced by assuming the material was rigid until it started to slip were
also low. The final model used to represent the fiber is shown in Fig. 4.10. The
forces for the simplified model during impact are shown in Fig. 4.11. In this model
all forces developed by the fiber are dissipative and no energy is returned to the
payload.
38
Figure 4.10 Fiber Model used for the Numeric Model
39
Figure 4.11 Stress as a Function of Density for the Model Used in the Numerical
Model
The result of this analysis is that if a stress and density relationship is
experimentally established for one density, the function f(p) will hold for other
starting densities provided the recoverable energy at the starting density is small.
For convenience in calculations the term p is related back to the Z axis in the
following equations:
40
4.25
* = ApA)
z
These equations are used in the numeric model presented in Section 6.
41
5. Effects Of Entrapped Air
During impact the air entrapped in the fiber generates pressures similar to the fiber.
These forces must be taken into account. As shown previously, the stress generated
during impact is a function of density. Therefore it is very convenient to relate
pressure of the entrapped air to changes in volume as was done for the fiber.
Assuming the process is isothermal and no air escapes during impact, the following
equation can be written from the ideal gas law:
PV = nRT = PV
1 o' 0 air 1 air r i
where:
Po = original absolute pressure in the container,
Vo_air== original volume of air in the in the container,
P = instantaneous air pressure in the container,
Vair = instantaneous volume of air in the container,
n = number of moles of air in the container,
5.2
42
R = gas constant, and
T = absolute temperature.
The fiber itself is assumed incompressible and occupies space in the matrix.
Therefore its volume must be taken into account. The volume of the solid material
is calculated from its solid densitv.
Psolid
where:
Vsoiid = volume occupied by the fiber,
Vo = bulk volume of the fiber,
po = original bulk density of the fiber, and
psoiid = density of a individual fiber.
The original volume of air is calculated from the total volume minus the volume of
solid material in the container:
V =V 
r 0 air v 0
Po
Psolid
Vo
5.4
43
The instantaneous volume of air is:
V =V Po. V
V air V V 0
P solid
5.5
where:
V = the instantaneous volume of the container.
Substituting these equations in Eq. 5.2 and converting to gauge pressure yields:
'v ^
v o v o
P = Pr,
P,
solid
V K
1
Psolid
5.6
As in the section on fiber stress the volume can be related back to the Z axis:
Z Z
p = p*
p,
solid
z Zn
Po
1
Ps,
solid
5.7
The assumption of no leakage during impact proved to be valid for axial fiber
designs. That was not the case for radial fiber designs. The following development
44
takes into account leakage. Leakage is assumed to be proportional to the difference
in the internal and external pressure via a constant K'.
where:
n = number of moles of air in the container.
Solving the differential equation yields:
/
n = \K'{PP0)dt + C 59
o
At t=0:
n= n0
5.10
Therefore:
n = n0 $K'{PP0)dt
0
n =
p v
1 0y 0_air
RT
i
K'\{PP0)dt
5.12
45
PV =
P V
1 (TO __!>
RT
 K'\(P PM
RT
5.13
Converting to gauge pressure yields:
P = Pn
K
0 air
RTK'
V,.,
P V
1 0 r 0 air 0
/ 'N
\(ppM ,
y
5.14
Let
a: = RTK'
and
Po
V
P so\\d
v.
zZn
Po
P solid
5.16
P = P0
77 ^
^0^0 {
P solid
zZn
Po
1
K
P V
1 0 v 0 air 0
i
\{p Po )dl
1
5.17
P
solid
46
Equation 5.17 is similar to 5.2 with the addition of the leakage term.
If leakage needs to be accounted for, the constant K would be varied in software to
reduce the error in the calculated results. This was done in some simulations to get
a better fit between predicted and experimental results for the payload rebounding
after impact. When testing fiber that had been layed in the radial direction the
leakage was significant because of the fiber orientation air could easily escape along
the boundaries. K in this case was made large enough that the effects of the
entrapped air were negligible. For axial fiber the air was effectively trapped and
therefore K=0. K is probably a function of the fiber density but it is outside of the
scope of this project to define the function.
47
6.
Numeric Methods
The equations developed for fiber stress and air pressure in the previous two
sections are a function of their initial conditions and the z dimension. The primary
container under consideration was cylindrical on the top and spherical on the
bottom. Since the container was symmetrical about the Z axis and the z dimension
was the primary concern, a cylindrical coordinate system was chosen. The
container was defined by the radius of the spherical bottom and the minimum z
dimension from the center of the container to the flat bottom of the fiber. This is
shown in Fig. 6.1.
48
The initial dimensions can be calculated as:
Z(r) ~ Z0(0) R2 r2 +R
During impact the dynamic dimensions are:
z(r) = Z0(r)Z0(0) + z(0)
49
Knowing the initial and the instantaneous dimensions the density can be calculated
by:
p(r) =
ZQ(r)
z(r)
P o
6.3
As shown in Section 4 bulk fiber stress is a function of the fiber density. This
relationship is experimentally determined in Section 7. The data is fit into a
polynomial.
Cfiherir) = Co +P(r)C, + p(r)2C2 + p(rf C, + p(r)*C4 + p(r)5C5 6.4
The equations for gauge air pressure from Section 5 are in the form of initial and
instantaneous position, thus the pressure as a function of radius is now
P(r) = P0
Z0(r)Z0(r)(
Psolid
z(r)Z0(r)
Po
1
K
P V
1 0' 0 air 0
J(PPM
1
p
solid
6.5
The integral for leakage is accounted for in the time steps that will be used later in
Eulers method for solving differential equations.
50
The force on the payload can then be calculated from the fiber stress and air
pressure.
2 xR
F= J j(
0 0
6.6
The integration was carried out numerically with radial bands of 3.17 mm (0.125
in.) as shown in Fig. 6.2. The majority of the rest of this report is based on test
containers with a flat bottom in order to easily compare the model with
experimental results. To simulate a flat bottom a radius of 25 m (1000 in.) was
entered for R.
51
Figure 6.2 Segments to Define the Inner Container
Acceleration during impact for the payload is
d2z F
dt2 ~ M '
52
The initial conditions at impact are:
v(0) = V2gS 6.8
and
z(0) = Z0(0).
6.9
Equations 6.2 through 6.9 are solved in an Excel spread sheet using Eulers method
with generally 0.2 millisecond time steps. Figure 6.3 shows the effect of step size
for a nominal container.
53
Figure 6.3 Effect of Time Step Size on the Predicted Acceleration
For higher accelerations needed for some of the points in the design of experiments
a 0.1 millisecond step was used because rates were as high as 6,000,000 g/s. Under
those conditions the peak acceleration was generally underestimated by the larger
time step.
The model became unstable only when the time steps were larger than 2 ms or if too
large a leakage constant was chosen. The leakage constant is zero for all the axial
fiber and 100 for the radial fiber calculations in this report. In the radial fiber
calculations the entrapped air was made negligible by the leakage constant.
54
Instability was determined by observing negative air pressures during the impact
compression cycle. The program could have been simplified by dropping the
entrapped air from the model for radial fibers, but this would have restricted the
versatility of the software.
A sample summary sheet of the calculations is shown in Fig. 6.4. The setup of the
spreadsheet is a balance between versatility and ease of programming. For this
specific problem the data entry and programming could have been simplified. The
purpose of the expanded flexibility was to be able to explore tangents to this
problem with little additional work. Expanded even further the program could be a
valuable and versatile tool to the packaging industry to insure safe shipment of their
material.
55
Figure 6.4 Summary Sheet from the Numerical Simulations
DROP HEIGHT
20 FT
XA5
DENSITY
SOLID DENSITY
6 #/FTA3
65 #/FTA3
XA4 XA3 XA2 X
0 0.0115 0.0952
Leakage
TIME STEP
0
0.2 MSEC
Z MIN INNER CONTAINER R 3 INCHES 2 INCHES MaxAcell= 417.3472 Min Z= 0.77636 500  400 i
WEIGHT 2.2 POUNDS maxstress 169.0782
8 ioo 0
PRESSURE 12.5 PSI
5 10
msec
15
MSEC
TIME
FORCE FORCE FORCE MAXIMUM STRESS
Z ACCEL VEL TOTAL FIBER AIR TOTAL FIBER AIR
0 3 0 36.22154 0 0 0 0
0.2 2.913068 17.0654 36.11232 37.54388 35.45602 4.287869 3.709232 2.858366 0.850866
0.4 2.826399 20.20036 35.98304 44.4408 37.83454 8.806261 4.409806 3.091014 1.318792
0.6 2.740039 23.53762 35.8324 51.78277 40.41095 13.57182 5.16575 3.347218 1.818532
0.8 2 654042 27.09633 35.65898 59.61192 43.2093 18.60261 5.983562 3 630485 2.353077
1 2.56846 30.89797 35.48124 67.97553 46.25729 23.91824 6.870735 3.944972 2.925763
1.2 2.483353 34.9667 3523745 76.92675 49.58678 29.53997 7.835937 4.295814 3.540323
1.4 2.398783 39.32973 34.98574 86.5254 53 23455 35.49085 8.889232 4.688299 4.200932
1.6 2.314818 44.0177 34.70402 96.83894 57.2431 41.79585 10.04234 5.130066 4.91227
1.8 2.231528 49.06524 34.39001 107.9435 61.66161 48.48193 11.30894 5.629356 5.679581
2 2.148992 54.51147 34.04113 119.9252 66.54704 55.57819 12.70507 8.19632 6.508747
2.2 2.067293 60.40058 33.65457 132.8813 71.96537 63.1159 14.24956 6.843196 7.40636
2.4 1.986522 66.78252 33.22716 146.9216 77.99304 71.12852 15.96456 7.584759 8.379805
2.6 1.906777 73.71369 32.75539 162.1701 84.71852 79.65159 17.87621 8.438877 9.437336
2.8 1.828164 81.25757 32.23535 178.7667 92.24402 88.72264 20.01532 9.427172 10.58815
3 1.750799 89.48545 31.66264 196.868 100.6872 98.38075 22.41824 10.57579 11.84245
3.2 1.674809 98.47684 31.03239 216.6491 110.183 108.666 25.12779 11.91633 13.21146
3.4 1.600331 108.3198 30.33914 238.3036 120.8849 119.6187 28.1942 13.4868 14.70741
3.6 1.527517 119.1106 29.57683 262.0434 132.9656 131.2778 31.67614 15.33272 16.34341
3.8 1.456533 130.9529 28.73873 288.0963 146.6173 143.679 35.6414 17.50817 18.13322
C
1.0648 2.7993
 Seriesl j
7.
Fiber Data
To determine the experimental fiber characteristics, a measured amount of fiber was
placed in a 102 mm (4 in.) round PVC tube and compressed with a threaded
plunger. Force was monitored with a strain gauge. Forces of over 4,400 N (1000
lb) were generated in the press. Defection was obtained by counting the turns of the
plunger. The raw data of force, turns on the plunger, and weight of the fiber were
entered into a spread sheet and stress and density were calculated. This information
was then graphed and a polynomial trend line fit to the data. An example of a graph
and polynomial for the radial fiber is shown in Fig. 7.1.
57
0 100 200 300 400 500
Density (kg/m3)
Figure 7.1 Stress Density for Radial Fiber
The fiber showed viscoelastic properties in that stress would slowly decay after the
move on the plunger was made. It is not known if the drop in stress is due to the air
entrapped in the fiber slowly escaping or due to the fiber itself. An arbitrary
decision was made to capture data 2 seconds after completion of the move. The rate
of decay is shown in Fig. 7.2.
58
700
600 t
T
T 500 !
in
0)
w
400 j
i
300 i
1
Figure 7.2 Viscoelasticity of Radial Fiber
Data taken 2 seconds after the plunger was moved may introduce considerable error
in the calculations. Assuming viscoelastic properties of the fiber and the regression
equation is correct, at 0.005 seconds the force might be 880 kPa (130 psi). In this
case error would be 45%. The viscoelastic curve does not appear to be a linear
logarithmic function for small times. Therefore the error induced may not be a large
as indicated. It might be possible to take similar data throughout the range of
densities and project them back to a 0.005 second time base for more accurate
results. That was not done for this project in that trends and insight of the processes
occurring at impact were more important than the absolute numbers. Work
10 100 1000
Time (s)
59
presented in the Design of Experiments, Section 8, will show the shift in stress due
to viscoelasticity is not significant.
60
8. Design Of Experiments
Test results on containers that were similar to the final design compared very well to
the results predicted in the mathematical model developed in the previous sections.
There was some concern whether the model was truly representative of the system
dynamics or if this was coincidental. To test the model, a design of experiments
was set up. The goal was to test the model at extreme conditions that would be out
of the range of normal designs and then examine their response surfaces. The
surfaces were then compared to each other by analyzing their equation coefficients
and their standard deviations. If the model held for these extreme designs, the
model is probably a reasonable representation of the system.
The major factors that are entered in the model are: 1) the height of the crush zone
Zo(0), 2) the density of the fiber, 3) the weight of the payload, and 4) the height of
the drop. To evaluate all four factors at three levels would have required 81
experiments, which is an inordinate amount of testing. Only the first two factors
were variables the contestants could change. Drop height and weight of the payload
were dictated by the rules of competition. Therefore a two factor and three level
design of experiments was performed.
61
It was expected that the response as these factors were varied would be highly non
linear. Therefore the data would be fit into a quadratic equation to account for the
system nonlinearities. To develop the surface response equation a central composite
face centered design was chosen.
DesignExpert was used to set up and analyze the data. To fit into the software
the factors were given names of only A, B, and coded values of1, 0, and +1.
These coded values and the actual values are shown in Fig. 8.1.
Factor A is the crush zone Zo(0). A nominal crush zone of 76 mm (3 in.) was
maintained as a center point. The lowest practical crush zone and still preserve the
accelerometer was 51 mm (2 in.). The 51 mm (2 in.) crush zone with the 64 kg/nT
(4 lb/ft3) resulted in nearly a 3000g acceleration and the accelerometer was rated at
4000g. That established the maximum crush zone at 102 mm (4 in.) to maintain the
design symmetry.
Factor B is density of the fiber. The lowest density that would support the payload
was 64 kg/m3 (4 lb/ft3). Below that density the crush zone could not be maintained.
The highest density possible by compressing the fiber by hand was 192 kg/nT
(12 lb/ft3). These two end points forced the center of the design to be 128 kg/mJ
(8 lb/ft3). To test the model at extreme conditions the fiber was in the radial
62
direction. This is the direction the fiber is the weakest, the most nonlinear and would
poorly entrap the air. While this design makes for a good test of the model, it would
make a poor shock absorbing package.
63
Figure 8.1 Design of Experiments Test Points
Factor B
Density
The design required 14 experiments to be run. Six of the experiments are at the
design center. The other eight experiments are around the periphery of the nominal
design. Unfortunately there was not enough fiber to run all 14 experiments in one
batch. To overcome the fiber shortage the experiment was divided into two blocks.
Blocking an experiment is a statistical way to compensate for changes in
experimental conditions when all the experiments cannot be run at the same time.
In this case there was a concern that used fiber would not respond the same way as
virgin fiber. Points in the blocks were chosen by the software. The software
automatically shifts the data to compensate for changes in experimental conditions.
Even though the fiber was reused for the second block, the mean for both blocks
was similar. Tests in each block were performed in random order.
The end result of the experiment is a surface response equation relating variables A
and B to acceleration. The equation is in the form of:
= Cmam + CA + ChB + C 2 A2 + C2 B2 + CahAB
8.1
where A and B are the coded variables ranging from 1 to +1. For each of the
constants in the equation, a standard deviation is reported by the software.
65
The container to be tested is shown in Fig. 8.2. To simplify the construction and
instrumentation a flat bottom was used in the experiment. The tube was an ABS
sewer pipe with an end cap glued in place. The payload was fabricated from a
pressboard disc and weighted with steel washers to meet the weight specification.
There was an 8.0 mm (.3 in.) gap around the disc to allow air to freely escape during
impact. The container was suspended 6 m (20 ft) above a concrete pad and released
by pulling on a spring clip. Photos of the test container, drop area, and release
mechanism are shown in the appendix in Photos A.l to A. 14.
66
67
Table 8.1 Summary of the Test Results by Averaging Two Adjacent Points
Standard Order Run Order Block Crush Height Density Test Calcu lations Differ ence % Error
1 5 1 2 4 2418 2121 297 14
2 2 1 4 4 485 654 169 25
j * j 1 0 z. 12 783 725 58 8
4 7 1 4 12 462 325 137 42
5 4 1 3 8 681 568 113 19
6 1 1 3 8 663 568 95 16
7 6 1 3 8 506 568 62 10
8 11 2 2 8 1036 1080 44 4
9 14 2 4 8 405 385 20 5
10 10 2 3 4 602 1070 468 43
11 12 2 3 12 570 446 124 27
12 13 2 J 8 895 568 327 57
13 18 2 3 8 631 568 63 11
14 9 2 3 8 535 568 33 5
Data from the experiments is shown in Table 8.1. The percent error was as large as
57%. This was not unexpected because in an ASTM 1596 (2.6) standard they warn
68
that testing errors of 50% are common. The ASTM test deals with drops of only 4
feet and uses equipment refined from years of use.
The design called for six test points at the center of the experiment. When these six
points are examined independently, their standard deviation is 138 g. Errors range
from +37% to 22%. Errors such as the ones above reinforce the need for a
mathematical model to gain understanding of the process of shock absorption in that
changes in performance could easily be masked by experimental error.
Experimental error is probably due to three major factors: inconsistencies in the
fiber, the container not hitting the concrete pad squarely, and high frequency
oscillation during impact of both the outer container and the payload. Another
source of error is determining the time when the impact event started. The time the
event started does not affect the peak acceleration but will be discussed later in the
report to try to reconcile the differences in the measured and calculated period.
Every attempt was made to keep the fiber as consistent as possible. It was
thoroughly mixed between experimental blocks, measured, and inserted in the
container the same way. Small differences in handling could easily change the fiber
orientation and thus its apparent characteristics. While not studied, it is also known
that humidity changes the characteristics of the fiber. Each test took about an hour
69
to set up and run. Over the course of the seven test runs in each block there were
significant shifts in temperature and probably in relative humidity during the day.
Rotation of the container is another source of experimental error. The container was
allowed to freefall with out guides. Instead of using a guide system, the container
was lengthened to increase its rotational moment of inertia and a release mechanism
was developed minimize movement at the start of the drop. These were major
improvements over the earlier systems used. While difficult to determine, it was
estimated that the container generally hit at 5 to 10 0 off vertical. After impact the
container would rotate another 180 0 to 3600 and bounce .6 m (2ft) to 1 m (3ft) in
the air. A guiding system would have helped reduce the error in the experiment.
During the competition the package was going to rotate and the more elaborate test
fixture was not warranted at this stage in the evaluation.
High frequency oscillation made interpretation of the data difficult. Sources of the
oscillation were vibration of the outer container during impact, vibration in the
payload, and vibration induced by the air escaping from the fiber. The outer
container was made from ABS sewer pipe. Early in the experimental stages PYC
pipe was used but it tended to shatter on impact. Limited data using PVC indicated
it produced less high frequency oscillation. A second source of vibration is in the
payload. The payload was constructed with an 8 mm (0.3 in.) air gap between its
70
outside diameter and the outer container. There was no guiding system between the
two parts. Therefore during impact and subsequent rotation the payload could be
vibrating between the walls of the outer container. A third source of vibration could
be compared to sitting on a whoopee cushion. Air probably does not escape from
the fiber in a smooth continuous flow. Instead the air escapes in bursts.
The effects of high frequency oscillation were made worse by the performance of
the data acquisition card and accelerometer. The data was collected on a 16 bit
analogtodigital conversion card with a 10 micro second conversion time. The
frequency response of the transducer was flat from 1 Hz to 100 kHz and it has a
resonant frequency greater than 1 MHz. Because of the violent nature of the impact,
the high frequency oscillations in the system, and the high frequency response of the
data acquisition card, the acquired data from the experiments was quite noisy. To
filter out the noise, data was sampled at 20 kHz and then filtered. The filtering
equation was:
a(t) =
a(t) + a(t 0.00005)
2
8.2
Even with the filtering, the maximum acceleration was subject to interpretation.
71
Acceleration in g
The experimental data and calculated results across the diagonal of the design of
experiments are shown in Figs 8.3 to 8.5. Several points are noteworthy:
1) The model tends to over estimate the period during impact by about 40
%. While an error this consistent and large is reason for concern, the
period is not a variable the contestants were concerned about.
2) There appears to be a 2 kHz natural frequency in the test equipment.
3) The escaping air is not uniform and the fibers do not appear to slide
smoothly across each other during compression thus generating
additional noise.
Figure 8.3 DOE Corner Point (1 (p= 64 kg/m3), 1 (Zo=51 mm)) Block 1 Run 5
72
Figure 8.4 DOE Center Point (0 (128 kg/m3), 0 (Zo=128 mm)) Block 1 Run 1
Time (ms)
Figure 8.5 DOE Corner Point (+1 (192 kg/m3), +1 (Zo=102 mm)) Block 1 Run 7
73
Even with those shortcomings the regression coefficients for both the model and test
data were quite similar. The coefficients for the response surface generated by these
two designs are with in .32a to 1.06a each other. Considering the severity and
difficulty of the test, the model appears to compare very well to the physical test
data. These coefficients and the standard error for the test data coefficients are
shown in Table 8.2.
Table 8.2 Summary of Regression Equations
Coefficient Theory Test Data Standard Error (a) Difference (a)
Intercept 566 623 101 0.56
A: distance 427 480 105 0.50
B: density 391 280 105 1.06
A2 178 228 155 0.32
Bz 203 94 155 0.70
AB 266 403 129 1.06
74
The equations for the response surfaces generated by the regression analysis are:
0 = 566427,4
(fix tan ce
3914?,te,,v+178,4
(fix tan ce
+ 203SL,,,
+ 266Ajjs tan ce B
8.:
for the model and:
a = 623480,4
(fix tan ce
'2805ffem.(V + 228,4,^.lanc.t + 944?,i,,
+ 403/1 R 8 4
r o.r
for the test data. The two equations are compared graphically in Fig 8.6 by taking
cross sections of the surface at both experimental extremes of crush zones and one
section thru the middle.
75
2500 
Figure 8.6 Cross Sections of the Response Surfaces Predicted by the Design of
Experiments
Cross sections of the response curves are similar for both the 51 mm (2 in.) and 76
(3in.) crush zones. Maximum difference on these two cross sections is 160 g. Both
cross sections show similar curvature and slope. The cross sections for the 102 mm
(4 in.) crush zone do not compare as well in that the maximum error is 300 g at the
64 kg/m3 (4 lb/ft3) density edge. The curves are similar in that they both predict a
broad minimum acceleration. This would be important in an optimization. The
model predicts the minimum acceleration to occur at B=.308 or Density =147 kg/m3
76
(9.21b/ft3). Test data predicts the minimum to occur at B=.654 or Density=86
kg/m3 (5.4 lb/ft3).
Reported in Table 8.2 is the Standard Error for each of the test data surface response
coefficients. The standard error is the standard deviation of the coefficient based on
the fit of the data to the best surface. In the next column is the difference between
the coefficients in terms of standard deviation. The worst case error is 1.06 a. An
error of this magnitude would be expected 29% of the time.
Knowing the response curves for both the model and test data are similar is
important for optimization. A known acceptable design can be used as a starting
point. Proposed design changes can then be evaluated and compared to a standard
design in very short time. Only when an acceptable design is developed is it
necessary to build a series of containers to be used in destructive testing. Another
option would be to run a second design of experiments around the predicted
optimum to verify the point.
While the surface response curves for the model and test data where similar, the
consistent error in the calculated period was both a concern and a matter of
curiosity. To try and reconcile the differences new variables were added to the
77
model and existing values varied in software. The new model predictions are
plotted against the test data for a center point on the design of experiments.
In the original model no viscous damping was used. All force was generated as a
function of position. To evaluate if viscous damping would improve the correlation,
a term proportional to velocity was added to the force calculation. The results are
shown in Fig. 8.7. While the term reduced the period, it also distorted the shape of
the acceleration curve by adding too much acceleration early in the impact event.
At the same time it reduced the peak acceleration well below the measured value.
Time (ms)
Figure 8.7 Predicted Acceleration with a Viscous Damping Term Added to the
Model
It was also suspected that viscous damping might be a function of density. To
evaluate the possible effects of this, a damping term was added that was
proportional to the fiber stress. This would reduce the viscous effects early in the
78
event when the velocity was the highest. The damping would increase as the fiber
was compressed and at the same time its total effect would be reduced because of
the decreased velocity. The effects of these changes to the model are shown in Fig
8.8. In this case there was still too much acceleration at the beginning of the cycle
and the peak acceleration was too low.
Figure 8.8 Viscous Damping Proportional to Stress Added to the Model
Another possibility to reduce the differences in the period was the viscoelastic
properties identified in the fiber section. To evaluate this theory the force developed
by the fiber was multiplied by a factor of 2.5 to more than compensate for any
effects of viscoelasticity. This also did little to improve the correlation of the model
79
and data. The peak acceleration was reduced and too much work was done early in
the cycle. The results are shown in Fig. 8.9.
Figure 8.9 Force of the Fiber Multiplied by 2.5 to Account for the Viscoelastic
Properties of the Fiber
A known potential problem in the model when it was develop was the assumption
that the fiber was infinitely rigid until it reached the yield point were the fibers
started to slip. Because of that assumption work is overestimated early in the event.
This would overestimate the acceleration at the beginning of the cycle and
underestimate the peak acceleration. The opposite assumption could be made in that
the starting stress is zero and will be offset by that amount during the entire event.
This would underestimate the starting acceleration and overestimate the peak. The
result of this theory is shown in Fig. 8.10.
80
Figure 8.10 Predicted Acceleration of the Model with the Fiber Force Offset to
Zero at the Starting Density.
One final explanation of the offset in the period is the start time of the event is not
correct. Data was taken continuously during the drop and for several seconds after
the impact. The starting point of impact was estimated to be when the signal from
the accelerometer moved off zero. If very little work was being done during the
first few milliseconds of impact, the acceleration would be quite low and therefore
the starting point missed. To evaluate this theory the accelerometer data was started
1.8 ms ahead of the assumed point of impact. The results are shown in Fig. 8.11.
The data fits the predicted curves with this shift reasonably well. Of interest is the
negative acceleration at the beginning of the event for which there is no explanation
81
Figure 8.11 Accelerometer Data Starting 1.8 ms Ahead of the Apparent Start
The most likely scenario is a combination of improperly picking the event starting
point for the experimental data and overestimating the fiber force at the beginning
of impact. Fig. 8.12 shows the results when the data is shifted 1.5 ms and the force
is offset to zero at the beginning of impact.
82
Figure 8.12 The Results Of Shifting the Accelerometer Data 1.5 Ms and Zero
Force Developed by the Fiber at the Time of Impact
In the next section axial fiber will be examined and the shift in period will not be as
great. This is probably due to the stiffness of the fiber in this orientation. It will be
shown in Section 11 that the axial fiber at the same density is four times stiffer than
the radial fiber. With the increase in starting stiffness the model assumption of a
rigid material is reasonably valid. The increased stiffness also means the
acceleration will be higher early in the cycle and therefore the event start will not be
as easy to miss.
83
9. Comparison or Predicted And Measured
Performance for Axial Fiber
Most of the work done prior to the design of experiments had been done with axial
fiber. The advantages of the axial fiber became obvious shortly after its inception.
With the initial focus on developing a competitive container, little testing of the
model was performed on the radial fiber until there was a need to put the theory of
stress and density to a more rigorous test. More information on the effects of the
fiber orientation is presented in Section 11.
84
Figure 9.1 Axial Fiber Test Container
To evaluate the model with a more reasonable design three containers were tested
with axial fiber. The container design is shown in Fig. 9.1. A fiber density of 128
kg/m3 (8 lb/ft3) and Zo=76 mm (3 in.) was used in the test in that they are close to
the container used in competition. The results of axial fiber test and calculated
values for the design are shown in Figs. 9.2, 9.3 and 9.4.
85
Acceleration (g
Figure 9.2
Test 1 with Axial Fiber at 128 kg/nT Zo =76 mm and a 6 m Drop
86
Acceleration (g)
Figure 9.3 Test 2 with Axial Fiber at 128 kg/m3, Zo =76 mm and a 6 m Drop
87
Figure 9.4 Test 3 with Axial Fiber at 128 kg/m3. Z0 =76 mm and a 6 m Drop
The calculations use a leakage constant of 0 for the axial fiber. The method of
construction is very effective at entrapping air in the fiber. For the radial fiber a
constant of 100 was used to almost eliminate the effects of entrapped air. This is
discussed further in Section 11.
As can be seen from Figs. 9.2. 9.3, and 9.4, the test data was consistent with peak
accelerations of 461 g, 441 g, and 436 g. This compares well with the predicted
acceleration of 451 g, an error of less than 4%.
88
As in the radial fiber example the period was about 1 ms shorter than predicted.
The error in estimating the period is about half of the radial fiber example. The
initial stiffness of the axial fiber is much higher than the radial fiber. If the theory of
the starting point of impact was missed due to low acceleration, it would make sense
that the error using axial fiber would be less. Further work would be required to
resolve this question.
89

PAGE 1
PREDICTION OF ACCELERATION DURlNG IMP ACT USING THE RELATIONSHIP OF STRESS AND DENSITY by LARRY JOEL WEINSTEIN B. S. E. E. UNIVERSITY OF TOLEDO, 1968 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science 2002 .. ,"' I i 1A. L 1 ".J
PAGE 2
by Larry Joel Weinstein All rights reserved.
PAGE 3
This thesis for the Master of Science degree by Larry Joel Weinstein has been approved by
PAGE 4
Weinstein, Larry Joel (M.S., Mechanical Engineering) Prediction Of Acceleration During Impact Using The Relationship Of Stress And Density Thesis directed by Assistant Professor Ron Rorrer ABSTRACT The purpose of this work was to develop an understanding of the dynamics of impact for a fiber filled container. To accomplish this, a theory was developed to relate fiber density to bulk fiber stress. The theory also included the air entrapped in the fiber media. The resulting equations were then solved using numerical methods. The predicted and measured accelerations were tested in a design of experiments using a central composite design to develop regression coefficients. No significant difference was found between the two sets of coefficients. This information was then used to evaluate container designs for competition sponsored by the Institute of Paper Science and Technology (IPST) and the US Department of Energy (DOE). This abstract accurately represents the content ofthe candidate's thesis. I recommend its publication. Ron Rorrer IV
PAGE 5
ACKNOWLEDGEMENTS I would like to acknowledge the following people for their valuable help in this project. Jim Weaver was a great lab partner in designing fabricating and testing the containers used in competition. Ron Rorrer for his patience even when he had doubts about the project. My wife, Sherry for her patience and help in editing this paper.
PAGE 6
CONTENTS Figures ...... ..... ............................... .... ... ... ......................... ...... ... ....................... viii Tables ......... ........................................ ..... .............................................................. xi 1. Introduction ........ . ... ................... ......... ......... .......... ... ..... .... ...... ... ......... ..... ....... 1 2. Existing Technology .............. ...... ..... ............................... .......... ...... ... ........... 7 3. Report Overview .............................................................................................. 16 4. Stress as a Function of Density ... ...... . .................................... ...... ................... 21 5. Effects ofEntrapped Air ........ .......................................................... ............... 42 6 Numeric Methods ............................ . . ......... .... .... ........ ...... . .... ....................... 48 7. Fiber Data .................................... ..... ... ................................... ........................ 57 8. Design of Experiments .................... .. ... ............................................................ 61 9. Comparison of Predicted and Measured Performance for Axial Fiber ............ 84 10. Peak Acceleration .............. .... .... .......... .... ......... ... .................... ..... ................. 90 11. Effects of Fiber Lay ............. ........................ ...... ...... ..... . ...... .................. ..... 96 12. Forces on the Inner Container .......... ... .............................. .......... ................. 1 0 3 13. Effects of Altitude ...................... ................................................................. 106 14. Recommendations for Future Work ............ ..................... .......... .... ......... ... 1 08 VI
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15. Conclusions .................................................................................................. 112 Appendix A. Project Photographs ....................................................................................... 113 References ................................................................ ............ ....... ..................... 127 Vll
PAGE 8
FIGURES Figure I I Initial Container Design for the Energy Challenge . ........ .... . . .... ....... . .... 2 2.1 StressStrain Relationships of Various Fiber Densities ............ ..... ............... 9 2 2 Slope of the StressStrain Curve ...... ....... . ....... . ...... ... . ..... ..... . .... .... .... ... I 0 2.3 Typical Cushioning Curves Used in Packaging .... .............. .... ............... . 14 3.1 Mass Spring Model For Predicting Acceleration .... ................. ............. .... 17 4.1 Micro Photograph of Fiberglass ......................... ... ...... ............. ........ . ........ 22 4.2 Density of Air as a Function of Pressure ........... ................ . ............ 24 4 3 Relationship of Stress and Density for Hammermilled Newsprint . ....... . 25 4.4 Coordinate System ..................................................... .............. 26 4.5 Energy Relationship of Radial Fiber Density and Work .... . ............... .. 29 4.6 Relationship of Stress and Density Assuming Stress is Zero at the Starting Density ............................. .............. ... ........... .. ............ 34 4. 7 Relationship of Stress and Density Assuming Stress is Zero at the Fibers' Natural Density ............................................. ... ............ 35 4.8 Ideal Model of the Fiber to Include Recoverable Energy ........ ... ........ .. 37 4.9 Stress and Density for the Model in Fig. 4.8 That Includes Recoverable Energy ..... ..... .............. .... . ............. ..... ................ .... .................................... 38 4.10 Fiber Model used for the Numeric Model.. . ....... .. .. . .. .. . ....... . .... . . 39 Vlll
PAGE 9
4.11 Stress as a Function of Density for the Model Used in the Numerical Model ......................................................................................... 40 6.1 Numeric Model Definitions ........................................................ .49 6.2 Segments to Define the Inner Container. .......................................... 52 6.3 Effect of Time Step Size on the Predicted Acceleration ......... ............... 54 6.4 Summary Sheet from the Numerical Simulations ............................. 56 7.1 Stress Density for Radial Fiber. .................................................... 58 7.2 Viscoelasticity for Radial Fiber. .................................................... 59 8.1 Design of Experiments Test Points ................................................ 64 8.2 Test Container for the Design of Experiments ................................... 67 8.3 DOE Comer Point (1 (p= 64 kg!m\ 1 (Z0=51 mm))Block 1 Run 5 ...... 72 8.4 DOE Center Point (0 (128 kg/m3), 0 (Z0=128 mm))Block 1 Run 1 .......... . 73 8.5 DOE Comer Point (+1 (192 kg!m\ +1 (Z0=102 mm)) Block 1 Run 7 ...... 73 8.6 Cross Sections of the Response Surfaces Predicted by the Design of Experiments ................................................................................................. 7 6 8.7 Predicted Acceleration with a Viscous Damping Term Added to the Model ........................... ..... ...................................... .................................... 78 8.8 Viscous Damping Proportional to Stress Added to the Model. ................. 79 8.9 Force of the Fiber Multiplied by 2.5 to Account for the Viscoelastic Properties of the Fiber. .................................. ........................... 80 8.10 Predicted Acceleration ofthe Model with the Fiber Force Offset to Zero at the Starting Density ......................................................... 81 8.11 Accelerometer Data Starting 1.8 ms Ahead of the Apparent Start ............. 82 IX
PAGE 10
8.12 The Results Of Shifting the Accelerometer Data 1.5 Ms and Zero Force Developed by the Fiber at the Time oflmpact. ......................... ........... 83 9.1 Axial Fiber Test Container. ........................................................ 85 9.2 Test 1 With Axial Fiber At 128 Kg/m3 Zo =76 mm and a 6 m Drop ........................................................ ..... ...... .................... ... 86 9.3 Test 2 With Axial Fiber At 128 Kg/m3 Zo =76 mm and a 6 m Drop ........................................................ ............. 87 9.4 Test 3 With Axial Fiber At 128 kg/m3 Zo =76 mm and a 6 m Drop ............ ......... ................................................ ..................... 8 8 10.1 Predicted Accelerations For a 6m (20ft.) Drop and a 60 mm 94 11.1 Comparison of Radial and Axial Fiber Stress to Density ...................... 97 11.2 Ratio of Axial to Radial Fiber Stress .............................................. 98 11.3 Spring Constant for Radial and Axial Fiber with a 1 kg Payload Dropped from a 6 m and Maximum Deflection held at 60 mm ................. 99 11.4 Leakage Paths For Radial Fiber. ................................................. 101 11.5 Leakage Paths For Axial Fiber. .................................................. 1 02 12.1 Failure Mode From Impact.. ..................................................... 1 04 13.1 Predicted Accelerations For a 6 m (20 ft) Drop in Denver Colorado and Atlanta Georgia . .................. ................................................ 1 06 X
PAGE 11
TABLES Table 8.1 Summary ofthe Test Results by Averaging Two Adjacent Points ......... 68 8.2 Summary of Regression Equations ............................................. 74 Xl
PAGE 12
1. Introduction In the summer of 1997, the United States Department ofEnergy (DOE) and the Institute of Paper Science and Technology (ISPT) ( 1 .1) solicited design proposals for an insulated and impact resistant container constructed of paper for their Energy Challenge 98. The universities submitting the top 10 proposals received a $2 000 start up grant and competed for a $15,000 first prize. The proposal, written by University of Colorado at Denver (UCD) students and Dr. Ronald Rorrer, Assistant Professor, placed in the top 10. Dr. Rorrer challenged his Designing with Composites class, ME 5114, to participate in the project. Graduate students Jim Weaver and the author agreed to work on this project. The challenge was to design and manufacture a container to hold 1050 cm3 ( 64 in. 3 ) of 0 C (32 F) water, a raw egg, and survive a 6 m (20ft) drop to a concrete surface using only recycled paper and chemicals customarily used in the paper making process. A design that met the criteria was built and tested by mid semester. Final judging was a complex scoring system that graded the submissions on insulating performance, weight, cost of materials, percentage of recycled paper leakage condition of the egg and the quality of written reports and video.
PAGE 13
The basic design is shown in Fig. 1.1. The container is constructed in three parts; an inner container to hold the water and egg, a lid to seal the system, and an outer container to both absorb the shock and insulate the water. The original design was very rugged and could be dropped many times from 6 m (20ft) without failure. At 4 kg (9 lb ), the team participants felt it was over designed and therefore not competitive against many of the top paper science schools in the United States. EIGHT 1.5" WIDE x 17.6" LONG EHTACLES FROW OLD NEWSPRINT '9.0f< NOTES: 1) ALL DIMENSIONS IN INCHES 2) AREA: BOTIOM 64 SIDES 298 TOP 79 TENTACLES 422 TOTAL 863 SQUARE INCHES HAND LAYED OLD NEWSPRINT WITH STARCH INNER CONTAINER HAS .090" WALL OUTER CONTAINER HAS .020" WALL LOOSE FILL HAI.IWERJ.IILLED OLD NEWSPRINT Figure 1.1 Initial Container Design for the Energy Challenge 2
PAGE 14
To make the design more competitive, a program of optimization was initiated. The major focus in the optimization was reducing the overall weight of the container. The weight affected several areas of scoring and therefore yielded the most improvement with a reasonable effort. At the same time the container was being cost reduced, the reliability to withstand the 6 m (20ft) drop had to be maintained This would have required destructive testing of many containers that would have been very time consuming to construct. Therefore a system to evaluate proposed changes without a threetoseven day fabrication period was needed. In addition to evaluating materials there were many questions about the system dynamics during impact the participants wanted answered. Questions to be answered: The following questions needed to be answered in the course of this study: I) What is the expected peak acceleration? Early estimates placed the average acceleration at impact to be I 00 g. This meant the peak acceleration would be on the order of 200 g because acceleration is negligible during the initial stages of impact. Only as the fibers are compressed do they generate a force. The generated force peaks at the bottom of travel. It is known by compressing the fiber by hand its stressstrain relationship is highly nonlinear. Therefore 3
PAGE 15
acceleration would be lower than predicted by a simple linear model during the initial stages of impact and higher than expected at the end. The question was how high were these accelerations. If one could calculate the peak acceleration for an acceptable known design density, then physical dimensions could be established for other materials and densities. 2) What effect does fiber orientation have on its ability to absorb shock? The normal way fiber is introduced into a package is with random orientation. The fiber is then compressed to the desired density. When the fiber is compressed most of the fibers end up laying in the radial direction, which is perpendicular to the direction of loading. As a load is applied, the fibers bend easily and very little force is generated until the fibers are compressed to nearly a solid mass. A novel approach was proposed midway through the project to orient the fibers in the axial direction. This would load the fibers in their strongest direction. Therefore less material could be used to maintain the system stiffness. A second advantage was the crushing strength of the new system appeared to be more uniform It was suspected that with this shock absorbing system the peak accelerations would be reduced. The question was how would this proposed system affect the performance of the container? 4
PAGE 16
3) What density should be used to absorb the impact and how large should the crush zone be? Constructing the crush zone out of low density material meant the initial forces developed would be low. As the material collapsed, the forces would progressively increase. If the crush zone was too small these forces would result in unreasonable acceleration thus destroying the inner container. As the length of the crush zone is increased, so is the container weight. A method to estimate the optimum combination of density and length of a crush zone was needed. 4) How will the inner container fail? One possibility is that it would fail because of the internal pressure. Another possibility is that it would fail due to the external forces decelerating the inner container. By understanding how the container would fail, there may be a better way to design the system. The inner container held 1 kg (2.2 lb) of water and was constructed out of used newsprint layed up in a paper mache' method. The lamination was 2.2 mm (0.09 in.) thick. The expected peak pressure was about 13 70 kPa (200 lb/in?) at the bottom of the container. The internal pressure was counteracted by stress generated by the fiber absorbing the energy of impact. 5) What effect would altitude have on the performance of the system? Would the system perform better in Atlanta Georgia than in the Denver Colorado area? All 5
PAGE 17
testing ofthe container was done at elevations in excess of 1500 m (5000 ft), however, the competition was going to be held at nearly sea level. It was known relatively early in the design stage that air trapped in the fiber played a major part in how the inner container decelerated. This report describes the development of the theory to describe the fiber, entrapped air, and a numeric model to answer these questions and test results used to verify the calculations. 6
PAGE 18
2. Existing Technology The most direct approach to answer the questions posed in the Introduction would be to assume the system was linear. Using classic linear vibration analysis the peak acceleration of a body dropped from a height h would be: 2 2gh2 + ( g '_ )2 amax =OJ/I 2.1 {J)/1 OJ/I {J)/1 = J! 2.2 where Wn is the system's natural frequency. The constant k is calculated from the system geometry and the material's equivalent modulus of elasticity E The stressstrain curves for three different densities of fiber are shown in Fig. 2 1 Two things are apparent from the curves. First is the curves are highly nonlinear and the second is they appear to have little in common with each other making generalization or normalization difficult. The modulus of elasticity can be defined in terms of the differential E=dcr /dE. The results of the differentiation on raw data 7
PAGE 19
are shown in Fig. 2.2. The irregularities in the curves are due to experimental error being accentuated by the differentiation process. The modulus of elasticity for the 240 kg/m3 ( 15 lb/ft3 ) fiber could be approximated at 2000 kPa with reasonable results. The modulus of elasticity for the 64 kg/m3 ( 4 lb/ft3 ) fiber varied from 50 kPa to 10,000 kPa over the range of expected strain making any simple linear assumptions invalid. It would have been possible to approximate the data by assuming the strain was linear up to a break point and linear thereafter with a new slope. This was considered after a different method had been developed and therefore not pursued. 8
PAGE 20
Stress Strain for Hammermilled Fiber 700 ,600 ___ ,_ i Ci 500 24d +tr,J(151blft3 ) I I ; 400 i1' t:3 128 kg/m3 ... (ij 300 __ 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Strain Figure. 2.1 StressStrain Relationships of Various Fiber Densities 9
PAGE 21
' I I 10000 1 64 kg/m (4 lb/ft3 ) 8000 ns I I I a. 6000 128 kg/m3j Q) Q. 0 C/) 4000 2000 0 Figure 2.2 0 0.2 (8 lb/te) 0.4 Strain Slope ofthe Stress Strain Curve 0.6 0.8 1 To complicate the analysis even further, the air entrapped in the fiber must be taken into account. Early estimates put the force generated by the air to be nearly as great as the fiber. The air pressure would follow the ideal gas law except for any leakage. Obviously, numerical methods could be used to solve the system dynamics. The method was strongly considered because of its straight forward approach. The major drawback to this method was experimentally developing a stressstrain curve for each density to be evaluated because it was felt there was too large a variance 10
PAGE 22
between the densities to accurately interpolate the data In reviewing methods to solve the problem, there was a persistent feeling a more basic approach to the problem was being overlooked A literature search was initiated to gain insight into the dynamics of impact. Little useful information was found Eshleman in his paper "High Performance Shock Isolation Systems" (2.1) states Modeling of dynamic systems is often given the least attention in the process of design and analysis. Specific fundamental research in this area has been rare; however, data and procedures on modeling can be found scattered over many texts and papers." This author found this to be true. There were two general categories of work done in this field. The first is in shipping of packages for the transportation industry. The second series of work was in transportation of nuclear waste Transporting nuclear waste is a sensitive, highly political topic, so significant amounts of public money have been put into research to determine the safety and possible modes of failure for these containers. Tim pert et al, in a series of two papers (2.2) (2.3) looked at transporting nuclear waste by rail in drums His method of analysis is based on a finite element mass spring system. The work was limited to accelerations of only 6 g and no test data was given. Butler (2.4) in another article used a finite element method in his analysis for drops of up to 55 m. The 11
PAGE 23
extreme was work published by Brown (2.5) where he looked at transporting waste by air. Drop heights as high as 1 0 km and velocities as high as 200 km/h at impact were examined. He, also, used a finite element method of analysis. The predicted acceleration and deflection generally agreed within 10% ofthe test data. The general trend in the work reviewed was to use finite element analysis. Each year there are thousands of disappointed customers when their packages arrive damaged from shipping, and consequently millions of dollars in insurance claims are filed. It is to the benefit of manufacturers, warehousers, and shippers to work together for the safe delivery of packages at the lowest cost. The Institute of Packaging Professionals (IOPP) is an organization dedicated to that cause. Their focus is on drop heights ofless than 1.2 m (48 in.) and accelerations generally under 75 g. Studies have shown 50% of the packages dropped will fall less than 0.3 m (12 in.). These typically occur as a package is being transferred from one conveyor to another, or as the package is manually placed on a truck. Occasionally packages will be dropped from waist high. It is rare to have a package fall more than 1.2 m (48in.). In reviewing the IOPP publication "The Best ofTranspack" (2.5), a compilation of the best papers presented over the last five years at their symposiums revealed little in modeling package performance. In two papers there were references to the 12
PAGE 24
cushioning manufacturer's "proprietary software". Based on the work presented it could be assumed that the software was a finite element system provided by the cushioning manufacturer with a proprietary nonlinear addin module to match the material proposed by the cushioning manufacturer. This software would be of little use for predicting the performance of recycled paper without the needed addin modules. A method available to the public for predicting cushion performance is ASTM 1596 (2.6). In the test, weights are dropped on a sample of cushioning material and the maximum accelerations are plotted as shown in Fig. 2.3 (2. 7). This is a very time consuming test in that a minimum of 25 points are required for each curve. Each point would require a new cushion to be fabricated. The standard states in 13 .1 "Differences produced by these factors, together with specimen performance variation, might exceed 50% of the ideal. However, use of different systems testing a consistent material by this method should produce data that agree within 1 0%". These results are with refined and expensive commercial equipment. Ground newsprint would hardly be considered a consistent sample. In addition to needing specialized equipment separate curves would be need for each density and thickness to be examined resulting in hundreds oftests. The most important drawback to this test method is that it only works for flat surfaces. The inner 13
PAGE 25
containers used in this project were spherical and therefore the cushioning curves would not apply . z 0 i= cf a: "" ...J w u w 0
PAGE 26
element analysis. Development of such a method would fill a needed gap, and lead to a more basic understanding of fiber dynamics. 15
PAGE 27
3. Report Overview The purpose of this section is to give the reader an overview of the procedures chosen to answer the questions posed in the Introduction. This will introduce the reader to some of the deviations from classical vibration theory taken by the author to model the nonlinear response at impact. The work presented is the result of years of the model evolving and being refined even subsequent to completion of the competition. Only the final version of the model is presented in that much of the work was undocumented, would add little to understanding the technique developed, and quadruple the volume of the report. It is interesting to note after several years of refinement and testing, the conclusions drawn from modeling the impact have not changed. In its simplest form the model can be reduced to a series of springs and a frictional damper. This is shown in Fig. 3.1. Force on the container during impact will be generated by both the fiber and the air entrapped in the fiber matrix. The air acts as a nonlinear spring and will obey the ideal gas law. The pressure generated by the entrapped air will be discussed in Section 5. The equations 16
PAGE 28
developed will account for the nongaseous material in the cylinder and leakage to the atmosphere. Fiber Section 4 MASS 1 Air Section 5 Figure 3.1 Mass Spring Model for Predicting Acceleration The model for the fiber is more complex and is addressed in Section 4. Some characteristics of the fiber are similar to a spring in that when compressed some energy is stored and will be returned to the payload. Most of the developed force is due to friction of the fibers rubbing together as they are being compressed. The effects of the frictional spring for paper fibers are quite small and will be ignored in the model. The model assumes all energy absorbed by the fiber will dissipated and not returned to the payload. 17
PAGE 29
The force developed is unidirectional in that it is significant only when the fiber is being compressed. When the load on the fiber is removed and it is expanding the internal stress quickly goes to zero and normal fibertofiber force is low. Therefore the frictional forces are also low. In the model presented force will be a function of position and only valid while fiber is being compressed. It is important to forewarn the reader that the fiber equation developed will be in terms of stress and not force. This was done for several reasons. The most important reason is that stress and fiber density will be shown to form a basic relationship. A secondary but very important reason to work with stress is that non uniform payload shapes can be evaluated. Finally, the units of stress are the same as pressure so they can be summed with the air pressure in the model. A second item to advise the reader is that density always refers to the bulk density of the fiber composite and not the density of an individual fiber. In the same way stress also refers to the stress of the fiber composite and not the stress on an individual fiber. The model developed is a macroscopic picture and not microscopic. The composite fiber characteristics are developed from a simple experiment that utilized a press to compress the material. Data consisting of stress and density are 18
PAGE 30
then fit into a polynomial equation to be used later in the model. The procedure is described in Section 7 Another deviation from classical theory is that strain is not directly used Instead the equations are related to density. The advantages of using of density will become apparent in the derivations for air pressure and fiber stress A second benefit of working in density is the equations become more linear. Numeric models are generally more stable with linear systems. To solve the differential equations at impact Euler's method is used to calculate the displacement at finite steps in time. From the initial conditions the position of the payload is known, the volume of the fiber and density can be calculated Knowing the fiber density, the fiber stress can be calculated. The position of the payload also yields the air pressure. From pressure and stress the total force on the payload is calculated using cylindrical coordinates. Total force is then used to calculate acceleration. The process is then repeated in an Excel spread sheet with a finite time step to approximate a new position and therefore a new fiber density and air pressure This process is described in Section 6 19
PAGE 31
To verify the model a twofactor design of experiments was run A somewhat unusual approach was taken for the experiment in that regression coefficients for the factors were developed separately for the model data and the experimental data. These regression coefficients were then compared to each other by looking at their differences in terms of standard deviations. The design of experiments and discussion of sources of error are described in Section 8. Once confidence in the model has been established it can be used to answer the questions posed in the Introduction. This is done in Sections I 0 thru 13. 20
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4. Stress As A Function Of Density The purpose of this section is to translate the classical approach of using a material's stressstrain relationship to the concept of stress as a function of density The concept will prove useful in that multiple stressstrain curves can be combined into a single curve that can be created from easy to obtain data. The curve can be approximated by a linear function if a high degree of accuracy is not needed and large changes in density are not required. While this paper is dedicated to solving the dynamic problem ofthe impact of a mass on a paper matrix, it should prove useful in many other fields. The problem is similar to baling ofhay, cotton, trash, packaging of fiberglass, or stuffing a down coat in a pack. In all the above cases, the package size is reduced by means of mechanical compression, thus raising the bulk density of the material. Package volume is inversely related to the mechanical force applied to the package. For a fixed mass, density is also inversely related to package volume. Therefore mechanical force, or equivalently stress, should have a direct relationship to the density. For each material there should be a function that relates stress and density for a fibrous material. 21
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A fibrous network has more in common with gases than it does with its solid counterparts. Both fibrous networks and gases are mostly open space. In a gaseous system the molecules are held apart by molecular forces. In a fibrous network the fibers are held apart by the stiffness of the fibers that cross and interfere with each other. This is illustrated in Fig. 4.1 which is a photograph of a fiberglass wool. Figure 4.1 Micro Photograph of Fiberglass (4.1) 22
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The ideal gas law states: PV=nRT 4.1 Rearranging the terms to solve for pressure which is analogous to stress in a solid system: P = !!_RT v The term n is proportional to the density of the gas p. v If the temperature and mass of the system is held constant: P=Kp 4.2 4.3 Thus for a gas pressure is linearly proportional to the density. Figure 4.2 is a graph ofthe relationship of air density and air pressure. The data was extracted from Crane ( 4.2). Crane claims an accuracy of 3%. Over a two decade range of density the relationship of pressure and density is linear. 23
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M 90 80 70 .E 60 en 50 c ':;+ 40 30 G) c 20 10 0 0 : I I +.. ! i ; i ,', 10 20 30 40 50 60 70 80 Pressure in atm Figure 4.2 Density of Air as a Function of Pressure Test data for two starting densities of hammermilled paper are plotted in Fig. 4.3 to show the relationship of stress and density. Even though the starting densities of the two samples differ by a factor of four they converge into a common curve This would lead one to suspect that there is a common function for all densities and stress. 24
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700 600 200 100 0 100 i f j 1 , I 400 Figure 4.3 Relationship of Stress and Density for Hammermilled Newsprint Fig. 4.4 shows the coordinate system used in this analysis The X andY dimensions are held constant so strain Ex and Ey are equal to zero. This would be analogous to dropping a package in a cardboard box where the sides did not rupture. 25
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z y X Figure 4.4 Coordinate System All deflection is assumed to occur in the Z direction. Since the mass ofthe packaging material is constant and strain will be in the Z direction, density and strain will be related by the following equations : 26
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M Po= AZ 0 p is the original density of the segment under investigation, where: M = mass of the segment A = area of the segment, and Zo = the original Z dimension. When the segment is compressed the new density can be expressed as M where 8z is the change in the Z dimension Equation 4 6 can be rewritten as : 27 4 5 4 6 4.7
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While strain and density are related, in this analysis the principle concern will be density. In a fibrous network the material is gathered and then mechanically compressed to the desired density. The density for a loose pile ofhammermilled old newsprint is 32 kg/m3 (2 lb/ft\ Beyond that density mechanical work must be utilized to compress the material further. Figure 4.5 is the relationship of the work required to compress a kilogram of hammermilled fiber. The curve is valid only when the bulk density ofthe sample is increasing (op>O). Ifthe material is allowed to expand, some of the energy used to compress the material will be recovered. Also included in Fig. 4.5 is the unrecoverable energy plotted against density. The unrecoverable energy could also be called the energy of formation. It is the energy required to compress the material to a given density with all external forces removed. To develop this information the fiber was compressed to various densities and the total energy was calculated from force and distance. The fiber was then allowed to expand to its new density. The energy returned by the expansion was calculated from product of estimating the average force as one half the peak force and the distance of the fiber expanded. The returned energy was then subtracted from the total energy to get the unrecoverable energy shown in Fig. 4.5 28
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, >. tn ... (I) c: w 900 800 700 600 500 400 300 200 100 0 1"' I i +1 1' ! Totai:Energy I 0 ' Energy 1 i 1 i '1! ____ : tJ 0 100 200 300 400 500 600 700 Density (kg/m3 ) Figure 4.5 Energy Relationship of Radial Fiber Density and Work The total energ y Utotal, of the fiber is represented by the following equation: utolal = U,mrf!C O\'erahl e + u recov e rah/e 4 8 The term Uunrecover able is the energy to form the material to a given density when all force is removed from the materiaL The term Urecov e r able is the recoverable energ y when the material expands as the force is released. 29
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The term Urecoverable is important in two situations. The first situation is if rebound of the payload is going to be considered. In this case maximum deflection and acceleration all occur at the bottom of travel just before the fiber starts to reexpand. What happens beyond maximum deflection is not considered in this study. Therefore Urecoverable can be ignored. In the second situation the recoverable energy must be considered when the initial density is greater than its natural density. The recoverable energy must be supplied to the material before it will start to yield and permanently change density. In this situation the energy ofthe payload is about 1200 J/kg (400 ftlbr/lbm) of paper. At the starting densities that were expected to be used in the final design, the recoverable energy was only 6.8 J (2.3 ftlbr/lbm ). Since the desired accuracy is about 1 0% the recoverable energy can be ignored. Based on the previous discussion the energy per unit mass can be written: U(p )=Uunrecoverable(P )+Urecoverable(P) =Utotai(P) 4.9 As long as the recoverable energy at p0 is small and dp >0, Eq. 4.9 can be written in differential form. 30
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dU =U'(p) dp During impact, work is absorbed by the media and used to change its density: W =JAartz z., or dW =A ariz 4.10 4.11 4 .12 The negative sign appears because during compression dz is negative. Equation 4.9 is work per unit mass. To equate 4.10 and 4.12, Eq. 4.12 must be divided by the mass. dw = A
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U'(a)dp =adz P oZo Density and z are related by the following equations: M p=Az dp M =A dz=zdp M Therefore: 1 (j z ()dp=U'(p)dp P o Z o P o Z o Since p = P o Z o z 32 4.15 4.16 4 .17 4.18 4 .19 4.20
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4.21 Solving Eq. 4.21 : cr=f(p) +C 4.22 The constant C would be defined by the initial conditions of the fiber. One of two assumptions can be made about the initial conditions: 1) the fiber at its compressed density before impact has zero stress, or 2) the fiber has zero stress at its natural uncompressed density. The assumption chosen will have an effect on the maximum predicted acceleration. The error induced by either assumption will be a function of the magnitude of the recoverable energy at the starting density. If one assumes the fiber at its original starting density has no stress: C=f(p) 4.23 The stress as a function of density for a 128 kg/m3 (8 lb/ft3 ) radial fiber sample is shown in Fig. 4.6 using this assumption. The effect of this assumption is to drop the entire strain density curve by 23 kPa (3.3 psi). This assumption is reasonably valid 33
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for fibers that are either very soft at the beginning of the compression or the starting density is very close to the natural uncompressed density of the fib er. In either case the force at the beginning of impact will be underestimated. Because acceleration is underestimated at the beginning of impact more energy will be absorbed during the last stage of impact. Due to the nonlinear nature of the fiber the peak acceleration is overestimated with this assumption. This assumption proved to be reasonably valid for fiber layed in the radial direction. 250 200 Ci 150 .:.: In 100 In en 50 0 50 ; I . ______ _ _ ______ ___________ t j 50 100 1so 200 250 3(!)0 Density (kg/m3 ) .. Figure 4.6 Relationship of Stress and Density Assuming Stress is Zero at the Starting Density 34
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The other assumption is that the fiber is at zero stress only at its natural density. If this assumption is made, the fiber matrix is assumed to be rigid until the fiber reaches its yield point. The effects of this are shown in Fig. 4.7 for the previous 128 kg/m3 (8 lb/ft3 ) example. Generally acceleration at the beginning of impact is overestimated and the final acceleration is underestimated. This assumption is valid for materials that are quite stiff at their starting densities such as axially layed fibers or material bonded together. Most of the work done on the project used axial fibers and therefore this assumption was chosen. For the design of experiments radial fiber was used and the two methods are compared. 250 200 +150 ca Q. 100 "' "' C1) I I :Starting density ... 50 C/) 0 50 5:o 100 150 Density (kg/m3 ) . Figure 4. 7 Relationship of Stress and Density Assuming Stress is Zero at the Fibers' Natural Density 35
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Physically what is happening in the fiber matrix is the fibers are locked together by friction. The fibers deform axially in bending but do not slide when small forces are applied. This is a linear region that any applied energy is returned when the force is released. As the stress is increased the fibers begin to slip causing a permanent change in density. This work is unrecoverable and begins the nonlinear portion of the stress strain relationship. The point the fibers start slipping is a function of their coefficient of friction and the normal fibertofiber force. The normal force is a function of both the direction the fibers are laying in respect to the applied load and the density of the matrix. As the density goes up both the normal fiber to fiber normal force due to the Poisson ratio and the number of fibers in contact with each other will increase. All of these factors result in a highly non linear response in this region. When the force is released the matrix will reexpand returning some energy. More fibers at the new higher density will be in contact with each other. This results in shorter unsupported spans bridges, and columns that will result in a stiffer matrix and a new equivalent spring constant for the recoverable energy portion of the fiber model. The preceding example can be modeled as shown in Fig. 4.8. The spring represents the linear portion of the example where the fibers are bending and there is no slippage. As the applied force is increased and exceeds the frictional forces of the fiber. energy is dissipated, the density is permanently changed and a new normal 36
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fibertofiber force is established. In this model a smooth transition is made between the two assumptions discussed earlier. MASS Fn Figure 4.8 Ideal Model ofthe Fiber to Include Recoverable Energy The relationship between stress and density for this model is illustrated in Fig. 4.9 with a load applied and then removed. Also on the graph are both assumptions to show how the addition of a spring makes the transition between them. This would be the most accurate model of the fiber. 37
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200 150 C'CS a.. 100 t/) t/) Q) .... en 500 100 150 200 Density (kg/m3 ) 250 Figure 4.9 Stress and Density for the Model in Fig. 4.8 That Includes Recoverable Energy 300 While the model presented is the most accurate, it was not pursued because the additional work required to determine the spring constant as a function of density was not warranted at this stage of the project. Because the starting densities were low, errors induced by assuming the material was rigid until it started to slip were also low. The final model used to represent the fiber is shown in Fig. 4.1 0. The forces for the simplified model during impact are shown in Fig. 4.11. In this model all forces developed by the fiber are dissipative and no energy is returned to the payload. 38
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MASS +F n=f(p) Figure 4.10 Fiber Model used for the Numeric Model 39
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140 120 ti 100 D.. :. 80 60 40 20 0 ,' 125 175 225 Density (kg/m3 ) Figure 4.11 Stress as a Function of Density for the Model Used in the Numerical Model The result of this analysis is that if a stress and density relationship is experimentally established for one density, the function f(p) will hold for other starting densities provided the recoverable energy at the starting density is small. For convenience in calculations the term pis related back to the Z axis in the following equations: Zo p=po Z 40 4.24
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4.25 These equations are used in the numeric model presented in Section 6. 41
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5. Effects Of Entrapped Air During impact the air entrapped in the fiber generates pressures similar to the fiber. These forces must be taken into account. As shown previously, the stress generated during impact is a function of density. Therefore it is very convenient to relate pressure of the entrapped air to changes in volume as was done for the fiber. Assuming the process is isothermal and no air escapes during impact, the following equation can be written from the ideal gas law : 5.1 5.2 where: Po= original absolute pressure in the container, V o_air = original volume of air in the in the container, P = instantaneous air pressure in the container V air= instantaneous volume of air in the container, n = number of moles of air in the container 42
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R = gas constant and T = absolute temperature. The fiber itself is assumed incompressible and occupies space in the matrix. Therefore its volume must be taken into account. The volume ofthe solid material is calculated from its solid density. v ___ti_v solid 0 5.3 P.m/id where: Ysolid= volume occupied by the fiber, V0 = bulk volume of the fiber p0 =original bulk density of the fiber and Psolid = density of a individual fiber. The original volume of air is calculated from the total volume minus the volume of solid material in the container : V o air = V o _f!_Q_ V o P mli d 5.4 43
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The instantaneous volume of air is: v v _fl_ v air = 0 5 5 Pm!id where: V = the instantaneous volume of the container. Substituting these equations in Eq. 5.2 and converting to gauge pressure yields: v _fl_v 0 0 __ :....P.:::..m:.:..::li':....' __ l 5 6 As in the section on fiber stress the volume can be related back to the Z axis: Z Z _ll_ 0 0 P mlid 1 5.7 7Z _ll_ 0 P m lid The assumption of no leakage during impact proved to be valid for axial fiber designs That was not the case for radial fiber designs. The following development 44
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takes into account leakage. Leakage is assumed to be proportional to the difference in the internal and external pressure via a constant K'. dn = K' (PP. ) dt 0 5.8 where: n = number of moles of air in the container. Solving the differential equation yields : I n =J K' (PP0 )dt + C 5.9 0 At t=O: n=no 5.10 Therefore: I n = n0 JK'(PP0)dt 5.11 0 5.12 45
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(PV J PV. = o o _air K' f(Pp }dt RT atr RT 0 0 5.13 Converting to gauge pressure yields: p = P0 Vor_"ir (1RTK' J(PP0 }dt)RTP0 } air PoVo_air 0 5.14 Let K= RTK' 5.15 and Z v 0 0 o air P solid = '==5.16 Vair zZo Psolid 5.17 46
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Equation 5.17 is similar to 5.2 with the addition of the leakage term. If leakage needs to be accounted for the constant K would be varied in software to reduce the error in the calculated results. This was done in some simulations to get a better fit between predicted and experimental results for the payload rebounding after impact. When testing fiber that had been layed in the radial direction the leakage was significant because of the fiber orientation air could easil y escape along the boundaries K in this case was made large enough that the effects of the entrapped air were negligible. For axial fiber the air was effectively trapped and therefore K=O. K is probably a function ofthe fiber density but it is outside ofthe scope of this project to define the function. 47
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6. Numeric Methods The equations developed for fiber stress and air pressure in the previous two sections are a function of their initial conditions and the z dimension. The primary container under consideration was cylindrical on the top and spherical on the bottom. Since the container was symmetrical about the Z axis and the z dimension was the primary concern, a cylindrical coordinate system was chosen. The container was defined by the radius of the spherical bottom and the minimum z dimension from the center of the container to the flat bottom of the fiber. This is shown in Fig. 6.1. 48
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Zo (r) Zo(Q) Figure 6.1 Numeric Model Definitions The initial dimensions can be calculated as: 6 1 During impact the dynamic dimensions are : z(r) = Z0 (r)Z0 (0) + z(O) 6 2 49
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Knowing the initial and the instantaneous dimensions the density can be calculated by: ( ) Z0.(r) P r =Po z(r) As shown in Section 4 bulk fiber stress is a function of the fiber density. This relationship is experimentally determined in Section 7. The data is fit into a polynomial. The equations for gauge air pressure from Section 5 are in the form of initial and instantaneous position, thus the pressure as a function of radius is now P(r) = P0 6.3 6.4 6.5 The integral for leakage is accounted for in the time steps that will be used later in Euler's method for solving differential equations. 50
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The force on the payload can then be calculated from the fiber stress and air pressure. 2trll F = f fCa1;,",(r) + p(r))rdrd() + Mg 6.6 0 0 The integration was carried out numerically with radial bands of 3.17 mm (0.125 in.) as shown in Fig. 6.2. The majority of the rest of this report is based on test containers with a flat bottom in order to easily compare the model with experimental results. To simulate a flat bottom a radius of25 m (1000 in.) was entered for R. 51
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Zo
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The initial conditions at impact are : v(O) = 6 8 and z(O) = Z0 (0). 6.9 Equations 6.2 through 6.9 are solved in an Excel spread sheet using Euler's method with generally 0.2 millisecond time steps. Figure 6.3 shows the effect of step size for a nominal container. 53
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700 600 C) c:: 500 0 .... 400 Cl) Cl) (,) (,) 300 <( .:.:: 200 C'G Cl) a. 100 0 0 Nominal 0.2 rins step 0.5 1 1.5 2 2.5 i Size of Time Steps (ms) Figure 6.3 Effect of Time Step Size on the Predicted Acceleration For higher accelerations needed for some of the points in the design of experiments a 0.1 millisecond step was used because rates were as high as 6,000,000 g/s. Under those conditions the peak acceleration was generally underestimated by the larger time step. The model became unstable only when the time steps were larger than 2 ms or if too large a leakage constant was chosen. The leakage constant is zero for all the axial fiber and 100 for the radial fiber calculations in this report. In the radial fiber calculations the entrapped air was made negligible by the leakage constant. 54
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Instability was determined by observing negative air pressures during the impact compression cycle. The program could have been simplified by dropping the entrapped air from the model for radial fibers, but this would have restricted the versatility of the software. A sample summary sheet of the calculations is shown in Fig. 6.4. The setup of the spreadsheet is a balance between versatility and ease of programming. For this specific problem the data entry and programming could have been simplified. The purpose of the expanded flexibility was to be able to explore tangents to this problem with little additional work. Expanded even further the program could be a valuable and versatile tool to the packaging industry to insure safe shipment of their material. 55
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.... \JCI = .., DROP HEIGHT 20FT nl 0'1 X"5 X"4 X"3 X"2 X c DENSITY 6 #/FP3 0 0 0.0115 0.0952 1.0648 2.7993 SOLID DENSITY 65 #/FP3 . r./1 c: 3 ZMIN 31NCHES MaxAcell= 417.3472 400 3 INNER CONTAINER R 21NCHES MinZ= 0 .77636 I 300  .. cSeri;81J Ill WEIGHT 2.2 POUNDS maxstress 169.0782 l:t= Q r./1 PRESSURE 12.5 PSI ::r 0 11 , Leakage 0 0 5 10 15 ..... TIME STEP 0 2 MSEC :::t> msec 0 I 3 ..... MSEC FORCE FORCE FORCE MAXIMUM STRESS ::r TIME z ACCEL VEL TOTAL FIBER AIR TOTAL FIBER AIR Vl z 0 3 0 36.22154 0 0 0 0 0\ c: 0 2 2 .913068 17.0654 36.11232 37.54388 35.45602 4 .287869 3 709232 2 .858366 0 .850866 3 0 4 2.826399 20.20036 35.98304 44.4408 37.83454 8 .806261 4.409806 3 .091014 1.318792 '"1 0 6 2 .740039 23.53762 35.8324 51.78277 40.41095 13.57182 5 .16575 3.347218 1.818532 r; 0.8 2.654042 27.09633 35.65898 59.61192 43.2093 18.60261 5.983562 3.630485 2.353077 r./1 1 2 .56846 30.89797 35. 46124 67.97553 46.25729 23.91824 6.870735 3.944972 2.925763 1 2 2 .483353 34.9667 35. 23745 76.92675 49.58678 29.53997 7 .835937 4 .295614 3 .540323 s:: 1.4 2 .398783 39.32973 34.98574 86.5254 53.23455 35.49085 8 .889232 4 688299 4.200932 6) 1.6 2 .314818 44.0177 34.70402 96.83894 57.2431 41.79585 10.04234 5 .130066 4 .91227 ..... ...... 1 8 2 .231528 49.06524 34.39001 107.9435 61.66161 48.48193 11.30894 5.629356 5 .679581 0 ::: 2 2 .148992 54.51147 34.04113 119.9252 66.54704 55.57819 12.70507 6.19632 6 .508747 ell 2 2 2.067293 60.40058 33. 65457 132.8813 71.96537 63.1159 14.24956 6.843196 7.40636 2 4 1 .986522 66.78252 33. 22716 146.9216 77.99304 71.12852 15.96456 7 .584759 8 .379805 2 6 1.906777 73.71369 32.75539 162.1701 84.71852 79.65159 17.87621 8.438877 9 437336 2.8 1 .828164 81.25757 32. 23535 178.7667 92.24402 88.72264 20.01532 9 .427172 10.58815 3 1 .750799 89.48545 31.66264 196.868 100.6872 98.38075 22.41824 10.57579 11.84245 3 2 1.674809 98.47684 31.03239 216.6491 110.183 108.666 25.12779 11.91633 13.21146 3 4 1 .600331 108.3198 30.33914 238.3036 120.8849 119.6187 28.1942 13.4868 14.70741 3 6 1 .527517 119.1106 29.57683 262.0434 132.9656 131.2778 31.67614 15.33272 16.34341 3 8 1 .456533 130.9529 28.73873 288.0963 146.6173 143.679 35.6414 1750817 18.13322
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7. Fiber Data To determine the experimental fiber characteristics, a measured amount of fiber was placed in a 102 mm (4 in.) round PVC tube and compressed with a threaded plunger. Force was monitored with a strain gauge. Forces of over 4,400 N (1 000 lb) were generated in the press. Defection was obtained by counting the turns of the plunger. The raw data of force, turns on the plunger, and weight of the fiber were entered into a spread sheet and stress and density were calculated. This information was then graphed and a polynomial trend line fit to the data. An example of a graph and polynomial for the radial fiber is shown in Fig. 7 .1. 57
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I700 600 500 cu Q. 400 .:.:: 200 100 0 ;9 ( 10T6)x_3 ___ o. ooo6x2 + o .1 04 x1." 1_8_4_ 7 1 i j R2 = 10.9997 I i I I __ __J__ _________ ,r.l I _j_ _ l ! I i '0 100 200 300 400 500 Density (kg/m3 ) Figure 7.1 Stress Density for Radial Fiber The fiber showed viscoelastic properties in that stress would slowly decay after the move on the plunger was made. It is not known if the drop in stress is due to the air entrapped in the fiber slowly escaping or due to the fiber itself An arbitrary decision was made to capture data 2 seconds after completion of the move. The rate of decay is shown in Fig. 7.2. 58
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700 . . ' i i y = 38.683Ln(x) + 677 18 600 tt"' ll. i 500 ill) 'stress at 2 seconds i ; t ; II) Q) ... en 400 ... 300 _______ _ t___ _________ ______ _:_ ' 1 10 100 1000 Time (s) Figure 7.2 Viscoelasticity of Radial Fiber Data taken 2 seconds after the plunger was moved may introduce considerable error in the calculations. Assuming viscoelastic properties of the fiber and the regression equation is correct at 0 005 seconds the force might be 880 kPa (130 psi). In this case error would be 45%. The viscoelastic curve does not appear to be a linear logarithmic function for small times Therefore the error induced may not be a large as indicated. It might be possible to take similar data throughout the range of densities and project them back to a 0.005 second time base for more accurate results. That was not done for this project in that trends and insight of the processes occurring at impact were more important than the absolute numbers Work 59
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presented in the Design of Experiments Section 8, will show the shift in stress due to viscoelasticity is not significant. 60
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8. Design Of Experiments Test results on containers that were similar to the final design compared very well to the results predicted in the mathematical model developed in the previous sections. There was some concern whether the model was truly representative of the system dynamics or if this was coincidental. To test the model, a design of experiments was set up. The goal was to test the model at extreme conditions that would be out ofthe range of normal designs and then examine their response surfaces. The surfaces were then compared to each other by analyzing their equation coefficients and their standard deviations If the model held for these extreme designs, the model is probably a reasonable representation of the system. The major factors that are entered in the model are: 1) the height of the crush zone Z0(0), 2) the density ofthe fiber, 3) the weight of the payload, and 4) the height of the drop. To evaluate all four factors at three levels would have required 81 experiments, which is an inordinate amount of testing Only the first two factors were variables the contestants could change. Drop height and weight of the payload were dictated by the rules of competition. Therefore a two factor and three level design of experiments was performed. 61
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It was expected that the response as these factors were varied would be highly non linear. Therefore the data would be fit into a quadratic equation to account for the system nonlinearities To develop the surface response equation a central composite face centered design was chosen. DesignExpert was used to set up and analyze the data. To fit into the software the factors were given names of only A, B, and coded values of 1. 0, and+ 1. These coded values and the actual values are shown in Fig 8.1. Factor A is the crush zone Zo(O). A nominal crush zone of 76 mm (3 in.) was maintained as a center point. The lowest practical crush zone and still preserve the accelerometer was 51 mm (2 in.) The 51 mm (2 in.) crush zone with the 64 kg / m3 ( 4 lb /ft3 ) resulted in nearly a 3000g acceleration and the accelerometer was rated at 4000g That established the maximum crush zone at 1 02 mm ( 4 in.) to maintain the design symmetry. Factor B is density of the fiber. The lowest density that would support the payload was 64 kg/m3 ( 4 lblft\ Below that density the crush zone could not be maintained. The highest density possible by compressing the fiber by hand was 192 kg / m3 ( 12 lblft\ These two end points forced the center of the design to be 128 kg / m3 (8 lb !ft\ To test the model at extreme conditions the fiber was in the radial 62
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direction This is the direction the fiber is the weakest, the most nonlinear and would poorly entrap the air. While this design makes for a good test of the model, it would make a poor shock absorbing package. 63
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... (JQ c ., QCI u 0 (/l ao ;:l 0 tTJ X '"0 0 :::!. 3 0 ;:l (/l l 0 0\ (/l ...... +:> '1:1 0 s (/l Fo_ctor B Density t (1,1> 1 Test 7 51 "" ." l Test o.t <LD 51 MM, 128 kg/1"13 t <1,D 1 Test kg/M3 ;7 51 1'11"1, / 1 Test o.t (0,1) 76 MM, 192 kg/M3 6 Test at 76 1'11'11 128 kg/1"13 1 Test o.t <0,D 76 MM, 64 kg/1'13 1 Test o. t <1,1> 102 MM, 192 kg/M3 1 Test o.t <1,0) 102 MM, 128 kg/M3 Fo_ctor A Crush zone 1 Test o.t
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The design required 14 experiments to be run. Six of the experiments are at the design center. The other eight experiments are around the periphery of the nominal design. Unfortunately there was not enough fiber to run all 14 experiments in one batch. To overcome the fiber shortage the experiment was divided into two blocks. Blocking an experiment is a statistical way to compensate for changes in experimental conditions when all the experiments cannot be run at the same time. In this case there was a concern that used fiber would not respond the same way as virgin fiber. Points in the blocks were chosen by the software. The software automatically shifts the data to compensate for changes in experimental conditions. Even though the fiber was reused for the second block, the mean for both blocks was similar. Tests in each block were performed in random order. The end result of the experiment is a surface response equation relating variables A and B to acceleration. The equation is in the form of: where A and B are the coded variables ranging from 1 to + 1. For each of the constants in the equation, a standard deviation is reported by the software. 65 8.1
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The container to be tested is shown in Fig. 8.2. To simplify the construction and instrumentation a flat bottom was used in the experiment. The tube was an ABS sewer pipe with an end cap glued in place. The payload was fabricated from a pressboard disc and weighted with steel washers to meet the weight specification. There was an 8.0 mm (.3 in.) gap around the disc to allow air to freely escape during impact. The container was suspended 6 m (20ft) above a concrete pad and released by pulling on a spring clip. Photos of the test container, drop area, and release mechanism are shown in the appendix in Photos A. I to A.l4. 66
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.35 f"'lf"'l 1 kg Po.yloo.cl with o.cceleroMeter o.nd 8 MM o.ir go.p Ro.cllo.l Lo.yecl Fiber _,.(horizonto.D Drop He ght 6 M Figure 8.2 Test Container for the Design of Experiments 67
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Table 8.1 Summary ofthe Test Results by Averaging Two Adjacent Points Standard Run Block Crush Density Test CalcuDiffer%Error Order Order Height lations ence 1 5 1 2 4 2418 2121 297 14 2 2 1 4 4 485 654 169 25 ,.., ,.., 1 2 12 783 725 58 8 ..) ..) 4 7 1 4 12 462 325 137 42 5 4 1 3 8 681 568 113 19 6 1 1 3 8 663 568 95 16 7 6 1 3 8 506 568 62 10 8 11 2 2 8 1036 1080 44 4 9 14 2 4 8 405 385 20 5 10 10 2 3 4 602 1070 468 43 11 12 2 3 12 570 446 124 27 12 13 2 ,.., 8 895 568 327 57 ..) 13 18 2 3 8 631 568 63 11 14 9 2 ,.., 8 535 568 33 5 ..) Data from the experiments is shown in Table 8.1. The percent error was as large as 57%. This was not unexpected because in an ASTM 1596 (2.6) standard they warn 68
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that testing errors of 50% are common. The ASTM test deals with drops of only 4 feet and uses equipment refined from years of use. The design called for six test points at the center of the experiment. When these six points are examined independently, their standard deviation is 138 g. Errors range from +37% to 22%. Errors such as the ones above reinforce the need for a mathematical model to gain understanding of the process of shock absorption in that changes in performance could easily be masked by experimental error. Experimental error is probably due to three major factors: inconsistencies in the fiber, the container not hitting the concrete pad squarely, and high frequency oscillation during impact of both the outer container and the payload. Another source of error is determining the time when the impact event started. The time the event started does not affect the peak acceleration but will be discussed later in the report to try to reconcile the differences in the measured and calculated period. Every attempt was made to keep the fiber as consistent as possible. It was thoroughly mixed between experimental blocks, measured, and inserted in the container the same way. Small differences in handling could easily change the fiber orientation and thus its apparent characteristics. While not studied, it is also known that humidity changes the characteristics of the fiber. Each test took about an hour 69
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to set up and run. Over the course of the seven test runs in each block there were significant shifts in temperature and probably in relative humidity during the day Rotation of the container is another source of experimental error. The container was allowed to freefall with out guides. Instead of using a guide system. the container was lengthened to increase its rotational moment of inertia and a release mechanism was developed minimize movement at the start of the drop. These were major improvements over the earlier systems used. While difficult to determine it was estimated that the container generally hit at 5 to 10 off vertical. After impact the container would rotate another 180 to 360 and bounce .6 m (2ft) to 1 m (3ft) in the air. A guiding system would have helped reduce the error in the experiment. During the competition the package was going to rotate and the more elaborate test fixture was not warranted at this stage in the evaluation. High frequency oscillation made interpretation of the data difficult. Sources of the oscillation were vibration of the outer container during impact, vibration in the payload and vibration induced by the air escaping from the fiber. The outer container was made from ABS sewer pipe. Early in the experimental stages PVC pipe was used but it tended to shatter on impact. Limited data using PVC indicated it produced less high frequency oscillation. A second source of vibration is in the payload. The payload was constructed with an 8 mm (0.3 in.) air gap between its 70
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outside diameter and the outer container. There was no guiding system between the two parts. Therefore during impact and subsequent rotation the payload could be vibrating between the walls of the outer container. A third source of vibration could be compared to sitting on a whoopee cushion. Air probably does not escape from the fiber in a smooth continuous flow. Instead the air escapes in bursts. The effects of high frequency oscillation were made worse by the performance of the data acquisition card and accelerometer. The data was collected on a 16 bit analogtodigital conversion card with a 10 micro second conversion time. The frequency response of the transducer was flat from 1 Hz to 100 kHz and it has a resonant frequency greater than 1 MHz. Because of the violent nature of the impact the high frequency oscillations in the system and the high frequency response of the data acquisition card, the acquired data from the experiments was quite noisy. To filter out the noise data was sampled at 20 kHz and then filtered. The filtering equation was: a(t) = a(t) + a(t0.00005) 2 Even with the filtering, the maximum acceleration was subject to interpretation. 71 8.2
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The experimental data and calculated results across the diagonal of the design of experiments are shown in Figs 8 3 to 8.5. Several points are noteworthy: 1) The model tends to over estimate the period during impact by about 40 %. While an error this consistent and large is reason for conc e rn, the period is not a variable the contestants were concerned about. 2) There appears to be a 2kHz natural frequency in the test equipment. 3) The escaping air is not uniform and the fibers do not appear to slide smoothly across each other during compression thus generating additional noise. 3000 r ...... ......... ,;. .... ... .... . . .. .... .. 2500 Calculated i r.Cl 2000 1 !+ j __ __.,___ ____ _____ ___; _ = r::: 1500 0 i ____ ___.___ _________ _____ ___ :;:; ca ... 1000 Cll (ij _ __j__ (.) (.) 500 <( 0 500 Time (ms) Figure 8.3 DOE Corner Point (1 (p= 64 kg/m3 ) 1 (Z0=51 mm)) Block 1 Run 5 72
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700 600 _500 C) s::: 4000 .. Q) "iii 0 0 < 2001000 100 _____ _2 _________ _4 ______ 6_ ___________ 8 _______ __10 Time (ms) Figure 8.4 DOE Center Point (0 (128 kg!m\ 0 (Z0=128 mm)) Block 1 Run 1 600. 500 400 C) s::: 300 0 .. n:s IQ) 200 "iii 0 0 100 <( 0 . 100 4 9 to Time (ms) Figure 8.5 DOE Comer Point (+1 (192 kg!m\ +1 (Z0=102 mm)) Block 1 Run 7 73
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Even with those shortcomings the regression coefficients for both the model and test data were quite similar. The coefficients for the response surface generated by these two designs are with in .32cr to 1.06cr each other. Considering the severity and difficulty of the test, the model appears to compare very well to the physical test data. These coefficients and the standard error for the test data coefficients are shown in Table 8.2. Table 8.2 Summary of Regression Equations Coefficient Theory Test Data Standard Difference (
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The equations for the response surfaces generated by the regression analysis are : a= 566427 Adinance 39lBdenw_v + l78A, ;;,,tance + 203B 3"""ity + 266A d i nanceBde n\'it_v 8.3 for the model and: for the test data. The two equations are compared graphically in Fig 8.6 by taking cross sections of the surface at both experimental extremes of crush zones and one section thru the middle. 75
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C) ........ 1 02 mm crush zone I Model Test 0 (128 kg/m3 ) Density Figure 8.6 Cross Sections ofthe Response Surfaces Predicted by the Design of Experiments Cross sections of the response curves are similar for both the 51 mm (2 in.) and 7 6 (3in.) crush zones. Maximum difference on these two cross sections is 160 g. Both cross sections show similar curvature and slope. The cross sections for the 1 02 mm (4 in.) crush zone do not compare as well in that the maximum error is 300 gat the 64 kg/m3 ( 4 lb/ft3 ) density edge. The curves are similar in that they both predict a broad minimum acceleration. This would be important in an optimization. The model predicts the minimum acceleration to occur at B=.308 or Density =147 kg/m3 76
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(9.2lb/ft3 ). Test data predicts the minimum to occur at B=.654 or Density=86 kg/m3 (5.4 lblft\ Reported in Table 8 2 is the Standard Error for each of the test data surface response coefficients. The standard error is the standard deviation of the coefficient based on the fit of the data to the best surface. In the next column is the difference between the coefficients in terms of standard deviation. The worst case error is 1.06 cr. An error ofthis magnitude would be expected 29% of the time. Knowing the response curves for both the model and test data are similar is important for optimization. A known acceptable design can be used as a starting point. Proposed design changes can then be evaluated and compared to a standard design in very short time. Only when an acceptable design is developed is it necessary to build a series of containers to be used in destructive testing. Another option would be to run a second design of experiments around the predicted optimum to verify the point. While the surface response curves for the model and test data where similar the consistent error in the calculated period was both a concern and a matter of curiosity. To try and reconcile the differences new variables were added to the 77
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model and existing values varied in software. The new model predictions are plotted against the test data for a center point on the design of experiments. In the original model no viscous damping was used. All force was generated as a function of position. To evaluate if viscous damping would improve the correlation, a term proportional to velocity was added to the force calculation. The results are shown in Fig. 8 .7. While the term reduced the period, it also distorted the shape of the acceleration curve by adding too much acceleration early in the impact event. At the same time it reduced the peak acceleration well below the measured value. 700 ,600 !' 500 400 : / calculjted 300 ! i I / : Q) a; u u < 200 4 Time (ms) Figure 8.7 Predicted Acceleration with a Viscous Damping Term Added to the Model It was also suspected that viscous damping might be a function of density. To evaluate the possible effects of this a damping term was added that was proportional to the fiber stress. This would reduce the viscous effects early in the 78
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event when the velocity was the highest. The damping would increase as the fiber was compressed and at the same time its total effect would be reduced because of the decreased velocity. The effects ofthese changes to the model are shown in Fig. 8.8 In this case there was still too much acceleration at the beginning of the cycle and the peak acceleration was too low. 700 , 600 500 c 400 0 ;; 300 I!! Q) Ci) 200u u < 100 0 100 'L4 __ __.,_ __ _ i i ____ ____;,_. ______ __IS. : .. : Time (ms) Figure 8.8 Viscous Damping Proportional to Stress Added to the Model Another possibility to reduce the differences in the period was the viscoelastic properties identified in the fiber section. To evaluate this theory the force developed by the fiber was multiplied by a factor of 2.5 to more than compensate for any effects ofviscoelasticity. This also did little to improve the correlation of the model 79
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and data. The peak acceleration was reduced and too much work was done earl y in the cycle. The results are shown in Fig. 8.9. t: 0 ;; e c Qj u u < 600 _, _ ____ :\ Calculated 500 400 300 200 100 '' _____ _ : _ ____ L _ ___ /'l 1 : 0 .. 100 Time (ms) Figure 8.9 Force of the Fiber Multiplied by 2.5 to Account for the Viscoelastic Properties ofthe Fiber A known potential problem in the model when it was develop was the assumption that the fiber was infinitely rigid until it reached the yield point were the fibers started to slip Because ofthat assumption work is overestimated early in the event. This would overestimate the acceleration at the beginning of the cycle and underestimate the peak acceleration. The opposite assumption could be made in that the starting stress is zero and will be offset by that amount during the entire event. This would underestimate the starting acceleration and overestimate the peak. The result of this theory is shown in Fig. 8.1 0. 80
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800 , __________ ____ ____ ____ .. _. ... Calculated : 700 ;' 6oo A1 500 +: 1t'tcI 400 1 e I Q) 300 ':+: Q) CJ CJ c( 200 100 , Time (ms) Figure 8.10 Predicted Acceleration of the Model with the Fiber Force Offset to Zero at the Starting Density. One final explanation of the offset in the period is the start time of the event is not correct. Data was taken continuously during the drop and for several seconds after the impact. The starting point of impact was estimated to be when the signal from the accelerometer moved off zero. If very little work was being done during the first few milliseconds of impact, the acceleration would be quite low and therefore the starting point missed. To evaluate this theory the accelerometer data was started 1.8 ms ahead ofthe assumed point of impact. The results are shown in Fig. 8.11. The data fits the predicted curves with this shift reasonably well. Of interest is the negative acceleration at the beginning of the event for which there is no explanation. 81
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700 600 500 c: 400 0 300 E Cl) Qj 200 t.) t.) <( 100 , 0 . .. ; . i 100 Time (ms) Figure 8.11 Accelerometer Data Starting 1.8 ms Ahead of the Apparent Start The most likely scenario is a combination of improperly picking the event starting point for the experimental data and overestimating the fiber force at the beginning of impact. Fig. 8.12 shows the results when the data is shifted 1.5 ms and the force is offset to zero at the beginning of impact. 82 j
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800 700 600 500 c 400 Q) 3oo r _ a; 200 <( 100 o +1 1 00 __ Time (ms) Figure 8.12 The Results Of Shifting the Accelerometer Data 1.5 Ms and Zero Force Developed by the Fiber at the Time of Impact In the next section axial fiber will be examined and the shift in period will not be as great. This is probably due to the stiffness of the fiber in this orientation. It will be shown in Section 11 that the axial fiber at the same density is four times stiffer than the radial fiber. With the increase in starting stiffness the model assumption of a rigid material is reasonably valid. The increased stiffness also means the acceleration will be higher early in the cycle and therefore the event start will not be as easy to miss. 83
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9. Comparison or Predicted And Measured Performance for Axial Fiber Most of the work done prior to the design of experiments had been done with axial fiber. The advantages of the axial fiber became obvious shortly after its inception. With the initial focus on developing a competitive container little testing of the model was performed on the radial fiber until there was a need to put the theory of stress and density to a more rigorous test. More information on the effects of the fiber orientation is presented in Section 11. 84
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Figure 9.1 Axial Fiber Test Container with o.cceleroMeter go.p To evaluate the model with a more reasonable design three containers were tested with axial fiber. The container design is shown in Fig. 9 .1. A fiber density of 128 kg/m 3 (8 lb/ft3 ) and Z0=76 mm (3 in.) was used in the test in that they are close to the container used in competition. The results of axial fiber test and calculated values for the design are shown in Figs 9 2, 9.3 and 9.4. 85
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C') c 0 :;; f! Q) Q) (.) (.) <( 500 I I 400 Test Data : 200 0 I I 2 I 6 100 ..... Time (ms) Calriulated ! ;.c' rI I i 7 8 g) 10 Figure 9.2 Test 1 with Axial Fiber at 128 kg / m3 Z o =76 mm and a 6 m Drop 86
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500 1 T"" I 400 Data __ c; 300 __Jc__ _________ l c 0 ; f! 200 .. C1) a; u 100 0 1 I 1 2 3 4 b s + 8 9 0 100 I : I i ____ j ____ ,_ .. _________ _____ W"'wlo : Time (ms) Figure 9.3 Test 2 with Axial Fiber at 128 kg/m3 Zo =76 mm and a 6 m Drop 87
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500 400 Ci 300 c: 0 :;; 200 Q) Q; (.) (.) <( 100 0 100 I (j) I ___ .. Time (ms) .. Figure 9.4 Test 3 with Axial Fiber at 128 kg / m3 Zo =76 mm and a 6 m Drop The calculations use a leakage constant of 0 for the axial fiber. The method of construction is very effective at entrapping air in the fiber. For the radial fiber a constant of 1 00 was used to almost eliminate the effects of entrapped air. This is discussed further in Section 11. As can be seen from Figs. 9.2. 9.3, and 9.4, the test data was consistent with peak accelerations of 461 g, 441 g, and 436 g. This compares well with the predicted acceleration of 451 g, an error of less than 4%. 88
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As in the radial fiber example the period was about I ms shorter than predicted. The error in estimating the period is about half of the radial fiber example. The initial stiffness of the axial fiber is much higher than the radial fiber. If the theory of the starting point of impact was missed due to low acceleration, it would make sense that the error using axial fiber would be less. Further work would be required to resolve this question. 89
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10. Peak Acceleration The first of the questions posed in the Introduction was "What is the peak acceleration of the inner container?" The ideal cushion would provide constant acceleration during the entire impact event. It might be possible to approach the ideal with fine layers of corrugated cardboard or possibly a honeycomb structure. The acceleration under these idea circumstances can be calculated by setting the energy gained from the fall to the energy absorbed in the cushion. 10. 1 Where: hdrop = height of the drop, a = acceleration by the cushion, and = distance the mass penetrates into the cushion. Solving Eq. 10.1 for acceleration: hdro p a =g L\zmax 10.2 90
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With a drop height of 6 m (20ft.) and a crush zone of 60 mm (2.4 in.) the expected deceleration is 1 00 g. This would be true only if acceleration was constant during impact. The time to decelerate can be calculated from Eqs. I 0.3 and I 0.4: I0.3 I0.4 For the 6 m (20ft) drop and a crush distance of 60 mm (2.4 in.) with the ideal cushion the time to come to a stop is 11 ms. It might have been possible to approach this acceleration, but it would have been expensive and time consuming. Because cost was a factor in the competition this route was not pursued. Another classic approach would have been to assume the fiber was a perfect spring. This could have been accomplished if a long crush zone was constructed with a large weight and cost penalty. The deceleration and position would be a sine function. The payload position would follow Eq. 10.5: 10.5 9I
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Velocity would equal to: 10.6 Knowing the initial conditions at impact and that v =zero at n /2: drop OJ='10.7 Acceleration would equal: 10.8 The acceleration will peak at n/2. Both and ro are known therefore : 2gh 10.9 The maximum acceleration by modeling the system as an ideal spring is twice what it would be with constant acceleration. With the same allowable crush zone as the 92
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earlier example the acceleration would be 200 g. Since the natural frequency of the system is known, the peak acceleration and zero speed would occur when cot is equal to rr./2. For the example of a crush zone of 60 mm (2.4 in.), peak acceleration and zero speed would occur at 8.7 ms. The acceleration as a function of time for the previous two examples, and containers with two different fiber lays are shown in Fig. 1 0.1. The original container with radial fiber had a calculated peak acceleration of 605 g with a 76 mm (3 in.) maximum crush zone and radial fiber at 130 kg/m3 (8lb!ft\ The radius of the inner container is 50 mrn (2 in.). Also shown in Fig. 10.1 is one of the final designs with axial fiber. The density was 96 kgm3 (6 lb/ft3 ) and the crush zone was held at 76 mm (3 in.) maximum. Peak acceleration for this design is 417 g. was calculated to be 62 mm (2.4 in.). The data can be summarized by the following statements: 1) Acceleration from a spring is twice that of a perfect cushion. 2) Acceleration from axial fiber is four times greater than that of a perfect cushion. 3) Acceleration from the radial fiber is six times greater than that of a perfect cushion. 4) Acceleration is reduced by one third when using axial fiber instead of radial fiber. 93
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A problem with using either of the fiber lays as a cushion is that very little work is being done in the first 4.5 ms By 4 5 ms the crush zone for the axial fiber has been reduced from 76 mm to 46 mm and the velocity is still 80% of the velocity at impact. Most of the work is done from 4.5 ms to 7 ms due to nonlinear characteristics of the fiber. The crush zone that was effectively stopping the payload was between z=46 mm (1.8 in.) to z=20 mm (0.8 in.). Most of the energy of impact is absorbed in only 26 mm (1 in.) ofthe cushion. 70J I : ,1 ..... 0 i : 0 1 2 3 4 5 6 7 8 9 10 11 12 lirre(rrs) Figure 10.1 Predicted Accelerations for a 6m (20ft. ) Drop and a 60 mm .6.Zmax 94
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The new design using axial fiber is superior in two respects. First the peak acceleration is greatly reduced. Second, the new design uses one third less fiber than the original design. A container with reduced cost, weight, and improved reliability due to lower peak acceleration were the major goals of the modeling. 95
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11. Effects Of Fiber Lay The second question posed in the Introduction is "What are the effects of fiber lay on the containers' ability to absorb the shock?". It is obvious from Fig. 10.1 that the axially layed fiber greatly reduces the peak shock on the inner container. It is also important to note the axial fiber example uses less material than the radial example. For competition at the Energy Challenge '98 in Atlanta, Georgia, axially layed fiber was the best choice to minimize shock, weight, and cost of materials. The improvement in performance comes from two effects. The first effect improving the shock absorption properties is that the axially layed fiber is stiffer than the radial fiber. This is shown in Fig. 11.1. 96
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"' a. ..: U) U) Q) L. en 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 I I .. . 0 1 I 100 fiber i t1 i i 200 300 400 500 600 700 Density (kgfmA3) Figure 11.1 Comparison of Radial and A x ial Fiber Stress to Densit y For a given density the force generated is 2 to 4 times greater with the axial fiber as compared to the radial fib e r This is s ho w n in Fig. 11. 2 97
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; 0 4.5 .... :t/) 4 t/) Q) t/) : !:; t/) 3.5 tJ) ....... 3 Q)t/) i :9 a; 2.5 iLL.C 2 .! LL :! 1.5 'I"'C 1 0 "' 0 0:: 0.5 ;; 0 0 t; ' ; !.. ' 1 i :1 L : r. _______ ____. _____ _ _______ i I _ __L 1 ++ 'i ++! 100 200 300 400 500 Density (kg/m3 ) 600 l i 700 Figure 11.2 Ratio of Axial to Radial Fiber Stress The increase in stiffness can be used in two ways. The first advantage of axial fiber is the density of the fiber can be reduced and still maintain the original stiffness and therefore good shock absorption properties. The lower density results in a lighter container at a reduced cost. The second ad v antage of the axial fiber is that more work is done on the payload in the early stages of impact. Shown in Fig. 11.3 is the spring constant as defined b y Eq 11. 1 for an ideal spring axial fiber and radial fiber. The spring constants between 0 mm and 20 mm (0.8 in.) are an estimate of the spring constant taking into account the recoverable energy for small changes in density. 98
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k= F Z0 z Displacements for all three examples are 60 mm (2.4 in ) with a 1 kg (2.2 lb) payload dropped 6 m (20 ft) 100 E z 90 rl _____ I i I 80 70 60 50 40 30 20 10 0 i +1___ ________ j_ 1 Axial tiber i i' 1 i i ____ !Radial fil:>__el : Ideal recoverable energy 0 10 20 30 40 50 60 Change in Z axis (mm) 70 I Figure 11.3 Spring Constant for Radial and Axial Fiber with a 1 kg Pa y load Dropped from 6 m and Maximum Deflection held at 60 mm. 11.1 The spring constant for both fiber lays is lower than an ideal spring through most of the crush zone. Because of the low spring constant less than ideal energy is being 99
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absorbed in this portion of the crush zone. The remaining energy is absorbed in the last 20 mm (0.8 in ) of the crush zone. The high energy absorption in a short distance results in the high accelerations seen in the previous sections The axial fiber spring constant is two to four times larger than the radial fiber spring constant in the first 40 mm (1.6 in.) of the crush zone This results in energy being absorbed more evenly throughout the crush zone with axial fiber. The second effect that is important to note is the air is entrapped more effectively in the axial fiber. The leakage paths for the two fiber lays are shown in Fig. 11.4 and 11.5. Radial fibers lay in the direction of the pressure gradient and do little to restrict the movement of air. Axial fibers are perpendicular the pressure gradient. In addition the axial fiber samples were formed with a continuous sheet wound into a spiral. The length ofthe spiral was about 450 mm (18 in.) from the center of the container to atmospheric air. This increases the escape path by nearly an order of magnitude. The restricted air path nearly doubles the total force on the inner container. Rarely did this sheet rupture during impact. 100
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Figure 11.4 Leakage Paths for Radial Fiber 101 Pa.yloa.d Ra.dia.l fiber pa.ra.llel to the pressure gra.dient
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Figure 11.5 Leakage Paths for Axial Fiber 102 fiber used for construction orient fiber forced through a 700 MM to esco.pe
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12. Forces On The Inner Container A key question posed in the Introduction and the one the contestants had the least understanding of was "How will the inner container fail? The depth of the water in the inner container was 117 mm ( 4.6 in.) deep. The static pressure at the bottom of the container was 1.14 kPa (0.166 psi). During impact the pressure was expected to increase to 345 (50 psi) to 690 kPa (100 psi). The walls ofthe inner container were fabricated from sheets of old newsprint and laminated to a thickness of 2.3 mm (0.090 in.). A mode of failure was initially thought to be due to the internal water pressure at impact and the walls would fail in tension. 103
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605 g Inner conto.lner filled with wo ter Rocllol fiber Failure due to a net 2040 kPa externo.l stress Figure 12.1 Failure Mode From Impact 4\7 g Fo.ilure due to o. net 683 kPo. externo.l stress In Fig. 12.1 the acceleration and maximum stress is summarized for the two principal fiber lays. The accelerations would result in internal pressures of 690 kPa ( 100 psi) for the radial fiber and 482 kPa (70 psi) for the axial fiber. External pressures due to the fiber and entrapped air are 2730 kPa (396 psi) and 1165 kPa (169 psi) respectively. In both case the inner container would fail with a net 2040 kPa (296 psi) or 683 kPa (100 psi) external pressure. Failure of the inner container would be due to the external forces in both cases. Knowing the failure mode and the effects of the fiber orientation, the final design was formulated. The containers used in competition would have the fiber in the 104
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axial direction. A second advantage of the axial fiber is that the density could be varied as a function of the radius. With the knowledge of the failure mechanism the density of the center of the container was reduced to lower the maximum external stress on the inner container. The effect of variable density was implemented with out modeling due to time constraints. 105
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13. Effects Of Altitude The final question in the Introduction was How will the container perform in Atlanta, Georgia as compared to the testing done in Denver, Colorado?". There was some concern that the shock at impact would be different at sea level as compared to the testing done in Denver, Colorado at an altitude above 5000 ft. To eliminate this concern a final design was run in the mathematical model. The results are shown in Fig. 13.1 .. I 500450 400 350 C) 300 c 0 Atlanta 404 g .. 250 cu .. Q) 200 a; CJ CJ 150 < 100 50 0 0 2 4 6 8 10 Time (ms) Figure 13.1 Predicted Accelerations for a 6 m (20ft) Drop in Denver Colorado and Atlanta Georgia 106
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The data indicates there will be little change in the peak acceleration for either location A design that performs well in the Denver Colorado area should function properl y in Atlanta, Geor g ia With the knowledge of the peak acceleration the effects of fiber lay, the failure mechanism of the inner container and the effects of altitude the contestants made a few minor changes to the design. It was unknown how the University of Colorado at Denver's (UCD) design would compare to the rest of the competition but the UCD contestants were fairly confident it would survive the 6 m (20ft) drop 107
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14. Recommendations For Future Work The model presented in this paper has served its purpose in that the University of Colorado at Denver easily won the Energy Challenge '98. With the mission accomplished, no future work is needed on this project. One may question the accuracy of the model in that errors of 50% are unreasonable in most fields. Even under controlled conditions, impact is a function of many random factors that are difficult to control. The dropping of a container most of the time is accidental, a random event in itself. A valid question is how accurate does the model need to be? If no additional accuracy is needed, no addition work should be put into the model. If the model has value in the science/art of packaging compressing loose material, compressing and expanding fiber such as fiberglass insulation, fiber permeability, and estimating the total energy of impact for a compressible body the work is just starting. Future work might include: 1) Explore the commercial feasibility of using hammermilled recycled newsprint for general packaging. While old newsprint has been used for years in packaging no known work has been done with reorienting it in the axial direction. As shown in Fig. 11.2 orientation of the fiber can double or 108
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quadruple the material stiffness The packaging material could probably be produced at onetenth the cost of standard packaging materials. The raw material would be low grade post consumer old news print that would normally go to the landfill. 2) Expand the capacity ofthe model for general packaging design. Current software appears to be either very restrictive in its capacity or very complicated to use The method presented in this report runs on simple spread sheets, can handle surfaces that are not flat, and characteristics of the packaging material are easy to input. After the software is refined it would be easy to install on low end computers for the plant floor and train operators to optimize the shipping containers. The results would be reduced shipping cost and product damage 3) Develop cushioning curves via the model and compare them to the standard published curves. The two systems should be very similar. If this can be verified for multiple materials there are commercial applications for the software. Instead of running hundreds of points on specialized expensive equipment a set of curves could be generated on simple equipment in a matter of minutes. 109
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Before any of the above could enter into the commercial market a significant amount of addition work would need to be completed. Some of the work would include: 1) Explore the viscoelastic properties of paper fiber. This would define the proper stress density relationship to use during impact. It could affect the developed stress by as much as 50%. This is shown in Fig. 7 .2. 2) Explore the relationship of permeability and density of paper fiber. In this report it is considered a constant. As the density of the material increases, the permeability would be expected to decrease. 3) Establish the stressdensity relationships for other compressible materials such as fiberglass, polyester fiber and the common packaging materials. 4) Develop the stressdensity relationship for the recoverable energy portion of the compression cycle. In real applications many small impacts may occur in this linear region where the energy of impact would be returned to the object. 110
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5) Rerun the DOE with better control of the angle the container impacts the ground and a higher sampling speed. 111
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15. Conclusions The model developed in this report has been tested against a designed experiment where no evidence was found to reject their validity. All evidence indicates the concept of a stress and density relationship for fiber is valid. The system developed has been a valuable asset to the contestants to gain a better understanding of the dynamics of impact and the evaluation of container designs. Without the model, test data would have been difficult to interpret because of the random errors. The final design of the container with the aid ofthis model had an expected peak acceleration of 400 g, fiber oriented in the axial direction an inner container that would fail due to external compressive forces, and would not be significantly affected by a change in altitude It proved to be a winning combination worth $15,000 to UCD. 112
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APPENDIX A. Project Photographs Photo A.l Laying out the hammermilled newsprint to be bonded and later reoriented in the axial direction 113
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Photo A.2 Rolling up the outer container to orient the fibers in the axial direction 114
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Photo A.3 Fabrication of the inner container 115
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Photo A.4 Compressing and pre drying the inner container with a vacuum 116
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Photo A.5 Final container components 117
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Photo A.6 One of the final assembled containers ready for loading the water and egg 118
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Photo A.7 Press to measure the density and stress relationship of the hammermilled paper 119
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Photo A.8 Form to orient the fiber in the axial direction 120
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Photo A.9 Axial oriented fiber ready for test 121
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Photo A.lO Test container components. From top to bottom; 1) Outer abs tube, 2) payload with accelerometer, 3) pin to establish the crush zone and limit the rebound. 122
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Photo A.ll A remote release mechanism at 10m (33ft) to insure consistent drops 123
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Photo A.12 A test container at 9 m (20ft) ready to be released 124
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Photo A.l3 Detail of the payload and accelerometer 125
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Photo A.l4 The landing zone for the test container 126
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References (1.1) Institute of Paper Science and Technology, 500 10111 Street NW, Atlanta GA 30318 (2.1) Eshleman R.L.: "HighPerformance Shock Isolation Systems" ASME Design Engineering and Technical Conference Cincinnati Ohio Sept 1973 (2.2) Timpert F.H.: "A New 20' ISO Box Container Qualified as a Type A Package", RAMTRANS, Vol8 No.1 pp 510 (1997), Nuclear Technology Publishing (2.3) Kausel E. Flessner H. C. Timpert F.H.: "Dynamic Analysis of a Container with a Single Layer of Drums Mounted on A Rail Car Suffering a Collision" RAMTRANS, Vol8 No. 1 pp 2126 (1997), Nuclear Technology Publishing (2.4) Butler N.: "Impact Strength of Containers for Carrying Radioactive Materials", The Nuclear Engineer, Vol33 No 5 pp138145 (1992) (2.5) Brown N.: "Specification of Test Criteria for Containers to be used in the Air Transport of Plutonium European Applied Research Report, Nuclear Science Technology, Vol3 Numbers 1 and 2 (1981) pp2336 (2.5) Fieldler R.M.: "The Best of Transpack" ( 1996) Institute of Packaging Professionals 481 Carlisle Dr, Herndon VA 20170 (2.6) ASTM Designation D 159678A Annual Book of ASTM Standards, Vol 15.09 (1990) 127
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(2. 7) Hanlon J. F.: Handbook of Package Engineering ", 2nd edition (1984) McGraw Hill Book Co, pp 163 (4.1) Photo from the Johns Manville Research Center, 10100 W Ute Ave, Littleton CO (4.2) Crane J : "Flow of Fluids Through Valves Fittings and Pipe" Technical Paper No.410 (1978), Crane Company 4100 S. Kodule Ave., Chicago IL 60632 128
