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This is to certify that the

thesis entitled

Predicting Shock Transmission Characteristics
of Convoluted Polyurethane Ester Cushions
Using Standard Cushion Curves

presented by

Manoch Srinangyam

has been accepted towards fulfillment
of the requirements for

Master degree in Jackaging.

 

 

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0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

 

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PLACE IN RETURN BOX to remove this checkout from your record.
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PREDICTTNG SHOCK TRANSMISSION CHARACTERISTICS OF CONVOLUTED
POLYURETHANE ESTER CUSHIONS USING STANDARD CUSHION CURVES

By

Manoch Srinangyam

A THESIS

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

MASTER OF SCIENCE

School of Packaging

1999

ABSTRACT

PREDICTING SHOCK TRANSMISSION CHARACTERISTICS OF
CONVOLUTED POLYURETHANE ESTER CUSHIONS USING
STANDARD CUSHION CURVES
By

MANOCH SRINANGYAM

The purpose of this study was to predict the peak acceleration G to falling weights
onto 2 and 4 pcf convoluted polyurethane ester cushions. The prediction method was
based on converting the convoluted cushion into the equivalent block cushion. The
convoluted shape (fingers) act to reduce the effective drop height. Standard published
cushion curves can be used to determine G level for the equivalent block cushion. Two
densities of the convoluted cushion of 2 and 4 pcf, three different drop heights of 16, 22
and 28 inches, and three different loadings of 17.8, 22.8 and 27.8 lbs were tested. The
results Show that the predicted peak acceleration was only about 20% different from the

experimental peak acceleration.

ACKNOWLEDGEMENTS

A very special thanks to Dr. Gary Burgess for his patient, guidance and humor while

acting as my advisor and helping me through the difficult times.

I also want to thanks Dr. S Paul Singh and Dr. Brian Feeny for their suggestions and for

serving on my committee.

Thanks to Chris Benner of Industrial Rubber & Supply Incorporation for his material

donation which made this project possible.

Finally, I would like to say thank you to Dr. Matt Damn, Jay Singh, and Krittika

Tanprasert for their help through out my thesis.

TABLE OF CONTENTS

 

 

 

 

TABLE OF CONTENTS ................................................................................................... iv
LIST OF TABLES ............................................................................................................... v
LIST OF FIGURES ........................................................................................................... vi
Chapter 1 Introduction and literature review 1
1.1ASTM D 1596-91 ............................................................................................. 2
1.2 Convoluted Cushions ........................................................................................ 2
1.3 Manufacturing and Properties of Convoluted Polyurethane Ester Cushion ..... 4
1.4 Damage Boundary Curve .................................................................................. 4
1.5 Cushion Curves ................................................................................................. 8
Chapter 2 Materials, Equipment and Test methods 12
2.1 Materials ......................................................................................................... 12
2.2 Cushion Testing .............................................................................................. 12
2.3 Compression Testing ...................................................................................... 21
Chapter 3 Results and discussion - _ -- - - 23
Chapter 4 Conclusions - 32
Appendix A: Convoluted cushion drop test results ......................................................... 36
Appendix B: Convoluted cushion compression test results ............................................. 39
Appendix C: Published cushion curves for polyurethane ester ....................................... 45
Appendix D: Predicted peak accelerations ...................................................................... 53
Appendix E: Error Analysis ............................................................................................. 55
References- - - - - - 58

 

iv

LIST OF TABLES

Table 1. Actual free fall drop height, impact velocity, and platen height ......................... 19
Table 2. Comparison of experimental and predicted G1 and G2 ....................................... 30
Table 3. Experimental G1 and G2 at actual free fall drop height 16 inches ..................... 37
Table 4. Experimental G] and G2 at actual free fall drop height 22 inches ...................... 37
Table 5. Experimental G1 and G2 at actual free fall drop height 28 inches ...................... 38

Table 6. Buckling force and deflection for convoluted polyurethane ester cushions ...... 40
Table 7. Predicted G] and G2 from actual free fall drop height of 16 inches for 0200

and 0400 convoluted cushions .......................................................................... 54
Table 8. Predicted Gland G2 from actual free fall drop height of 22 inches for C-200

and C-4OO convoluted cushions .......................................................................... 54
Table 9. Predicted G1 and G2 from actual free fall drop height of 28 inches for 0200

and C-400 convoluted cushions .......................................................................... 54
Table 10. Error between predicted and actual G1 at free fall drop height of 16 inches. 56
Table 11. Error between predicted and actual G 1 at free fall drop height of 22 inches. 56
Table 12. Error between predicted and actual G1 at free fall drop height of 28 inches. 56
Table 13. Error between predicted and actual G2 at free fall drop height of 16 inches. 57
Table 14. Error between predicted and actual G2 at free fall drop height of 22 inches. 57

Table 15. Error between predicted and actual G2 at free fall drop height of 28 inches. 57

LIST OF FIGURES

Figure l. Convoluted cushion ............................................................................................. 3
Figure 2. Manufacturing process for convoluted polyurethane cushions ........................... 5
Figure 3. Damage boundary curve ...................................................................................... 7
Figure 4. Cushion curves .................................................................................................... 9
Figure 5. C-200 and C-4OO convoluted polyurethane ester cushions ............................... 13
Figure 6. Photograph of cushion tester ............................................................................. 14
Figure 7. Schematic of cushion tester ............................................................................... 15
Figure 8. Typical shock pulse using convoluted cushion ................................................. 17
Figure 9. Experimental design .......................................................................................... 20
Figure 10. The compression tester .................................................................................... 22
Figure 11. Typical shock using block cushion .................................................................. 24
Figure 12. Equivalence between convoluted and block cushions ..................................... 29
Figure 13. Particular example .......................................................................................... 33
Figure 14. Force versus deflection for C-ZOO, test 1 ......................................................... 41
Figure 15. Force versus deflection for C-200, test 2 ......................................................... 42
Figure 16..Force versus deflection for C-400, test 1 ......................................................... 43
Figure 17. Force versus deflection for C-400, test 2 ......................................................... 44

vi

CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

One of the jobs of a packaging engineer is designing a package to protect
products. The goal is to ensure that the consumer obtains an undamaged product. From
the time the product is manufactured until it is carried and ultimately used, the product is
subjected to some form of handling and transportation. During this process, the product
can be subjected to many potential hazards. One of them is the damage caused by shocks
(Baumdraher, 1995). In order to design a product/package system to protect the product,
the peak acceleration or G force to the product that causes damage needs to be
determined. In addition, it is sometimes necessary to determine the critical velocity
change of the product as this also affects cushion design. The damage boundary curve
contains this information and can be generated by following the American Society for
Testing and Materials (ASTM) designation: D 3332-93, Standard Test Method for
Mechanical-Shock F ragility of Products, Using Shock Machine. Once the product
fragility is known, the cushion curves for selected cushion materials need to be consulted.
For widely used cushion materials, the cushion curves are readily available. If not,
standard cushion curves can be obtained by a standard test method as stated in ASTM D
1596-91, Standard Test Methods for Dynamic Shock Cushioning Characteristics of
Packaging Material. .

The purpose of this study was to predict the peak acceleration G delivered to a

weight falling on convoluted polyurethane cushions under different static loadings.

The prediction method was based on converting the convoluted cushion into an
equivalent block cushion. The finger part acts to reduce the effective drop height and
standard published cushion curves can be used to predict the G level for the equivalent

block cushion.

1.1 ASTM D 1596-91

This standard test method describes a procedure for obtaining dynamic shock
cushioning characteristics of packaging materials. The data can be obtained by flat
dropping a falling guided platen assembly onto motionless block-shaped cushion samples
from different heights, static loadings and thickness of the cushion samples and recording
the peak acceleration from an accelerometer attached to the platen. The data obtained
may be used for producing dynamic cushion curves for the particular packaging material
being tested. Inch-pound units are employed as the stande (ASTM D 1596-91, 1994)

suggests.

1.2 Convoluted Cushions

Convoluted cushions shown in Figure l have been developed to conform to odd
shaped products: for example, glassware, electronic components, and small appliances.
The finger part will hold the product in place and may help to lower the peak
acceleration.

Unlike block-shaped cushions, the convoluted cushion has two different

thicknesses; the thickness of the finger tip and the block part. Therefore, standard

 

 

Figure 1. Convoluted cushion

cushion curves are not applicable for this cushion. This creates the need to develop

a method to predict peak acceleration.

1.3 Manufacturing and Properties of Convoluted Polyurethane Ester Cushion

The manufacturing process for convoluted polyurethane cushions is shown in
Figure 2. The continuous dispensing method is used heavily in the manufacture of
flexible and rigid low-density foams. In continuous dispensing, the foam ingredients are
rapidly mixed by machine and poured onto a moving belt. The foam rises and gels as it
moves along the belt. The resulting foam slab, or bun, is sent through rollers and cut into
two pieces of equal thickness. The finger parts of each piece are on the sides that face
each other.

Polyurethane can range from very soft to very hard, from very flexible to very
rigid, or from the hydrophilic type which soaks up water to types which repel water.
Polyurethane falls mainly into two categories, esters and ethers. Esters are generally the
tougher of the two but will eventually hydrolyze and degrade when exposed to water.
Ethers do not hydrolyze or biodegrade even on prolonged exposure such as direct burial,
but are not as tough and are less resistant to chemicals and oils (Markley and Gillis, 1994;

Polyurethane Foam Association, 1991).

1.4 Damage Boundary Curve
A damage boundary curve (DBC) is a plot of the critical acceleration, Gp, on the

ordinate (y-axis) and the critical velocity change, AVp, on the abscissa (x-axis)

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of the input shock required to damage a product. The damage boundary curve is shown
in Figure 3. The standard test procedure for determining the two fragility parameters is
ASTM D3332-93.

The product is first mounted on a shock table and subjected to a series of drops
using plastic programmers (high G, short duration shock) from higher and higher drop
heights until the product breaks. The critical velocity change to the product is the area
under the shock pulse just before the damage occurs. The product can tolerate any
acceleration without damage as long as the velocity change to the product
is less than this critical velocity change. When the velocity change is more than the
critical velocity change to the product, the product may or may not be damaged
depending on the peak acceleration to the product (Granthen, 1991).

To obtain the critical acceleration of the product, an identical new product in the
same orientation is mounted on the shock machine and subjected to a series of drops
using gas programmers (low G, long duration square wave). The table drop height of the
shock machine was adjusted and fixed so that the velocity change is at least 1.57 times
the critical velocity change of the product. The gas pressure is increased until the product
breaks. The critical acceleration is the value measured just before damage occurs. The
product can tolerate any velocity change, AV, or any long duration shocks without
damage as long as the peak acceleration to the product is less than this critical
acceleration.

It is important to note that the particular DBC is valid only for the specific

product tested and for one orientation.

 

 

Peak acceleration to product (G's)

Critical

Velocity

Change Damage Region

<————>

 

\

Critical acceleration

v

 

Velocity change to product (in/sec)

 

Figure 3. Damage boundary curve

 

1.5 Cushion Curves

A cushion curve is a plot of peak acceleration on the y-axis and static stress on the
x-axis for a specific drop height and cushion thickness. It is valid only for the specific
material tested as shown in Figure 4. It is important to mention that all standard
published cushion curves are for block-shaped cushions which have uniform thickness. It
should be noted that as static stress increases, the peak acceleration decreases until it
reaches its lowest value and then starts to increase. The high peak acceleration at low
static stress is due to the insufficient energy to compress the cushion while the high peak
acceleration at high static stress is because the cushion bottoms out (Burgess, 1997).

There are many types of cushioning materials which are used in packaging.
Cushions may be divided into two categories based on structure. These are open-cell and
closed-cell foams. An open-cell foam is composed of a network of intercommunicating
cells where air can flow between each cell and also out of the foam. Open-cell foam
absorbs energy by letting the air escape from the foam. A closed-cell foam is composed
of individual bubbles of air trapped and join together to form the cushion. The trapped
air inside each cell absorbs energy from compression. Closed-cell foam is stiffer,
stronger, and produces higher peak acceleration than open-cell foam.

In theory, a product will be damaged if both the peak acceleration and velocity
change of the shock to it in a drop fall within the damage region of the DBC. Otherwise,
the product will be fine and no cushion is needed. For instance, if the velocity change to
the product is less than the critical velocity change, the product will have no damage,
regardless of the peak acceleration. However, it is usually the case that the velocity

change is more than the critical velocity change. To ensure that the product is not

PEAK ACCELERATION (G's)

PEAK ACCELERATION (0'!)

180

160

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120

100

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GRAPH 2: POLYURETHANE ESTER

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damaged in this case, a cushion is needed to keep the peak acceleration lower than the
critical acceleration.

There are many factors that need to be considered when selecting a cushion.
They are type of material, density, shape, bearing area, and thickness. For irregular
shapes like convoluted cushions, it is difficult to decide on the appropriate thickness to
use on the cushion curves. If the thickness of only the block part is used when
determining the peak acceleration from the cushion curve, the G level will be higher than
actual because the fingers also provide cushioning. It will be over protection since the
thickness of the finger part was ignored. On the other hand, if the thickness used to
determined peak acceleration from the cushion curve includes the height of the fingers,
the product might be damaged since it will receive a peak acceleration higher than what
the cushion curves say. This is the reason why the predicted peak acceleration of a
convoluted cushion needs to be developed.

To date, not much research has been done in the area of designing and predicting
characteristics with convoluted cushions. Granthen (1991) studied the shock
transmission characteristics of ribbed expanded polypropylene cushions using standard
cushion curves for flat plank cushions. Three drop heights of 18”, 30”, and 42”, three rib
heights of 1.5”, 2.0”, and 2.5”, three rib angles of 5°, 15°, and 25°, and three static
loadings were tested. Granthen examined seven mathematical models to predict the G
level. Each model is based on converting the ribbed cushion into an equivalent plank
cushion so that published cushion curves can be employed to predict the peak

acceleration. It was concluded that the “Equivalent Volume” model that weighs the

10

varying cross sectional areas of the ribbed cushion and the plank section of the cushion

equally gave the best predicted peak acceleration.

11

CHAPTER 2

MATERIALS, EQUIPMENT AND TEST METHODS

2.1 Materials

The cushion material used in this research was convoluted polyurethane ester
foam. The material is a low density open-cell structure, so that air can penetrate the
intracellular spaces, made by Industrial Rubber & Supply Incorporated, Washington.
Two different densities having the same shape were used in this research. They are
designated as C-200 and C400 meaning 2 and 4 pounds per cubic foot. The C-200 foam
has a lighter color than C-400. The base dimensions of the convoluted cushion samples
were 8 inch x 8 inch. The thickness is designated as 23/4 inch over l‘/4 inch, meaning that
the overall height is 2% and the block part (base) is 1‘/4 inch thick. Photos of the
convoluted polyurethane ester cushion for 2 and 4 pounds per cubic foot are shown in

Figure 5.

2.2 Cushion Testing

A Lansmont Model 23 Cushion Tester was used to perform all drop testing
following ASTM D1596. As shown in Figures 6 and 7, a piezoelectric accelerometer
was mounted on the platen of the cushion tester. The signal from the piezoelectric
accelerometer was carried by a shielded cable to a Kistler piezotron charge amplifier and
then to a twelve bit analog card on an IBM AT compatible 80286 computer. The

software which was used by this computer to analyze the shock pulse was Test Partner

12

 

C 200 c 400

Figure 5. C-200 and C-400 convoluted polyurethane ester cushions

             

  

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Figure 6. Photograph of cushion tester

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Lifting mechanism
Guide rods

Ballast weight
Platen
Accelerometer
Chain

Convoluted cushion
Seismic mass

. Charge amplifier

0. Test partner

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Figure 7. Schematic of cushion tester

15

version 2 from Lansmont Corporation. The shock pulses were displayed on a VGA

computer monitor. A photograph and a schematic of the Lansmont Model 23 Cushion

Tester are shown in Figure 6 and 7, respectively.

TEST PROCEDURE

The drop tests onto the convoluted polyurethane ester cushion samples were

conducted as follows:

1.

9.

Mount the piezoelectric accelerometer on the platen and hook it up with the Kistler
piezotron charge amplifier.

Turn on all switches which control the cushion tester, computer, and open the valve
on nitrogen tank which supplies air brakes.

Set up the computer program to monitor and analyze the shock pulse.

Make sure that the guide rods are clean and have a thin film of lubrication. Friction
between the guide rods and the platen will slow down the platen. Because of friction,
the platen must be raised higher than the actual free fall drop height. The relationship
between the actual free fall drop height and the height of platen will be explained
later.

Insert a convoluted polyurethane ester cushion test sample underneath the platen.

Put a variety of ballast weights on the platen to get different static loadings.

Raise the platen and drop it onto the cushion sample.

Afier each drop, Test Partner will show the shock pulse on the monitor. A typical
shock pulse from a convoluted cushion is shown in Figure 8.

Record the two peak acceleration values for the finger part and block part.

10. Repeat steps 3 through 9 for all drop heights, static loadings and types of cushions.

16

Acceleration, G’s

A

(32

 

Block

G1

 

 

> Time, ms

F x
\

Finger compression

Figure 8. Typical shock pulse using convoluted cushion

17

ADJUSTING PLATEN DROP HEIGHT

According to step 4, due to friction between the guide rods and the platen, the
platen must be raised higher than the desired free fall drop height as stated before. In the
absence of any resistance, products, regardless of their size, weight, or composition, drop
toward the ground with nearly constant acceleration (Resnick and Halliday, 1977). Since
the free fall drop height of a packaged product is not too great, just 1-6 feet height, the
acceleration can be assumed constant throughout the drop. Since impact velocity is what
really matters, it can be calculated from drop height in a free fall where there is no

resistance using

v2 = 2gh (1)

Where V is impact velocity
h is free fall drop height
g is acceleration due to gravity (386.4 in/secz)

On the cushion tester, the platen falls with an acceleration less than g and so in
order to obtain the same impact velocity V, the platen was raised and must be higher than
the actual free fall drop height. In order to find the correct platen drop height, the platen
was dropped onto an Ethafoarn 220, closed-cell polyethylene foam, and the shock pulse
was recorded. Test Partner sofiware was used to calculate the impact velocity which is
the area under shock pulse from the start of the shock pulse to its peak. This procedure
was repeated by moving the platen up and down until the impact velocity corresponded to

the desired free fall drop height as in equation (1). The average values of platen height

18

corresponding to the impact velocity and the actual free fall drop height are shown in

Table 1.

Table 1. Actual free fall drop height, impact velocity, and platen height.

 

 

 

 

 

 

 

 

Actual free fall drop height Impact velocity Platen height
(in) (in/sec) (in.)
16 111.2 25
22 130.4 36
28 147.1 44
DROP TESTS

Using the cushion tester, two different kinds of foams, C-200 and C-400, were
dropped from three different drop heights, 16 in, 22 in and 28 in. At each drop height,
three different loadings were tested: 17.8, 22.8 and 27.8 lbs. Ten replicates were
performed for each loadings. After each drop, the G levels corresponding to finger
compression and block compression from the shock pulse were recorded. The
experimental design is shown in Figure 9.

The results of the drop tests on the convoluted polyurethane ester cushions are
shown in Tables 3 through 5 in Appendix A. These experimental results are compared to

predictions using mathematical modeling later.

19

 

Weight of Platen and

ballasts
(lbs.)
Drop height (in)
/v 17.8
16 <: 22.8
TYPe of cushiV 27.8
. - 17.8
C-ZOO —> 22 4 22.8

/ \ \‘ 27.8
17.8

28 ——/—'> 22.8

Convoluted Cushion \ 27.8

/' 17.8

/ 27.8
/v 17.8

C-400 ———> 22 ——> 22.8

\> 27.8
/v 17.8

28 —’ 22.8

\> 27.8

Figure 9. Experimental design

20

 

2.3 Compression Testing

The Lansmont Model 152-30TTC Compression Tester shown in Figure 10 was
used to perform compression testing. The compression tester was used to measure the
force and deflection required to compress the cushion in a static situation. This was done
to evaluate the contribution of the fingers to overall cushon stiffness. As the cushion
starts to compress, the force increases rapidly to a nearly constant level. During this time,
only the fingers compress, not the block part. This is a small force compared to the force
required to compress the block part. For purposes of modeling, this finger buckling force
was assumed to be constant. Two samples of both convoluted cushions, C-200 and C--
400, were compression tested. The buckling forces and deflections were then recorded
and printed out as shown in Figures 14 through 17 in Appendix B. For each type of
cushion, the average buckling force and deflection were then calculated as shown in

Table 6 in Appendix B and used in mathematical modeling later.

21

 

Figure 10. The compression tester

22

CHAPTER 3

RESULTS AND DISCUSSION

This chapter identifies the physical characteristics of convoluted polyurethane
ester as a packaging cushion material using the cushion tester and compression tester
results. It uses these properties in a mathematical model which converts the convoluted
cushion into an equivalent block cushion. This allows published cushion curves to be
used to predict G levels.

Typically, the shape of the shock pulse from block-shaped cushion looks like a
half sine wave or symmetrical bell. In reality, the area under the right side of the bell,
which represents the rebound velocity, will be smaller than the area of the lefi side since
the product loses energy during the impact. There are three important points on the bell
which are the start, the peak and the end points. At the start, the product makes contact
with the cushion and the instantaneous acceleration is zero. The product will obtain the
maximum acceleration and has momentarily stopped at the peak point. Finally, at the end
point, the product breaks contact and the acceleration is zero again. Figure 11 imitates a
typical shock pulse from a block cushion.

Unlike the block-shaped cushion, the convoluted cushion shock pulse is almost as
same as the block shaped cushion but with a raised leading edge. Figure 8 imitates the
shape of shock pulse from a convoluted polyurethane ester cushion. The shock pulse has

two peak acceleration points. The first peak is much lower than the second peak and stays

23

 

Acceleration, G’s

A

G2

 

 

 

Time, ms

Figure 11. Typical shock using block cushion

24

constant for a short time before it rises. This short time happens while the falling weight
compresses the fingers of the cushion. From the shock pulse in Figure 8, during
compression of the fingers the acceleration stays constant and the block part stays
uncompressed. The second peak is much higher than the first and the shock pulse drops
immediately after it reaches the peak. This peak occurs because the block part of the
cushion has been compressed.

, To predict the peak acceleration which occurs during finger compression,
Newton’s second law was employed. Newton’s second law states that a net force acting
on a body gives it an acceleration which is in the direction of the force and has a

magnitude inversely proportional to the mass of the body (Ohanian, 1985),

5:3 (2)
m

where a is acceleration
F is force acting on the body (lbs)
m is mass of the body

By applying the Newton’s Second Law, equation (2) can be rewritten as

F = ma = mgGl = WGl (3)

Where g is acceleration due to gravity
W is weight of the product (lbs)

G, is the acceleration as a multiple of gravity

From the shock pulse in Figure 8, G; is constant. Since W is also constant, the
product of WGI, which is F, is constant as well. In other words, the compression force
stays constant during finger compression. This makes sense because the mode of

deformation of the fingers is buckling, and the buckling force is typically independent of

25

the amount of compression. To predict the acceleration G), the compression tester was
used to determine the buckling force and corresponding deflection of the fingers, as
discussed earlier. The buckling force acts on the falling weight and produces the
acceleration G. Since the buckling force is proportional to the number of fingers and
since the number of fingers is proportional to the base area of the convoluted cushion, the
ratio of buckling force to base area should be a constant, independent of the size of the

cushion. The buckling stress is defined as

BS = (4)

F
X
where BS is buckling stress

F is the buckling force

A is the base area of the cushion sample tested

The buckling stress is a constant value for the specific shape and density of the

convoluted cushion. Therefore, equation (4) can be rewritten as

WG
BS = —-'— (5)
A
By definition, the static stress is the stress on the cushion created by the weight of the
product resting on top of the cushion. The static stress or “static loading” as it is

sometimes called on the cushion curves is defined as

SL = — (6)

where SL is static loading
W is weight of the product

A is area of the cushion

26

Since the buckling stress is known and the static loading can be calculated, the

acceleration G can be predicted from equation (7) as

B_S_
SL
The G1 prediction from equation (7) will be compared with the experimental data later.

G1 = (7)

The principle of conservation of energy (Resnick and Halliday, 1977) states that

SW = AB;( (8)
where 23W is the total work done on the product

AB, is the change in kinetic energy

The situation at the end of finger compression (beginning of block compression) can be

analyzed using equation (8),

mg(h,+f)—mgG,f = émv2 (9)

where m is the mass of product
g is acceleration due to gravity
h is drop height
f is finger height
G1 is the constant acceleration during finger compression

v is velocity of the product as the block part begins to compress

Equation (9) can be rewritten as

v2 = 2g[hl —(GI —l)f] (10)

The fingers can now be viewed as having the net effect of slowing down the product
before it starts to compress the block part of the cushion. The product therefore appears

to have fallen from a reduced height. The reduced drop height, h2, can be calculated as

27

 

v2 = 2gh2 (11)
where v is the same impact velocity
g is acceleration due to gravity

h2 is the reduced drop height

Setting equation (10) equal to equation (1 1),

28112 = 281111 - (Gr-1m (12)

112 = h1—(G1-I)f (13)

Since the part of the shOck pulse in Figure 8 corresponding to the compression of the
block part of convoluted cushion is just like the kind of shocks appropriate to cushion
curves, the convoluted cushion drop can be viewed as an equivalent block cushion drop
from the reduced height h2. The published cushion curves can now be used to determine
the peak acceleration of block part of the convoluted cushion which is called G2 in Figure
8. The equivalent block cushion is the block (base) portion of the convoluted cushion.
This equivalence is shown in Figure 12. This was done and the predicted peak
acceleration was compared with the experimental data.

Since published cushion curves have discrete values for the thickness of the
cushion, static stress and drop height, interpolation usually needs to be used for

predicting G2. The percent error from the mathematical model can be obtained as

(Gup_ Glare)
%error = x100 (14)

CH)

 

where Gexp is the peak acceleration from experiment

Gpm is the predicted peak acceleration

28

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This equation can be applied for errors in both G1 and G2. However, G 1 is typically very

small, usually less than 5 G. This is comparable to the error associated with the kinds of

accelerometers used to capture shock pulses. It is therefore likely that the percent errors

on G] calculated using equation (14) will be falsely high. The acceleration of most

interest, G2, is much larger and so errors are much more likely to be due to modeling than

to instrument precision.

The comparison between experimental and predicted G1 and G2 is presented in

Table 2. The detail of experimental and predicted values are shown in Appendixes.

Table 2. Comparison of experimental and predicted G1 and G2

 

 

 

 

 

 

 

 

 

 

 

 

 

Cushion Drop Static G1 G2
118$? 2386:: Exp.‘1 Pre.‘2 % Error Exp.’l Pre.‘2 % Error
0.28 2.82 3.75 -32.98 30.11 28.89 4.05
16 0.36 2.74 2.94 -6.57 34.62 33.11 4.36
0.43 2.47 2.44 1.21 41.00 38.41 6.32
0.28 7.77 3.75 51.74 53.87 42.22 21.63
0200 22 0.36 5.93 2.92 50.76 61.67 51.14 17.07
0.43 4.90 2.44 50.20 73.17 58.90 19.50
0.28 6.76 3.75 44.53 68.37 66.67 2.49
28 0.36 7.14 2.92 59.10 82.59 74.68 9.58
0.43 5.46 2.44 55.31 99.86 87.52 12.36
0.28 4.30 4.86 -1302 27.81 31.05 -11.65
16 0.36 2.79 3.78 -35.48 32.15 33.80 -5.13
0.43 1.76 3.16 -7955 37.36 35.17 5.86
0.28 9.30 4.86 47.74 49.41 39.04 20.99
0400 22 0.36 5.77 3.78 34.49 58.42 45.15 22.71
0.43 4.04 3.16 21.78 69.12 54.33 21.40
0.28 8.02 4.86 39.40 64.74 53.70 17.05
28 0.36 7.06 3.78 46.46 78.37 61.33 21.74
0.43 5.66 3.16 44.17 93.34 72.17 22.68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’1 Experimental
’2 Predicted

3O

 

The calculated peak acceleration of the finger part, G1, are shown in Tables 7
through 9 in Appendix D and then compared with experimental values in Tables 10
through 12 in Appendix E. The calculated peak acceleration of the block part, G2, are
also shown in Tables 7 through 9 in Appendix D and then compared with experimental

values in Tables 13 through 15 in Appendix E.

31

CHAPTER 4

CONCLUSIONS

The goal of this thesis was to be able to predict the peak acceleration in dynamic
loadings of convoluted polyurethane ester cushion using published information for block-
shaped cushions made of the same material.

Suppose, for example, a 27.80 lb product was dropped onto a convoluted
polyurethane ester C-200 cushion from a drop height of 16 inches as shown in Figure 13.
The base dimensions of the convoluted cushion is 8 inches x 8 inches. The thickness is
designated as 2% inches over 1% inches. From Appendix B, the buckling stress for the
C-200 material is 1.05 psi. The static loading is 27.80/64 = 0.43 psi. According to
equation (7), the predicted G1 is 1.05/0.43 = 2.44. In the actual drop, the average
experimental G1 was 2.47 as shown in Table 3, Appendix A. In this case, the error
between predicted and actual G; is [(2.47-2.44)/2.47] x 100 = 1.21 %.

To predict G2, the reduced drop height needs to be determined using equation
(13). The reduced drop height is 16—[(2.44-1)x1.50] = 13.84 inches. Knowing the type
and density of the cushion, the equivalent block cushion thickness of 1.25 inches, the
static loading and drop height, the peak acceleration can be determined from published
cushion curves. Since there are no published cushion curves for this material at the
reduced drop height (13.84 inches), interpolation needs to be used. From graph 1 in
Appendix C, for the C-200 cushion, drop height of 12 inches, static loading of 0.43 psi
and cushion thickness of 1.25 inches, the peak acceleration is 33.28 G. From graph 2 in

Appendix C, at drop height of 18 inches, static loading of 0.43 psi and cushion

32

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33

thickness of 1.25 inches, the peak acceleration is 50 G. Therefore, the interpolated
acceleration at a drop height of 13.84 inches is 38.41 G. This value is the predicted G2.
The average experimental G2 was 41.00 as shown in Table 3, Appendix A. In this case,
the error between predicted and actual G2 is [(41.00-38.4l)/41.00]x100 = 6.32%
OTHER PREDICTIONS

1. The buckling stress also represents the largest static stress that can be used to
take advantage of the fingers ability to reduce the drop height. In general, if BS is larger
than SL, G1 > 1 and from equation (13), the reduced drop height will be less than the
actual drop height. In the case that BS is equal to SL, G1 = 1, and the reduced drop
height will be equal to the actual drop height. To obtain an advantage using the
convoluted cushion, packaging engineers need to make sure that the buckling stress is
more than the static loading.

2. In the case where the actual drop height is not greater than f(BS/SL-l) the
product will compress only the fingers, not the base. Substituting h) = f(BS/SL-1) into
equation (13) gives h2 = 0. The shock to the product consists of G1 only, with G1 =
(BS/SL). This will not likely damage a product because G1 is usually about 5 G which is
very small. In this case, even though the product is dropped on to the convoluted
cushion, no compression of the block part occurs. This is one of the advantages of
convoluted cushions.

3. A compression tester is not required to find the buckling stress BS. The
buckling stress can be obtained from a recorded shock pulse by measuring G1 and using

equation (7) BS = G; x SL.

34

ERRORS

Tables 10 through 12 in Appendix E show the predicted G1 at each drop height
and comparisons with experimental G. values. The results show that the percent errors
between the experimental and predicted peak G1 were very large, up to 80%. The reason
of these large errors is partly due to the sensitivity of accelerometer, which was not
reliable at low accelerations. Another source of error is the fact that the number and
shape of the fingers per unit area varies from sample to sample. The predicted G1 is in
theory independent of the drop height since it came from buckling stress divided by static
stress.

Tables 13 through 15 in Appendix E show the predicted G2 at each drop height
and comparisons with the experimental G2 values. The results show that the percent error
between the experimental and predicted G2 was about 20%, which is good because
published cushion curve are usually no more accurate than about i15%. At lower drop
height like 16 inches, the predicted G2 tended to be over estimated. On the other hand, at

higher drop heights, 22 and 28 inches, the predicted G2 tended to be under estimated.

35

APPENDIX A

CONVOLUTED CUSHION DROP TEST RESULTS

36

Table 3. Experimental G. and G2 at actual free fall drop height 16 inches

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

G’s
C-200 C-400
Weight 17.8 lbs 22.8 lbs 27.8 lbs 17.8 lbs 22.8 lbs 27.8 lbs
G1 02 GI G2 G1 G2 G1 02 G1 G2 G1 G2
1 2.49 26.71 2.13 33.73 2.15 40.14 5.10 30.11 2.80 29.00 2.76 36.46
2 3.09 27.31 2.58 34.13 2.79 40.81 5.33 27.49 3.01 35.46 1.83 32.70
3 2.91 27.88 2.87 33.83 2.32 39.46 4.21 24.98 2.89 30.30 1.80 43.05
4 2.01 35.97 3.13 34.85 3.32 41.34 4.81 31.04 2.98 29.50 1.85 36.77
5 2.69 27.90 2.94 35.00 3.11 41.39 3.43 26.39 3.09 37.29 1.17 33.58
6 2.91 27.08 3.17 34.69 2.26 41.60 3.93 25.96 2.46 31.19 1.59 41.96
7 2.99 36.09 2.19 34.79 2.19 40.80 3.74 30.64 2.56 29.86 1.54 37.11
8 2.86 27.82 2.47 35.17 1.92 41.78 3.24 25.78 2.76 35.92 1.77 34.13
9 2.96 27.77 2.87 34.83 2.20 40.61 4.23 25.58 2.32 31.23 1.72 42.70
10 3.30 36.54 3.09 35.15 2.46 42.08 5.00 30.15 3.04 31.75 1.57 35.16
Avg. 2.82 30.11 2.74 34.62 2.47 41.00 4.30 27.81 2.79 32.15 1.76 37.36
SD 0.36 4.22 0.38 0.53 0.45 0.80 0.73 2.40 0.26 2.97 0.41 3.87
Table 4. Experimental G. and G2 at actual free fall drop height 22 inches
G’s
C-200 C-400
Weight 17.8 lbs 22.8 lbs 27.8 lbs 17.8 lbs 22.8 lbs 27.8 lbs
G. G2 G. G2 G. G2 G. G2 G. G2 G. G2
1 7.73 46.60 5.88 62.38 5.52 71.94 8.95 45.29 5.37 53.73 4.58 75.50
2 5.92 60.19 6.07 60.43 5.67 71.62 10.67 45.07 6.59 64.07 3.90 64.92
3 8.26 48.60 6.15 60.63 4.57 73.59 9.73 52.42 5.54 56.88 4.25 61.02
4 8.32 49.23 5.81 61.06 4.99 73.73 8.92 48.87 6.32 54.88 4.14 77.30
5 6.00 65.10 5.92 62.27 5.31 73.94 9.47 47.07 5.67 64.38 3.40 66.25
6 8.96 52.16 5.85 61.88 4.67 71.65 8.96 54.76 5.34 57.57 4.79 62.95
7 8.62 50.58 5.73 62.56 4.72 73.47 8.68 49.84 5.64 54.87 3.62 75.85
8 6.49 64.73 5.89 61.72 4.47 73.70 9.98 47.50 6.06 65.19 3.71 66.43
9 8.89 50.53 5.88 61.98 4.38 74.29 9.40 54.54 5.51 57.92 3.97 62.92
10 8.54 50.99 6.08 61.78 4.71 73.79 8.23 48.76 5.69 54.66 4.08 78.01
Avg. 7.77 53.87 5.93 61.67 4.90 73.17 9.30 49.41 5.77 58.42 4.04 69.12
SD 1.19 6.83 0.13 0.73 0.45 1.02 0.71 3.49 0.42 4.45 0.43 6.73

 

 

37

 

 

Table 5. Experimental G. and G2 at actual free fall drop height 28 inches

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- G’s
C-200 C-400

Weight 17.8 lbs 22.8 lbs 27.8 lbs 17.8 lbs 22.8 lbs 27.8 lbs
G1 G2 G1 G2 G1 G2 G1 G2 G1 G2 G1 G2
1 6.49 69.06 6.61 86.95 6.42 97.76 8.85 56.85 6.53 87.21 6.45 101.15
2 6.90 67.83 8.95 82.74 4.94 100.05 8.22 62.25 6.56 72.43 5.37 86.22
3 7.00 69.17 7.86 81.88 5.29 98.39 8.31 71.73 7.78 76.71 6.67 101.32
4 6.81 68.18 7.17 83.41 6.25 99.86 8.50 62.19 7.06 88.09 5.05 84.61
5 6.64 67.97 7.00 81.57 5.64 98.71 7.64 65.80 7.82 73.81 5.19 89.73
6 6.94 68.14 6.73 81.95 4.52 99.34 7.86 72.48 7.58 75.89 6.55 104.60
7 7.19 70.04 7.33 83.82 4.59 100.23 8.24 63.16 6.68 73.36 4.39 87.45
8 6.97 68.31 6.55 84.89 5.03 102.10 7.26 63.55 7.45 75.78 5.03 102.82
9 6.20 66.66 6.43 79.05 5.69 101.65 8.02 63.82 6.46 86.23 6.09 84.64
10 6.50 68.29 6.75 79.64 6.22 100.55 7.33 65.54 6.72 74.21 5.76 90.86
Avg. 6.76 68.37 7.14 82.59 5.46 99.86 8.02 64.74 7.06 78.37 5.66 93.34
SD 0.30 0.91 0.77 2.35 0.69 1.38 0.51 4.60 0.54 6.22 0.77 8.15

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

38

 

APPENDIX B

CONVOLUTED CUSHION COMPRESSION TEST RESULTS

39

From the attached force versus compression curves, the following buckling forces have
been identified.
The average buckling force for 0200 = (70+64)/2 = 67 lbs.

The average buckling force for C-400 = (88+86)/2 = 87 lbs.

Table 6. Buckling force and deflection for convoluted polyurethane ester cushions

 

 

Cushion Average Deflection of fingers, Equivalent block
buckling force, lbs inches thickness, inches
C-200 67 1.50 1.25
C-400 87 1.25 1.50

 

 

 

 

 

 

The buckling stresses for the two types of cushions based on Table 6 and a base area of 8

inches x 8 inches are:

II
II

BS 67/64 1.05 psi for C-200

BS

87/64 = 1.36 psi for C-400
The deflection of the fingers, the equivalent block thickness and buckling stress (BS) will

be used to calculate h2, predict G2 and G ., respectively, in Appendix D.

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44

APPENDIX C

PUBLISHED CUSHION CURVES FOR POLYURETHANE ESTER

(United State Department of Defense, 1978)

45

PEAK ACCELERATION (G's)

PEAK ACCELERATION (G's)

 

100 i 1
160 2 V
140

120 /

100

 

 

 

 

 

 

 

 

 

 

 

 

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I‘ I T I *
.02 .10 .04 .06 .080.1 -0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS WIA (psi)
GRAPH I: POLYURETHANE ESTER

2 pcf. 12" Drop Height

180 l V1
1'1

 

 

160

140

120 I Z
W, / L

 

 

 

 

 

 

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.02 .13 .04 .06 .08 0.1 0.2 0.3 0.4 0.5 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 2: POLYURETHANE ESTER

2 pct, 18” DropHeight

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ll

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

46

PEAK ACCELERATION (G’s)

PEAK ACCELERATION (G's)

 

180

 

160

 

140

120 /

100 I71:-
.. \ / / //

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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-'Z . 3 .04 .06 .00 0.1 0.2 0.3 0.4 0.6 0.01.0 2.0

STATIC STRESS wm (psi)
GRAPH 3: POLYURETHANE ESTER
2 pcf, 24" Drop Height

160

140

120

100

60

40

20

     

0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS wa (psi)
GRAPH 4: POLYURETHANEESTER
2 pct 30"DropI-leigllt

47

PEAK ACCELERATION (G’s)

PEAK ACCELERATION (G’s)

 

180

160 V
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STATIC STRESS W/A (psi)
GRAPH 5: POLYURETHANE ESTER

2 pcf, 36” Drop Height

180

160

140

120

100

60

40

20

 

. .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 6: POLYURETHANE ESTER
2 pcf, 42” Drop Height

48

PEAK ACCELERATION (G’s)

PEAK ACCELERATION (G’s)

 

180

 

160

 

 

140

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120

100

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.113 .04 .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 7: POLYURETHANE ESTER
2 pcf, 48” Drop Height

 

. .04 .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 8: POLYURETHANE ESTER

4 pcf, 12”D.opHeig,h1

49

PEAK ACCELERATION (G‘s)

PEAK ACCELERATION (0'5)

180

160

140

120

100

60

40

20

 

- .04 .05 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 9: POLYURETHANE ESTER
4 pct, 18" Drop Height

180

160
140
120
100
80
‘60
‘0

20

 

. . .04 .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 10: POLYURETHANE ESTER
4 pcf, 24" Drop Height

50

PEAK ACCELERATION (G's)

PEAK ACCELERATION (0‘5)

180

160

140

120

100

60

40

20

180

160

140

120

100

60

40

20

 

.04 .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 11: POLYURETHANE ESTER

4 pcf, 30" Drop Height

 

.04 .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 12: POLYURETHANE ESTER
4 pcf, 36" Drop Height

51

PEAK ACCELERATION (G’s)

PEAK ACCELERATION (G's)

180

160

140

120

100

60

40

20

180

160

140

120

100

60

40

20

 

.04 .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 13: POLYURETHANE ESTER
4 pcf, 42” Drop Height

 

.04 .06 .08 0.1 0.2 0.3 0.4 0.6 0.81.0 2.0

STATIC STRESS W/A (psi)
GRAPH 14: POLYURETHANE ESTER
4 pcf, 48” Drop Height

52

APPENDIX D

PREDICTED PEAK ACCELERATIONS

53

Table 7. Predicted G. and G2 from actual free fall drOp height of 16 inches for C-200

and C-400 convoluted cushions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cushion Weight, lbs BS, psi SL, psi G. h2, in. G2
17.80 1.05 0.28 3.75 11.88 28.89
0200 22.80 1.05 0.36 2.92 13.12 33.11
27.80 1.05 0.43 2.44 13.84 38.41
17.80 1.36 0.28 4.86 11.18 31.05
C-400 22.80 1.36 0.36 3.78 12.53 33.80
27.80 1.36 0.43 3.16 13.30 35.17
where SL = W/A
G. = BS/SL
112 = h1-(G1-I)f

G2 was obtained from cushion curves for equivalent block cushion.

Table 8. Predicted G.and G2 from actual free fall drop height of 22 inches for C-200

and C-400 convoluted cushions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cushion Weight, lbs BS, psi SL, psi G. h2, in. G2
17.80 1.05 0.28 3.75 17.88 42.22
C-200 22.80 1.05 0.36 2.92 19.12 51.14
27.80 1.05 0.43 2.44 19.84 58.90
17.80 1.36 0.28 4.86 17.18 39.04
C-400 22.80 1.36 0.36 3.78 18.53 45.15
27.80 1.36 0.43 3.16 19.30 54.33

 

Table 9. Predicted G. and G2 from actual free fall drop height of 28 inches for C-200

and C-400 convoluted cushions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cushion Weight, lbs BS, psi SL, psi G. h2, in. G2
17.80 1.05 0.28 3.75 23.88 66.67
C-200 22.80 1.05 0.36 2.92 25.12 74.68
27.80 1.05 0.43 2.44 25.84 87.52
17.80 1.36 0.28 4.86 23.18 53.70
C-400 22.80 1.36 0.36 3.78 24.53 61.33
27.80 1.36 0.43 3.16 25.30 72.17

 

54

 

 

 

APPENDIX E

ERROR ANALYSIS

55

Table 10. Error between predicted and actual G. at free fall drop height of 16 inches.

 

 

 

 

 

 

 

Cushion Static stress, psi Experimental G. Predicted G. % Error
0.28 2.82 3.75 -32.98
C-200 0.36 2.74 2.92 -6.57
0.43 2.47 2.44 1.21
0.28 4.3 4.86 -13.02
0400 0.36 2.79 3.78 -35.48
0.43 1.76 3.16 -79.55

 

 

 

 

 

 

 

Table 11. Error between predicted and actual G. at free fall drop height of 22 inches.

 

 

 

 

 

 

 

Cushion Static stress, psi Experimental G. Predicted G. % Error
0.28 7.77 3.75 51.74
C-200 0.36 5.93 2.92 50.76
0.43 4.90 2.44 50.20
0.28 9.30 4.86 47.74
C-400 0.36 5.77 3.78 34.49
0.43 4.04 3.16 21.78

 

 

 

 

 

 

 

Table 12. Error between predicted and actual G. at free fall drop height of 28 inches.

 

 

 

 

 

 

 

Cushion Static stress, psi Experimental G. Predicted G. % Error
0.28 6.76 3.75 44.53
0200 0.36 7.14 2.92 59.10
0.43 5.46 2.44 55.31
0.28 8.02 4.86 39.40
C-400 0.36 7.06 3.78 46.46
0.43 5.66 3.16 44.17

 

 

 

 

 

 

 

56

Table 13. Error between predicted and actual G2 at free fall drop height of 16 inches.

 

 

 

 

 

 

 

 

Cushion Static stress, psi Experimental G2 Predicted G2 % Error
0.28 30.11 28.89 4.05
C-200 0.36 34.62 33.11 4.36
0.43 41.00 38.41 6.32
0.28 27.81 31.05 -11.65
C-400 0.36 32.15 33.80 -5.13
0.43 37.36 35.17 5.86

 

 

 

 

 

Table 14. Error between predicted and actual G2 at free fall drop height of 22 inches.

 

 

 

 

 

 

 

 

Cushion Static stress, psi Experimental G2 Predicted G2 % Error
0.28 53.87 42.22 21.63
C-200 0.36 61.67 51.14 17.07
0.43 73.17 58.90 19.50
0.28 49.41 39.04 20.99
C-400 0.36 58.42 45.15 22.71
0.43 69.12 54.33 21.40

 

 

 

 

 

Table 15. Error between predicted and actual G2 at free fall drop height of 28 inches.

 

 

 

 

 

 

 

 

Cushion Static stress, psi Experimental G2 Predicted G2 % Error
0.28 68.37 66.67 2.49
C-200 0.36 82.59 74.68 9.58
0.43 99.86 87.52 12.36
0.28 64.74 53.70 17.05
0400 0.36 78.37 61.33 21.74
0.43 93.34 72.17 22.68

 

 

 

 

 

57

 

 

 

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