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THESIS

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'This is to certify that the

dissertation entitled

SHOCK RESPONSE SPECTRUM “ND. FATIGUE DAMAGE:
A NEW APPROACH TO PRODUCT FRAGILITY TES'PING

presented by

' MATTHEW PAUL DAUM

has been accepted towards‘fulfillment
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SHOCK RESPONSE SPECTRUM AND FATIGUE DAMAGE:
A NEW APPROACH TO PRODUCT FRAGILITY TESTING

By

Matthew Paul Daum

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
School of Packaging

1999

ABSTRACT

SHOCK RESPONSE SPECTRUM AND FATIGUE DAMAGE:
A NEW APPROACH TO PRODUCT FRAGILITY TESTING

By

Matthew Paul Daum

Product fragility assessment in packaging has for years been based on modeling fragile
components as linear, undamped spring/mass systems inside a rigid frame. This led to
the development of the Damage Boundary Curve (DBC) which evaluates the velocity
change and deceleration of an input shock for its damage potential to a product. The
DBC has some limitations because of assumptions used in its derivation, including its
reliance on square shaped input shocks, requirement of multiple test specimens, and the
use of a large, often expensive, shock test equipment. The Shock Response Spectrum
(SRS) was borrowed from other engineering disciplines and applied to fi'agility issues in
packaging in part to simplify data gathering for DBCs. SRS uses the response of a
component to an input shock rather than the input shock itself, and can be used to
eliminate the need for testing whole products, extracting the same fragility information as
traditional DBC testing without a shock table. However, a serious limitation in
traditional DBC and SRS-generated DBC curves is the assumption that failure of the
component occurs in a brittle mode, meaning that all shocks before failure have no effect
on the product. Most products are in fact not brittle in nature, but ductile, and so failure
may be a cumulative effect. This results in a DBC which must incorporate not only

velocity change and deceleration, but number of cycles to failure as well.

The purpose of this study was to develop a mathematical model expanding the
SRS technique to account for the ductile nature of many products, using an elastic,
perfectly plastic model as an idealization of the stress-strain curve. Components
exhibiting brittle behavior can also be handled with this model since the brittle stress-
strain curve can be considered to be a special case of the elastic, perfectly plastic curve.
Two new testing procedures to obtain product fragility information are outlined and
demonstrated on four products, including one method that does not require a shock table.
Software was also developed to demonstrate the new SRS-fatigue algorithm and to prove
its applicability.

Results show good agreement using the predictions from the new SRS-fatigue
model and the new test procedures for finding product fragility information. The result
may be considered a new approach to fragility testing which includes not only velocity
change and deceleration information, but number of cycles to failure as well. By finding
the characteristic material properties of the component, predictions about its response to

any kind of input shock can be made.

C0pyright by
Matthew Paul Daum

1999

Dedicated to my loving and supportive wife, my best friend, April.

To my Lord and Savior Jesus Christ, in whom rests all knowledge and truth, and who
gives purpose and meaning to life.

ACKNOWLEDGMENTS

Thank you, Dr. Burgess, for helping an ordinary student reach an extraordinary goal.

Thank-you also to my committee members, Dr. S. Paul Singh, Dr. Brian Feeny, and Dr.

Robb Clarke.

vi

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... x
LIST OF FIGURES ........................................................................................................ xi
CHAPTER 1 ................................................................................................................... 1
INTRODUCTION AND LITERATURE REVIEW .......................................... 1
CHAPTER 2 ................................................................................................................... 19
THEORETICAL DEVELOPMENT .................................................................. 19
2.1 FINDING MATERIAL PROPERTIES x0, x1 AND F0 .................... 19
2.1.1 Plastic Programmer Method For Determining
Material Properties ............................................................. 21
2.1.2 Freefall Method For Determining Material Properties ...... 27
2.1.3 Sources Of Experimental Error .......................................... 29
2.2 SR8 AND DUCTILE MODEL ........................................................ 30
2.3 NUMERICAL SOLUTION .............................................................. 35
2.4 GENERATING FATIGUE DBC CURVES ..................................... 36
2.5 TEST TO VERIFY PROGRAM ALGORITHM ............................. 36
CHAPTER 3 ................................................................................................................... 39
MATERIALS AND TEST METHODS ............................................................. 39
3.1 MATERIALS .................................................................................... 39
3.2 METHODS ....................................................................................... 40
3.2.1 Determination Of Natural Frequency ................................ 40
3.2.2 Definition Of Failure .......................................................... 46

vii

3.2.3 Test Method For Finding Material Properties From
Plastic Programmer Testing ............................................... 46

3.2.4 Validation Of Results From Plastic

Programmer Testing ........................................................... 47
3.2.5 Test Method For Finding Material Properties From
Freefall Drops .................................................................... 50
3.2.6 Validation Of Results From Freefall Drops ....................... 50
3.2.7 Method For Generating Fatigue DBCs .............................. 50
CHAPTER 4 ................................................................................................................... 52
RESULTS ........................................................................................................... 52
4.1 RESULTS FOR MATERIAL PROPERTIES FROM
PLASTIC PROGRAMMER TESTING ........................................... 52
4.2 RESULTS FOR VALIDATIN G MATERIAL PROPERTIES
FOUND FROM PLASTIC PROGRAMMER TESTING ................ 54
4.3 RESULTS FOR MATERIAL PROPERTIES FOUND
IN F REEFALL DROPS .................................................................... 57
4.4 RESULTS FOR VALIDATING MATERIAL PROPERTIES
FOUND FROM FREEFALL DROP TESTING .............................. 59
4.5 RESULTS FOR GENERATING FATIGUE DAMAGE
BOUNDARY CURVES ................................................................... 59
4.6 DISCUSSION OF ERRORS AND IMPACT ON RESULTS ......... 65
CHAPTER 5 ................................................................................................................... 69
CONCLUSIONS AND FUTURE WORK ......................................................... 69
5.1 CONCLUSIONS ............................................................................... 69
5.2 FUTURE WORK .............................................................................. 71

viii

APPENDICES

APPENDIX A RECOVERING OTHER MODELS ......................................... 74

APPENDIX B MATERIAL PROPERTIES FROM FREEFALL TESTING ...77

APPENDIX C SRS PROGRAM CODE ........................................................... 79

REFERENCES ............................................................................................................... 99

ix

LIST OF TABLES

Table 1. Variation in Observed Number of Drops to Failure for Test Models .............. 26
Table 2a. Comparison of Values, Trapezoid Pulse ........................................................ 38
Table 2b. Comparison of Values, Half-Sine Pulse ........................................................ 38
Table 3. Natural Frequency of Test Models .................................................................. 46
Table 4. Failure Modes and Determination Method ...................................................... 46
Table 5. Freefall Drop Height Test Configurations ....................................................... 48
Table 6. Summary of Material Properties From Plastic Programmer Testing .............. 53

Table 7. Actual and Predicted Drops to Failure, F reefall Testing, PMMA Beams ....... 56

Table 8. Actual and Predicted Drops to Failure, Freefall Testing, Sheet Metal ............ 56
Table 9. Summary of Calculated Material Properties From Freefall Testing ................ 58
Table 10. Comparison of Calculated AVcr and Ger ...................................................... 59
Table 11. Data For Constructing Fatigue DBCs ............................................................ 60
Table 12. Variation of x0 and x1 From Static Deflection Test ....................................... 66
Table 13. Values For Number of Drops to Failure ........................................................ 66
Table 14. Effect of N on Material Properties x0 and x1 ................................................. 67

LIST OF FIGURES

Figure 1. Product With Fragile Component, Modeled As Spring/Mass System ........... 2
Figure 2. Traditional Damage Boundary Curve ............................................................ 2
Figure 3. SRS Plot From Commercial Software By Lansmont Corporation ................. 6
Figure 4. Shock Response Plot From Commercial Software
By Lansmont Corporation .............................................................................. 6

Figure 5. Dynamic Force Deflection Curve, Including Damping ................................. 9
Figure 6. Material Parameters Describing Ductile Failure ............................................ 13
Figure 7. Multiple Stretching and Releasing of a Ductile Component .......................... 13
Figure 8. Generalized DBCs Incorporating Fatigue Damage ........................................ 16
Figure 9. Simple Spring/Mass/Dashpot Model .............................................................. 20
Figure 10. Effect of Small Changes in Input Shock Velocity Change on Number

of Drops to Failure ........................................................................................ 31
Figure 11. Spring Force in Four Regions of Interest ..................................................... 33
Figure 12. PMMA Beam With Mass Attached to Wood Fixture .................................. 41
Figure 13. Metal Beam With Mass Attached to Wood Fixture ..................................... 42
Figure 14. Spring With Attached Mass .......................................................................... 43
Figure 15. LaserJet Printer Mounted Onto Wood Base, Plastic

Programmer Testing ...................................................................................... 44
Figure 16. LaserJet Printer Sheet Metal ......................................................................... 45
Figure 17. LaserJet Printer Mounted Onto Wood Base, Freefall Testing ..................... 49
Figure 18. Static Force Versus Deflection Curve for PMMA Beam ............................. 55
Figure 19. DBCs for PMMA Beams .............................................................................. 61
Figure 20. DBCs for Metal Beams ................................................................................ 62

xi

Figure 21. DBCs for 119 Spring .................................................................................... 63

Figure 22. DBCs for LaserJet Printer Sheet Metal ........................................................ 64

xii

CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

Fragility assessment in packaging has for years been based on modeling fragile
components as linear, undamped spring/mass systems inside a rigid frame (the
“product”). In Figure l, the component consists of a weight W on a linear spring having
stiffness k (lbs/in) which fails when reaching some predetermined permanent
deformation, or by breaking. Failure is usually taken to be complete fiacture of the
component in question. According to accepted theory [1], an “input” shock pulse to the
product must have a critical velocity change, AV“, and a critical deceleration, Ger, in
order to transmit the shock to the component and cause failure. Modeling components as
simple linear spring/mass systems led to the development of the Damage Boundary
Curve (DBC), which shows pictorially the combination of velocity change and
deceleration of an input shock pulse needed to damage the component (Figure 2).

To find AVcr and Ger, controlled drop tests on at least two whole products must be
done to find each parameter independently. Shock machines were developed for this
purpose, and are common equipment in packaging test labs. A shock machine consists of
a rigid, heavy table that can be dropped from any height onto one of two different
surfaces. The type of surface determines the characteristics of the shock pulse. Short
duration (about 2 ms), high G shock pulses are produced when the table is dropped onto a
hard elastic surface, called “plastic programmers”, and are shaped like half-sine waves.
Long duration, low G pulses are produced when the table is dropped onto a piston inside

a cylinder containing an inert gas, called the “gas programmers”, and are trapezoidal in

 

 

 

 

 

 

 

 

 

1 l
l l
.1 r A
Product )1 . y
I i
Fragile 8:13
Component - __> g)
‘ <::;_'_..:::> k
a;
(:“_::43 I
C r-
C1?) x
(h “1)
l <-_ a 7,) .

 

 

 

l ' 7 7777777

Figure 1. Product With Fragile Component, Modeled As Spring/Mass System.

 

 

 

 

G ‘ *
W/
Gcr w“ \““
l >
AVcr AV

Figure 2. Traditional Damage Boundary Curve.

shape. ASTM D 3332-93 (98) “Mechanical-Shock Fragility of Products, Using Shock
Machines” [2] was written to use the plastic and gas programmers to find critical velocity
change and the critical deceleration values respectively for the product. ASTM D 3332
requires a sample product be fixed to the table and subjected to repeated drops of
increasing intensity (increased drop height) on the plastic programmers. The critical
velocity change is recorded from the input shock pulse when damage first occurs to the
component. A new sample, in the same orientation as the previous one, is mounted to the
table, and subjected to repeated drops of increasing intensity (increased gas pressure) on
the gas programmers. The critical deceleration value is recorded from the input shock
pulse that first causes damage. Often the values used for critical velocity change and
critical deceleration are averages of the shock causing damage and the preceding shock.
The DBC is then constructed as shown in Figure 2. The small rounded knee has a known
shape [1] but is usually squared off for simplicity.

The DBC is used to evaluate the damage potential of an input shock to the
product. This is done by comparing the input shock’s velocity change and G level to the
AVcr and Gcf found on the graph. If both velocity change and G level from the input
shock are greater than AVcr and Gcr on the DBC, damage is predicted for that component.
Since the packaging community is familiar with the DBC, AV" and Go, will also be used
in this study to eventually describe damage.

The DBC has some limitations because of assumptions used in its derivation that
make its use less than ideal. The common DBC shown in Figure 2 really only applies to
square shaped inputs. This is because GC, is defined by a square shaped input pulse,

which was used because it is more severe than other waveforms. At best then, the DBC

generated with a square shaped input is a very conservative description of failure (the
DBC for it envelopes other input pulse shapes), leading to overprotection and more costly
packaging solutions in many cases. Another limitation of the traditional DBC is its
reliance on an expensive shock table for generation. Generally speaking, it is difficult to
create and control square shaped shock pulses, even with the aid of specialized
machinery. The actual shock is trapezoidal at best. Developing DBCs then relies on
equipment that must be bought or rented, usually at significant cost. The traditional DBC
spelled out in ASTM D 3332 also requires at least two whole products to be damaged for
each orientation of testing. Testing for all six orientations can require a minimum of 12
products — an impossible requirement especially for high priced, schedule-driven
products in the electronics industry. The use of many products can be quite expensive,
and only data relating to velocity change and deceleration is captured. DBCs are also
sensitive to the faired G values of the trapezoid shock pulses, resulting from equipment
limitations and high frequency noise superimposed on the pulse (ringing). Non-
trapezoidal shocks would cause the critical acceleration line of damage to vary widely as
a function of velocity change. An even more serious limitation is the ignored effect of all
the previous impacts leading up to damage, in both the plastic and gas programmer parts.
This will be discussed later.

In the early 90’s, the concept of Shock Response Spectrum (SRS) was borrowed
from other engineering disciplines and applied to fragility issues in packaging [3 -6]. SRS
takes any input shock pulse and predicts the response of an ideal spring mass system with
a known natural frequency. This is a departure from the traditional method, which

focused on information about the input shock to the product, not the response of the

component. Commercial software packages generate plots that show peak G response of
a component to any input shock to the product, which is a function of only its natural
frequency. These plots usually range from 3 Hz to about 10,000 Hz (Figure 3). They can
also calculate only the entire time response for a single component with a particular
natural frequency; this is simply called the “Shock Response” (Figure 4). Many
commercial software packages are available that perform this analysis, relying on the
speed and power of modern computers to quickly execute the complex algorithm. This
approach eliminated some of the difficulties inherent with traditional testing, including
“ringing” of input shock pulses, and the frequent difficulty in measruing the response of a
critical element, which is often very small and delicate.

Newton showed the relationship between SRS and the DBC, and proposed a
procedure for determining fragility using shock response to trapezoidal input pulses [1].
Work was also done to use SRS in establishing shock response tolerance limits with free
fall drop testing [5].

To simplify data gathering for DBCs, SRS can be used, eliminating the need for
testing whole products. Using only one product and any input shock that causes damage,
it has been shown that it is possible to extract the same AVC, and Gcr using SRS as
traditional DBC testing [6]. Deflection damage criteria was also used in conjunction
with SRS, but only with the linear, elastic model inherent in the SRS algorithm.
Assuming the component fails when the spring deflection reaches some critical amount,
(1, and assuming an ideal linear, undamped spring/mass system, the relation between

component properties and whole product fi'agility parameters (AVcr, Ger) is:

 

Frequency ( Hz )

Figure 3. SRS Plot From Commercial Software By Lansmont Corporation.

 

—686
5 .0 vases/div

Figure 4. - Shock Response Plot From Commercial Software By Lansmont Corporation.

Ave, = 21rfnd (1)

2 f 2d
.=(—7§)— (2)
g
_L 53:
f.—2fl W (3)

where fn is the natural frequency of the component. The utility in using SRS for DBC
generation is that if the component properties fn and d are known, the DBC can be
constructed without the need to destroy any products. Even if d is unknown, only one
product need be damaged to get it, since fn can usually be found non-destructively by
conducting a resonance search on a vibration table [2] or by simply observing the
vibration of the component using an accelerometer attached to the product. All of this
can be done without the aid of a shock table, resulting in a streamlined and economical
method for generating DBCs.

The most serious limitation in traditional DBC and SRS-generated DBC curves is
the assumption that failure occurs in a brittle mode, ignoring the possibility of fatigue
failure, since all shocks before damage occurs are considered to have had no effect on the
product. In reality, however, most products are not brittle in nature, but ductile. ASTM
D 3332 assumes a brittle model, but also does describe an alteration to the basic method
called the “staircase” method, which attempts to account for the effect of multiple shocks.
Several products are tested sequentially. The first product is tested at a level near its

estimated failure point, and subsequent products are tested at levels higher or lower than

the previous one, depending on if the product fails or not. The average AVcr and Gcr are
then calculated. The data extracted from this test requires many more whole products to
be destroyed than the traditional method and yet still only yields Gcr and AVCr required
for one drop. Recognition is also given to the premise of ductile failure in ASTM D775-
80 (86), “Drop Test for Loaded Boxes” [2], which describes the number of drops from a
particular height to cause failure. However, the only data extracted is number of drops
versus drop height, and is specific to that test specimen and its package system. Nothing
can be said of the inherent properties of the test specimen itself. The motivation for this
study is the recognition that most materials are ductile in nature, not simply brittle, and a
more complete accounting needs to be made between AVcr, Gcr and number of drops to
failure.

The brittle model assumes a linear, spring/mass/dashpot model of an internal
component which fails at the elastic limit of the spring. Glass and tempered steel are
examples of materials that generally fail in this mode. However, this model does not
adequately describe most plastic and soft metal components encountered in many of
today’s electronic and home appliance products. Many products are manufactured with
materials that exhibit plastic deformation (ductile in nature) and so are susceptible to
fatigue damage. Soft steels, aluminum and many plastics behave this way. Simply
bending these materials back and forth will demonstrate the principle. For an ideal
ductile component, as shown in Figure 5, force is proportional to displacement
(compression of material) up to the elastic limit (marked EL), and compression occurs
under constant force after that point up to the break point (BP). This is a much different

model than a simple linear spring/mass system, where the elastic limit and break point are

Spring
Force

 

 

 

 

 

_ EL Ductile Material BP
F0 “I”— /* Q
/ Brittle Material
// EL, BP
7 j J

/ l _ l a
Displacement

x0 x1

Figure 5. Dynamic Force Deflection Curve, Including Damping.

the same. Burgess recognized the need for fragility testing to include ductile failure
modes, and has proposed rigid, perfectly plastic, and elastic, perfectly plastic models for
components [7,8], which led to the conclusion that the number of drops to failure is also
important.

The purpose of this study is to develop a mathematical model expanding the SRS
technique to account for plastic deformation and fatigue damage based on a simplified
version of the Bauschinger effect [9] and an idealization of the stress—strain curve (see
Figure 5), with the preposal that this simplified description is sufficient for a wide range
of common manufacturing materials in use today. The idealized stress-strain curve in
Figure 5 is usually understood in the context of static testing conditions, and behavior of
several materials have been experimentally verified to be approximated by this model
[10]. This study will make use of this same idealized stress-strain relationship, but in the
context of dynamic conditions. This distinction is important, since the material properties
in the model will be generated and used in dynamic conditions, the normal environment
for package/product damage. Components exhibiting brittle behavior can also be handled
with this model because the brittle stress-strain curve can be considered to be a special
case of the elastic, perfectly plastic curve: one where EL and BP are the same. If a
component cannot be described as brittle, lab testing using the DBC or conventional SRS
concepts can yield perplexing results. For instance, a product may not fail at a certain
drop height one time, but does fail at the same level the next. For a truly brittle product,
this should not happen. A component that is described as brittle can be subjected to
repeated shocks without damage, providing the shock input level is below a critical value.

The first shock above this level will cause damage, regardless of the number of previous

10

shocks below this level. An example is a rubber ball hitting a glass window. Only when
the force of impact exceeds the critical resisting force of the window will there be
damage, regardless of the number of previous hits below this level. For the same rubber
ball thrown against an aluminum garage door, a small dent would occur on the first
throw, and continue to enlarge with repeated throws.

This study is an enhancement to the traditional SRS approach for determining
product fragility because it includes a method for predicting component failure due to
repetitive shocks. Important material properties describing the elastic/perfectly plastic
deformation behavior of the component are extracted from initial tests, and are used to
predict not only the deceleration response to transient input shocks, but fatigue deflection
as well. This idea of fatigue deflection centers around the traditional S-N curve [11],
which shows number of cycles to failure versus force applied. The result may be
considered a new approach for constructing DBCs using SRS, which includes not only
velocity change and deceleration information, but number of cycles to failure as well.

Traditional DBC generation requires a shock machine with gas and plastic
programmers. However, using SRS and a ductile model, all relevant information for
constructing a robust fatigue DBC can be extracted using only the plastic programmers,
or even more significantly any input shock, such as those generated from fieefall drops.
Using the component’s material properties, the component’s response to shock pulses
from any source can be evaluated in terms of deceleration and fatigue damage
(displacement).

The software developed in this study (“PROGRAM”) allows the user to input the

component material properties and a digitized shock pulse, then calculates and displays

11

the shock response, predicted percent damage, and the predicted number of identical
drops to cause failure (failure pre-defmed by the user). Several commercial software
packages exist that are able to produce Shock Response Spectrums, given an analog
shock input. However, there are none known that incorporate a fatigue model.

There are three important parameters needed to describe ductile damage, as
shown in Figure 6. The displacement x0 and spring force F0 denote the elastic limit, the
point corresponding to the end of a proportional relationship between applied force and
displacement. The displacement x1 denotes the end of the perfectly plastic region, and
corresponds to failure of the material. Again, the model in this study is assumed to
describe material behavior under dynamic loading conditions.

Stretching a ductile material up to the elastic limit and releasing it results in
complete recovery of the material to its original size and shape, and the material returns
to zero displacement. Stretching the ductile material past the elastic limit but short of the
break point results in permanent deformation when the load is completely removed. As
the load on the material is being released, the force versus displacement relationship
follows a path parallel to the elastic loading line, designated as elastic unloading, as
shown in Figure 7 [9]. As a result, it does not return to its original length, but has some
permanent displacement corresponding to the point where the unloading line crosses the
displacement axis. Multiple stretching and releasing of the ductile material past the
elastic limit gives a force/deflection history as shown in Figure 7, following path 0-1-2-3
in the first cycle, 3—4-5-6 in the second cycle, 6-7-8-9 in the third cycle, and 9-10-11 in
the fourth. At point 11 in this example, the break point is reached, and the material fails.

For a product being dropped, this would correspond to failure after the fourth drop,

12

 

   
 

 

 

 

 

 

Spring
Force
EL BP
F0 “
/<---A-- J“ *' Unloading line
. .1 4 ,
’ l
/ x0 x1 Deflection
/
/
/ /
/ /
4 + I
Figure 6. Material Parameters Describing Ductile Damage.
A
Force
EL 13p
1 2 4 5 7 8 10 11
F0 —— / / r ,7/ 7 J
/ / / /
1
l //
. p / I % ’
O 1 3 3 6 6 9 9 '
x0 x1 Deflection

Figure 7. Multiple Stretching and Releasing of a Ductile Component.

l3

assuming each input shock pulse was identical. A large enough drop could cause the
component to follow the path O-l-ll, and break in one drop. This theory of damage
accumulation is based on a linear damage rule, also known as the Palrngren-Miner
hypothesis [11]. This rule is widely used in the generation of S-N curves because of its
simplicity and the experimental fact that other much more complex cumulative damage
theories do not always yield a significant improvement in failure prediction reliability.
The elastic, perfectly plastic stress-strain curve in Figure 6 can be used to modify
the traditional DBC by accounting for fatigue through the number of drops to failure [8].
Using the plastic and gas programmers, critical velocity change, critical deceleration and

nrunber of drops to failure are related by:

 

 

B
.V - 2t. _)
c. g +N (4)
2A
Ger: —-—— fl (5)
g 2A+B/N
where:
(ozx2
A: °
2g (6)
B-mzx‘2’(fl I] 7
g x0 ()

14

w=27;f. = —F°—g (8)

and
g = 386.4 in/sec2 (gravity)
W = component weight
N = Number of drops to failure

x0, x1 and F0 are spring properties, as shown in Figure 6

Figure 8 shows generalized DBC’s for a product with certain A and B values. Now
critical velocity change and critical deceleration will depend on the number of drops to
failure (N).

It is important to point out that the spring properties when found are interpreted as
dynamic properties. Obtaining x0 and x1 from dynamic testing automatically accounts for
another material property, damping. This means that natural frequency in Equation 8,
which normally defines natural frequency for a linear undamped spring/mass system,
now relates to a linear damped spring/mass system due to its relationship to dynamic
properties F0 and x0. The results in Equations 4 through 8 will be used as a check against
the methods developed for SRS-generated DBCs in this study. The brittle model results
(Equations 1 and 2) can be recovered from the above equations by putting x1 = x0. When
x0 = O, the rigid, perfectly plastic model is recovered (see Appendix A). As B in
Equations 4 and 5 approaches zero, the effect of N on AVcr and Gcr is reduced, and so a
large number for A or a small number for B indicates a more brittle component.
Conversely, a small value for A or a large value for B indicates a more ductile

component. A and B may be regarded as two material properties associated with the

15

 

 

 

 

 

 

 

AV

Figure 8. Generalized DBCs Incorporating Fatigue Damage.

16

component that replace x0, x1 and F0. This allows for simpler expressions describing
AVCr and G6,. The use of A and B also helps to point out the relationship between fatigue
theory and the observations of ASTM D 775. To show this relationship, recall velocity
change from a plastic programmer drop is equivalent to impact velocity in a free fall drop

[12]. Thus,

AV,,=AV -_- 2gh (9)

table

Substituting into Equation 4 and solving for the equivalent free fall drop height, h, yields:

h=A+— (10)

Plotting Equation 10 gives the hyperbola—shaped relationship describing drop height and
number of drops to failure observed in ASTM D 775. A and B therefore have physical
meaning related to free fall drop height. As number of drops to failure gets large, drop
height approaches A, and so A is the smallest drop height required for failure. When
number of dr0ps to failure is one, drop height is A plus B.

An inherent limitation associated with predicting material failure due to fatigue
are variations in the material itself. It is widely recognized that scatter in material
properties is common, and can be wide, when describing damage from stress-strain
applications [13]. Statistical methods are necessary to account for these material

variabilities [10,13]. These issues will be addressed later in this study.

17

Finally, some early data exists showing qualitatively the effect of number of drops
on damage and generation of DBCs. Singh [14] used light bulb filaments and bricks to
experimentally demonstrate that DBCs can be generated for different number of drops
causing failure at different energy inputs (shock table velocity change). It is interesting
to note that bricks appeared to follow the same pattern, an unexpected result since bricks
would normally be considered brittle. The implication is that a fatigue model might be a
good fit for crack propagation failure. In any case, no physical explanation was given for
the observations, but the results do lend support for the premise of ductile failure and the

incorporation of that information into DBCs.

18

CHAPTER 2

THEORETICAL DEVELOPMENT

2.1 FINDING MATERIAL PROPERTIES x0, x1 AND F0

The first objective when defining fragility using an elastic/perfectly plastic model is
to determine the material properties x0 and X], which correspond to the elastic limit and
break point, respectively (See Figure 6). Intuition suggests finding x0 and x] from a
single static force/deflection test. However, practice has shown that viscoelastic
materials can behave quite differently under dynamic rather than static loads [15]. In
drop situations, events causing damage occur in dynamic modes, so care must be taken
to determine x0 and x1 accordingly. Also, recall that the ductile model in Figure 6 is
assumed to apply to dynamic, not static conditions. To demonstrate how material
behavior can be quite different under dynamic versus static conditions, consider a simple
spring/mass/dashpot model under static loading (Figure 9). The force exerted on the
mass is F = kx+c5c, where k is the spring constant and 5c is velocity. In a static
compression test, only the spring force kx is active and in a dynamic test, both the
spring force and the dashpot force at are present. The dashpot force can be
considerable, giving very different results. In view of this, x0 and x1 can be found
experimentally from shock tests using both gas and plastic programmers [8]. There are
two other methods that may also be used to find x0 and x1, each with advantages over
using both gas and plastic programmers. The first is to use only the plastic
programmers, eliminating the need for the gas programmers. The second method is to

use freefall drop testing, eliminating the need for a shock table altogether.

19

 

 

 

414:

 

l3
l:

>5>5>>)

E_K

C,
N, .
:/\ ../\ /\ (“way/>\

k

 

23

:7; f

CC

// /

/

/

._

/

,7

 

Figure 9. Simple Spring/Mass/Dashpot Model.

20

2.1.1 Plastic Programmer Method For Determining Material Properties
The first method, to be called the “plastic programmer method,” relies on the
nature of the shock produced on the plastic programmers. In general, short duration

shocks of about two milliseconds can be produced. This corresponds to an “input shock

pulse fiequency” of ——1—— = 250 Hz. As long as this fi'equency is larger than the
2(.002 sec)

natural frequency of the spring/mass system, which is usually the case, the peak response

of the spring/mass system will occur after the input shock is over because the mass will

not have had time to react during the input shock. This observation was used in the
derivation of Equation 4. In practice, the input shock pulse frequency need be only about
two times larger than the spring/mass natural frequency to make this statement [12]. For

a two millisecond half sine pulse, this would apply to spring/mass components of 125 Hz

or less. This covers many products.

To find x0 and x1 using the plastic programmer method, the material properties A
and B in Equation 4 should first be found. A suggested procedure is as follows:

1. Place a new product on the shock table and set the table for a low level drop. Raise
and drop the table, recording the velocity change. If no damage is found, repeat the
drop at the same height until failure occurs, keeping track of the number of drops.
This number will be known as the number of drops to failure, N, at this velocity
change. It is important to note there may be some ambiguity as to what number to
use for N when damage does occur. Since damage accumulates, choosing N to be the
last drop when failure is observed is too high and choosing N to be one drop less than
this is too low. Thus, N is probably somewhere between the two drops. We must

make some rationale as to which N to use. Choosing the drop before damage is

21

observed clearly is not a good strategy since the objective is to reach failure. A
reasonable approach might be to average the number of drops (use 4.5 when damage
occurs on the 5th drop). However, since number of drops to failure in a practical
sense is limited to integer values (we only observe the 4th and 5th whole drops), and
since damage is the clear objective, choosing the drop when failure is observed
provides a straightforward albeit “conservative” method. For accurate results, this
low level dropping should yield many drops, maybe 20, before failure. It is likely AV
will vary slightly from drop to drop, and therefore the average AV and corresponding
N becomes the first data point (AVavg, N). Because of material variability, it is
recommended to record AV and N for several units at each table height, and to use
average AV and average N for generating the data point.

2. Having established damage at a particular height (AV) from Step 1, take a new
product and mount it on the table. Choose a higher drop height from which to dr0p.
Proceed as before, dropping the table repeatedly and recording AV and the number of
drops to failure. This will establish failure at a higher energy input. For this drop
height, the average AV and number of drops to failure become the second data point.
At least two data points are needed, since there are two unknowns, A and B. This
step may be repeated for different drop heights, establishing other data points, in
which case A and B will be found using regression over all the data points.

3. To find A and B, first convert all of the plastic programmer velocity change data into

equivalent free fall drop heights, h using Equation 9 cast as:

equivalent 9

22

AV; .
hequimlcnt = _Tgb!_ (1 1)

Doing so will give the data points expressed as pairs of h and N rather than AV and N.
Equation 10 now expresses a linear fit between h and UN which can be solved using

standard regression techniques as follows. The discrepancy between the true value of h

and the predicted value A + B/N is
h—A—£=error (12)
N

Minimizing the sum of the squares of the residuals (errors) for “m” data points

In B 2
SSE—;(h—A—7V—] (13)
requires that
a B
52(5513) _ 2th —A ’NJO)‘ 0 (14)
a B 1
a—B(SSE) .. 22(1: — A “NJINJ _ 0 (15)

Solving simultaneously for A and B gives:

23

A = (16)

 

B: (17)

 

nag—2,1,2}:

R = (18)

[mm—[zilszhZ—(zht

 

 

 

where all summations in 14 — 18 are over the number of data points, and R is the
correlation coefficient. Squaring R will give the proportion of drop height variability that
is explained by the linear regression model. As R2 approaches unity, the better the model
explains the residuals.

One last piece of information is needed before finding x0 and x1: component
natural frequency, which can be found non-destructively. A simple method is to place
the unit on a vibration table and perform a sine sweep test [2] noting the frequency
corresponding to resonance. If available, using an electromagnetic vibration table is
recommended since table performance tends to be more accurate than hydraulic-driven
tables, especially at higher frequencies. Other methods for finding natural frequency are
given in previous works [6]. Now x0, x1 and F 0 can be found by rearranging Equations 6,

7and8:

24

 

x0 = 2 (19)
0)
x, =x0 +—“>:i (20)
w x0
F0 =w2xo K (21)
g

By definition, the SSE calculated in Equation 13 represents the variance in

freefall drop height. Dividing by m and taking the square root gives

S = LEE. (22)
m

where S is the standard deviation in the predicted equivalent free fall drop height [16].
This will be a measure of both the model error and experimental error. A small
experimental error is expected, since velocity change of the shock table can generally be
held to within +/- 5%.

In the minimization of SSE in Equation 13, a high number of drops to failure is
weighed equally with small number of drops to failure. This may prove troublesome if
the effects of material variability are pronounced. Because packages are normally
dropped only a few times from heights that cause failure [17], less importance can be
placed on high number of drops to failure. Greater importance can be given to small N in

Step 3 of Section 2.1 by weighting residuals,

25

ib—A—éé) -W,.=SSE (23)

o

l=l

where W, = Ni , for example. Other weighting schemes might be considered, such as

taking W} = P. , the probability of dropping the package a certain number of times. This

I

kind of data does not readily exist, making the justification for using a certain Pi no more
or less arbitrary than other weighting schemes. Still, this might be a useful improvement
if such data is reliable and available, or if testing in Steps 1 and 2 yields a large difference
between number of drops to failure at the respective drop heights. The method chosen
for this study was to use Equation 13. The reason is shown in Table 1. For each of the
four models tested in this study, the observed number of drops to failure varied similarly
regardless of drop height. This suggests that the materials tested behaved with similar

variance under the conditions tested.

Table 1. Variation in Observed Number of Drops to Failure for Test Models.

 

 

 

 

 

 

Table Height 1 Table Height 2 Table Height 3 Average
Model 1 49% 46% 51% 49%
Model 2 23% 12% 23% 19%
Model 3 19% 9% N/A 14%
Model 4 50% 36% N/A 43%

 

 

 

 

 

26

 

2.1.2 Freefall Method For Determining Material Properties

The second method investigated in this study was to obtain material properties x0
and X] from any input shock - in this case, shocks from freefall drops. This simple
method eliminates the shock table altogether, and relies solely on freefall drop testing
with the test product placed in a cushioned package. The procedure for dropping and
recording number of drops to failure is similar to the one described in 2.1.1, with one
notable exception: the SSE to be minimized is the difference between the known number
of drops to failure and the predicted number for the given input shock pulse. The
predicted number of drops to failure uses the algorithm described in detail in the next
section. Since the predicted number of drops requires x0 and x1 to be known beforehand,
an educated guess on their values is made, and then the correct x0 and x1 are then found
by an iterative method: systematically varying them until the SSE is minimized. A
summary outline can be found in Appendix B. This approach must be used since the
shock pulses in a typical freefall drop test usually are not short enough in duration to be
considered spikes to the component, thus nullifying the relationship in Equation 10. The
advantage of this approach is its simplicity: the shock table is eliminated and only
freefall drops need to be done. The apparent disadvantage of not knowing x0 and x, can
be eliminated by estimating their starting values as in plastic programmer testing, since
the values are expected to be similar. Even if nothing is known about x0 and x1, the
program will still find the appropriate values, though this may be a lengthy process. A
potential disadvantage of this method is the importance of accelerometer placement, since

SRS assumes the shock it analyzes is in fact the true imput shock to the component. It

27

may be difficult to determine the proper accelerometer location, depending on the test

model. The procedure, however, is straightforward:

1.

Place an accelerometer on the frame of the product, and choose a drop height from
which to drop the packaged product. Repeatedly drop the product until failure
occurs, recording number of drops to failure, and capturing the shock pulses from the
accelerometer. As with the plastic programmer method, there may be some
ambiguity as to what number to use for N when damage occurs, but the same
conclusion for choosing N is reached for freefall testing — choose N to be when
failure is observed. The choice of cushion material is important, since number of
drops to failure is recorded with the assumption that each shock pulse is similar: the
cushion (and to a lesser extent the corrugated container) must recover most of its
cushioning properties between drops. This eliminates certain cushion materials since
performance degrades rapidly after the first drop. Therefore it is best to choose a high
quality resilient cushion that performs consistently after multiple drops — neoprene is
recommended if it is available. Of course the cushion could be replaced after each
drop, but this may prove unnecessarily time consuming, and small variations in AV
and G will still occur between drops. As in the plastic programmer method, material
variability of the critical component is to be expected, so it is recommended to test
more than one unit at this height, and take the average number of drops to failure as
N. The number of drops to failure and corresponding input shock pulse become the
first data set.

Having found damage at a particular freefall drop height from Step 1, take a new

product in a cushioned package and choose a higher height from which to dr0p.

28

Proceed as before, dropping the package repeatedly, recording number of drops to
failure. This will establish failure at a higher energy input. The number of drops to
failure and corresponding input shock at this drop height becomes the second data set.
Two data sets are needed since there are two unknowns x0 and x1 which must be
solved for. This step may be repeated for different drop heights to establish other
data sets.

. A program following the outline in Appendix B can now used by entering the number
of drops to failure observed fiom each height and representative digitized shock
pulses from each height. If variability in AV and G between repeated drops is a
concern, it is possible using existing hardware and software to reconstruct an average
resultant input shock for the program, giving one representative pulse for each drop
height tested. Executing the program returns the values for x0 and x1 corresponding
to the minimized SSE, as well as the calculated values for material properties A and
B. AVcr and Ge, for each N is also calculated and displayed. The program could also

be extended to draw the corresponding fatigue DBCs.

2.1.3 Sources of Experimental Error

There are two sources of experimental error which can account for part of the SSE

of Equation 13: variation in material properties among the individual specimens tested,

and fluctuations in AV produced by either the shock table or in fi'eefalls during testing.

Variation in the material is evident fi'om the large variations in the number of drops to

failure shown in Table 1. Traditional S/N curves readily acknowledge the problem of

(scatter in collected data [10,13], and materials can exhibit a range of values during testing

29

for many reasons. For high molecular weight plastics, engineering therrnoplastics and
therrnosets, molecular weight distribution is especially critical. Even small variations can
lead to noticeable performance changes [18]. Impurities found in the material can also
create voids, weakening some samples, or providing additional reinforcement. Residual
stresses, from mechanical fabrication or thermal or mechanical treatment can also cause
variation. Number of drops to failure will be affected by these factors. Material
anisotropy is also a concern, especially when parts can be manufactured randomly with
respect to the property directions.

Small changes in AV can also affect number of drops to failure, but usually only
for large N. As an illustration, consider Figure 10. If the input shock puts the

compression of the spring just past the elastic limit at point A, the number of drops to

 

6‘

failure is If the input shock is changed only slightly, it is conceivable the

compression of the spring can be pushed firrther past point A, say 28. Then the ntunber

of drops to failure is $3 , half as many as before.
a

2.2 SRS AND DUCTILE MODEL
Having found x0 and x], the ductile model can now be added to the SRS
algorithm, providing a means for predicting fatigue failure in response to any input

shock. Applying Newton’s second law to the spring/mass system in Figure 1 gives:

30

 

 

 

 

 

Input Shock

 

 

 

Input Shock Changed Slightly

 

Spring
Force ,
Po nt A
mi /
F0 C" '
l//
\> / 1 1 >
time
X 0 x1
Deflection
Spring 4
F 2
orce E e Pth A
L i
F0 -- f——o——c
1‘ i >
time X0 x,
Deflection

Figure 10. Effect of Small Changes in Input Shock Velocity Change
on Number of Drops to Failure.

31

F = mj} (24)

where F is the spring compression force and m = W/g is the mass. Damping is not
explicitly included in Equation 24 since the properties in Equations 19 through 21 are
obtained under dynamic test conditions. As such, F depends only on the spring

compression 2, which in Figure 1 is:

z=x_y (25)

where x(t) and y(t) are the positions of the base of the product and the component relative
to their static positions, respectively, and the dot notation in Equation 24 indicates

derivatives with respect to time. Substituting for y from Equation 25, and noting that

F = F (2) , Equation 24 reads

 

(26)

It is advantageous to use Equation 26 over Equation 24 since it involves the acceleration
of the product it , which is easily obtained experimentally using an accelerometer.

The spring force in Equation 26 changes depending where on the curve in Figure
11 one is at. There are four regions of interest. Region 1 is the elastic loading region. A

linear relationship is assumed between spring force and displacement so that

32

Spring

 

 

 

 

 

Force
EL Regron 2 BP
F0 g + + -— ——— 4
r /
//’
Region 1
Region 3
1 1 1%
l / | 1
x0 Zmax xl
/ / Deflection
/
/ /
/ , Region 3
/
/
Region 4

Figure 11. Spring Force in Four Regions of Interest.

33

F(z) = kz ifz S x0 (27)

where k is the spring constant, as shown in Figure 1. The spring force will be described
by Equation 27 as long as z is less than or equal to x0. Region 2 is the perfectly plastic

region, and is described by a constant spring force

F(z) =170 ifXOSsz1 (28)

Equation 28 is valid if z is greater than x0. In a dynamic situation, the spring force
remains constant until 2' reaches zero, indicating the onset of unloading and
corresponding to the maximum deflection Zmax. Assuming spring compression does not
reach the break point X], Region 3 describes elastic unloading, parallel to the elastic

region. The spring force in this elastic unloading region is described by

F12) = F. — kc... — z) (29)

Region 4 is similar to Region 2, and is known as the plastic unloading region, if it even
occurs during a normal shock.
If the compression of the spring ever exceeds x1, failure occurs. If compression of

the spring is below x0, no permanent damage occurs. If compression is between x0 and
. . z - x . . . . .
x1, damage Will be the fractron —"‘”‘—1 , accumulatrng lrnearly. By provrdmg any input

x1_xo

shock, it , the response 2 can be calculated using Equation 26 and 2 can be propagated in

34

time using concepts developed in the next section. Now both peak G and displacement
can be determined and used to assess damage potential for both brittle and ductile
components.

As in traditional SRS, the reaction of the mass during the input shock duration is
known as the primary response, and the reaction of the mass after the duration of the
input shock pulse is known as the residual response. The larger of the two responses is

reported as the maximum shock response deceleration value.

2.3 NUMERICAL SOLUTION

The equations above for finding deceleration and displacement must be solved
numerically on a computer since 56 will be provided at discrete points in time from a
digitized shock pulse. Since instantaneous spring compression 2 is necessary in order to
evaluate the spring force and hence 'z' , a time step approach was used to solve the

equations of motion. From elementary physics [19], if the acceleration '2' at time t is
assumed to be constant over the time step At, and z(t) and 2':(t) are the initial

displacement and velocity respectively, the spring compression 2 and rate of compression

2': at the next time (t + At) are:
z(t + At) = z(t) + z'(t)At + % 2'(t)At2 (30)

z'(t + At) = 2(2) + 'z'(t)At (31)

35

When working with digitized shock pulses, it is customary to use a “sampling
frequency”, which denotes the rate at which the computer picks off instantaneous values
from the shock pulse to use in evaluation. Since it is known, and At is obtained from the
reciprocal of the sampling frequency, the output deceleration 2' can be calculated by
substituting Equations 30 and 31 into Equation 26 and solving. The program for this
algorithm (PROGRAM) was written in Visual Basic 5.0, and can be found in Appendix

C.

2.4 GENERATING FATIGUE DBC CURVES

Generating DBCs incorporating fatigue requires three pieces of information: AVG,
number of drops to failure, and G6,. Using the plastic programmer method, number of
drops to failure and AVcr will have already been defined in Section 2.1, Steps one through
three. Gcr can then be found using Equation 5. Using the freefall method, number of
drops to failure has already been observed, and AVcr and G, can be calculated from

Equations 4 and 5. The data is plotted similarly as shown in Figure 8.

2.5 TEST TO VERIFY PROGRAM ALGORITHM
To verify that the model works properly, a hand check was done to compare the

results from two pulse shapes easy to evaluate by hand. The pulse shapes are a square

wave and a half sine pulse. The solution of Equation 24 for the square pulse (56 = Gg) is:

GW
2(1) — T (33)

36

which satisfies the initial conditions z = z' = 0. The solution to Equation 24 satisfying

initial conditions z = z' = 0 for the half-sine pulse (56 = Gg sinwt ) is:

 

x(t): Gg sinart (34)

where G is peak G of the half-sine pulse. 33,333 Hz was used for the sampling frequency
to force the PROGRAM to interpolate many values for the input shock 16. Table 2a
summarizes the results for a 20 G, 20 millisecond square pulse. Table 2b summarizes the
results for a half-sine pulse with a peak G of 200 and a duration of 2 milliseconds. Only
three points were chosen from the square pulse and 9 from the half-sine, enough to
adequately describe the pulses. Natural frequency used was 17 Hz, and the values for x0
and x1 were taken to be .787 inches and 1.93 inches, respectively. These values are
representative of small plastic beams with a mass attached to one end. The results in
Tables 2a and 2b show good agreement. The small discrepancies can be attributed to
computer round-off error and the interpolation between the relatively few data points

given to the program for describing the pulses.

37

Table 2a. Comparison of Values, Trapezoid Pulse.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Spring Hand Check Computer
Time (s) .01626 .01626
At Elastic Limit Position (in) .78700 .78783
Velocity (in/sec) 71 .51514 71 .49242
Time (s) .02000 .02004
At End of Input Position (in) 1.04900 1.04913
Shock Velocity (in/sec) 66.76000 66.73914
Time (s) .02748 .02748
At End of Plastic Position (in) 1.29760 1.29869
Loading Velocity (in/sec) 0.00000 .50099
Table 2b. Comparison of Values, Half-Sine Input Pulse.
Spring Hand Check Computer
Time (s) .00200 .00201
At End of Input Position (in) .09751 .09635
Pulse Velocity (in/sec) 97.29128 96.48929
Time (s) .01070 .01071
At Elastic Limit Position (in) .78710 .78193
Velocity (in/sec) 48.91088 49.59233

 

 

 

 

38

 

 

CHAPTER 3

MATERIALS AND TEST METHODS

3.1 MATERIALS

To capture analog shock inputs, software and hardware from Lansmont
Corporation called Test Partner 2 (TP2), were used. TP2 converts analog shock pulses
captured from accelerometers into digital data. For this study, the shock pulses were
translated into a .csv format to be imported by the Program. Equipment used in this study
included the following:

Calipers: Mitutoyo, Model CD-6” BS, serial number 0010283

Accelerometers: PCB Triax, Model 356A11, serial numbers 4201, 4202

Charge Amplifiers: PCB Model 482A16

Shock Table: Lansmont Model 65/81, s/n 57-681-0016

Vibration Table: Lansmont Touchtest System

Model 10000-10, size 152 cm, Hydraulic power supply

Drop Tester: Lansmont Model PDT-56E Freefall Tester

Mndsls

Four different spring/mass models were used for testing, and are described below:

1. PMMA beams. These beams were constructed of polymethyhnethacrylate (PMMA)
material, commonly known as PlexiglassTm. The material was obtained and cut into 6
5/8” x 1 3/8” x 1/4” pieces. A 9/32” hole was drilled on either end. To one end a
steel mass was attached, and the opposite end was attached to a wood test fixture, as

shown in Figure 12.

39

2. Metal Beams. Two-inch zinc-plated steel mending plates were mounted to a fixture
as shown in Figure 13. A mass consisting of 3 separate brass pieces was attached to
the opposite end of the beam using a small bolt and wing nut. Deflection
measurements were measured from the center of mass, an assumption used in
Equations 24 and 26 relating to Newton’s second law.

3. Springs. As shown in Figure 14, a small spring was suspended from a long bolt and
held in place using a series of nylon spacers and wing nuts. A lead-filled aluminum
mass was attached to the opposite end of the spring. The springs were stock items
fi'om the local hardware store, known as 119 Springs.

4. Sheet Metal. Hewlett Packard 5L LaserJet printers were used as a product test.
Removing the plastic housing and laser unit, the product was mounted onto a wood
base as shown in Figure 15. The sheet metal on the bottom side was the area of

interest, specifically the area shown in Figure 16.

3.2 METHODS
3.2.1 Determination of Natural Frequency

To find natural fi'equency of each model, two methods were employed. The first
was to place the test specimen on the vibration table and watch for resonance as the table
swept from 3 to 300 Hertz [2]. All four models were tested this way. The second method
was to mount an accelerometer onto the component, capture the oscillations after setting
it into motion, and calculating natural frequency from the response. All models except
the sheet metal were tested in this manner. The sheet metal proved to be difficult to

analyze, due to multiple resonant frequencies and noise superimposed on the

40

 

Figure 12. PMMA Beam With Mass Attached to Wood Fixture.

41

 

 

Figure 13. Metal Beam With Mass Attached to Wood Fixture.

42

 

Figure 14. Spring With Attached Mass.

43

 

Figure 15. LaserJet Printer Mounted Onto Wood Base, Plastic Programmer Testing.

Area of Interest

 

Figure 16. LaserJet Printer Sheet Metal.

45

waveform after excitation. Therefore, natural frequency was determined only by
observing resonance on the vibration table during the sweep test. The reported natural
frequencies in Table 3 are combined averages from the two methods since results were

similar in each test.

Table 3. Natural Frequency of Test Models.
PMMA Metal Beam 119 Spring Sheet Metal
Vibration Sweep 1 1 133 84 33

 

 

 

 

 

 

 

 

 

3.2.2 Definition of Failure
The table below describes the failure mode for each test model. The failure mode

was chosen to be easily quantified.

Table 4. Failure Modes and Determination Method.

 

 

 

 

 

 

 

 

Method of Failure
Description of Failure Determination
PMMA Beams Complete break Visual inspection
Metal Beams Permanent deflection of 2.92 mm Digital calipers
119 Spring Permanent deflection of 1.63 mm Digital calipers
Printer Sheet Metal Permanent deflection of 3.50 mm Digital calipers

 

3.2.3 Test Method for Finding Material Properties From Plastic Programmer Testing

The material properties A, B, x0 and x1 were found for all four components using
the shock table plastic programmers, as outlined in Section 2.1. The test models were
mounted on the shock table, and subjected to shocks on the plastic programmers at

specified levels. Average N and AV were recorded from the drops and used for the data

46

 

points. Equations 6 through 8, 19 and 20 were then used to find the material properties.
30 specimens were tested at each of three different shock table heights for the PMMA
and metal beams. Thirty 119 Springs were tested at each of two different heights. Five
printers and eight printers, respectively, were tested at two different heights for the sheet
metal properties. Fewer printers were tested due to the limited number of samples

available.

3.2.4 Validation of Results From Plastic Programmer Testing

Freefall drops were used to validate the material properties found from the plastic
programmer test method. Testing was done using the PMMA beams and sheet metal
specimens. First, the values for x0 and x1 obtained from shock machine drops were
inserted into the PROGRAM. The specimens were then subjected to freefall drops in
cushioned boxes. Freefall drops were used to change the impact shock to the specimens
as much as possible from the shock machine drops, thereby testing the model and
program under very different conditions than were used to get material properties.
Freefall drops were also chosen because they are simple and easy to perform. Dropping
the packages repeatedly in flat-bottom drops until failure occurred (same failure as
defined in the plastic prograrmner test), the number of drops to failure was recorded at
each height. The observed number of drops to failure was then compared to the predicted
number of drops from the PROGRAM and the digitized input shock obtained from the
freefall drop.

For the PMMA beams, thirty separate beams mounted in the wood test fixture

were tested using the free fall drop test machine from heights of 24, 26 and 30 inches. A

47

corrugated box was constructed to hold the wood test fixture and resilient foam (Dow
Ethafoame 220, Figure 17). The input shock was captured by TP2 from an
accelerometer mounted on the floor of the wood test fixture, and then translated into a
.csv file format.

For the printer sheet metal, the unit was mounted onto a wood base, and placed
into a box with foam for the freefall testing. Drops from 30 inches and 48 inches were
done using the freefall drop tester. Because of limited number of units available, only
three units were tested at 48 inches, and five units at 30 inches. More units were tested at
the lower height since number of drops to failure varied more. Ethafoam 220Tm was used
as the cushion in the 30 inch drops, and a low density convoluted polyurethane cushion
was used for the drops at 48 inches. These materials were chosen because of their
availability and lower cost compared to neoprene. The accelerometer was mounted as
shown in Figure 17, to capture the “input” pulse to the sheet metal. As mentioned before,
accelerometer placement is very important since SRS assumes the pulse it is analyzing is
an input to the component. Table 5 summarizes the test configurations used for the

fieefall drops.

Table 5. Freefall Drop Height Test Configurations.

 

 

 

 

 

 

 

F reefall Drop Cushion Thickness Cushion Bearing
Height Area
PMMA Beams 24 inches 2 inch 18 in2
PMMA Beams 26 inches 1 inch 48 inlr
PMMA Beams 30 inches 1 inch 48 113
Sheet Metal 30 inches 1 inch 15 in7
Sheet Metal 48 inches 1.75 inches 154 in2

 

 

 

 

48

 

Accelerometer location

 

Foam cushions

Figure 17. LaserJet Printer Mounted Onto Wood Base, Freefall Testing.

49

3.2.5 Test Method for Finding Material Properties From Freefall Drops

The material properties A, B, x0 and x1 were found for the PMMA beams and
printer sheet metal using the freefall drop method outlined in Section 2.1 and the program
in Appendix B. The procedure began by choosing the same freefall drop heights outlined
in 3.2.4, and noting the number of drops to failure at each height. Representative analog
shock pulses captured by accelerometer from each of the drop heights were digitized by
TP2 and entered into the program using the .csv format. Using the observed number of
drops to failure, starting values of x0 and x1 obtained from the plastic programmer
method, and the .csv formatted input shock pulse, the program in Appendix B calculated

the values for x0 and X].

3.2.6 Validation of Results From F reefall Drops

To validate the material properties found from the freefall testing, the calculated
x0 and x1 values from 3.2.5 were used with Equations 4 through 8 to calculate AVC, and
Gcf for the same number of drops to failure, N, as the plastic programmer test in 3.2.4.
AVcr and Gcr calculated from material properties found from freefall testing were then
compared to AVcr and Gcr calculated from material properties found from the plastic

programmer testing.

3.2.7 Method for Generating Fatigue DBCs
Fatigue DBCs were constructed for the four products using AVcr and Gcr

calculated from the plastic programmer testing in 3.2.4. The fatigue DBC curves are

50

identical to traditional DBC curves, with the addition of number of drops to failure at a

particular AVcr and Ger.

51

CHAPTER 4

RESULTS

4.1 RESULTS FOR MATERIAL PROPERTIES FROM PLASTIC PROGRAMMER
TESTING

The material properties A, B, x0 and x1 for the four models were calculated from
the plastic programmer testing method (3.2.3) and are summarized in Table 6. For the
PMMA and metal beams, the higher B values in relation to A indicate more ductile
materials, an expected result. Conversely, for the 119 Spring the smaller B value
indicates a more brittle material, also as expected. The A and B values for the sheet
metal indicate a somewhat ductile material. As described earlier, the value of A
represents the smallest drop height in inches required for failure, and appears reasonable
for each model. From Equation 10, adding the values of A and B gives the drop height
for failure in one drop. The value for the PMMA beams appears somewhat high in a
practical sense. However, the velocity change from the plastic programmer testing for
each N was quite high to establish failure, and the equivalent freefall height causing
damage in one drop is an extrapolation from this data.

The x0 and x; values for the metal beams, springs and sheet metal in Table 6 all
appear reasonable. Based on the size and shape of the actual models, the x0 and x1 values
are plausible deflections. However x0, and especially x], appear to be grossly overstated
for the PMMA beams, given the fact the beams are less than seven inches in length to
begin with. This raises the question whether the calculated material property values have
real physical meaning, or are simply the result of a mathematical fit of a particular model

to observed data. More importantly, if the material properties do not

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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53

have real physical meaning, will the predictions of the PROGRAM using these material
properties give inaccurate results when using them to predict number of drops to failure
for different shock pulses. The x0 and x1 values are expected to be slightly higher in a
dynamic test compared to a static test, and so performing a static deflection test will
provide some basis for rationalizing the values found in Table 6. Several PMMA beams
were tested in a static deflection test, and a representative curve is shown in Figure 18.
The breakpoint x1 reported is .55 inches, and x0 can be estimated to be roughly .25
inches. These static values are about eight and fifteen times less than the dynamic x0 and
x, values in Table 6. Either the material properties for the PMMA beams were
erroneously calculated from the plastic programmer testing, or the beams do not fit the
idealized elastic, perfectly plastic model. To eliminate the possibility of calculation or
data gathering error, the same material properties were found from freefall drop tests a
few sections ahead in 4.3. Regardless of a lack of physical meaning, the calculated
material properties were used with the elastic, perfectly plastic model to predict number
of drops the failure for other types of input shocks by performing freefall drop testing as

described in the next section.

4.2 RESULTS FOR VALIDATING MATERIAL PROPERTIES FOUND FROM
PLASTIC PROGRAMMER TESTING
Shock pulses from freefall drop testing with the PMMA beams were entered into
the PROGRAM to predict number of drops to failure using the material properties
reported from plastic programmer testing in Table 6. The predicted number of drops to

failure from the PROGRAM were compared to the observed number of drops to failure,

54

Connrntinq PCX Inaqv...

Sample ID: PHHR Bean

Sample # Peak Force Defl 9 Pk Temp Humidity
8 146.3 70. 5

8.55 8 57.

0.075 In/Div

 

Figure 18. Static Force Versus Deflection Curve For PMMA Beam.

55

and the results are summarized in Table 7. Representative shock pulses were entered into
the PROGRAM from the 30, 26 and 24 inch drop heights. 30 separate beams were tested
at each drop height.

Similarly for the sheet metal, representative shock pulses from the 30 and 48 inch
drop heights were entered into the PROGRAM, along with the material properties from
Table 6, to predict number of drops to failure. Actual number of drops to failure was

compared with the predicted number, and is summarized in Table 8.

Table 7. Actual and Predicted Drops to Failure, Freefall Testing, PMMA Beam.

 

 

 

 

30 inches 26 inches 24 inches
Method Avg. St. Dev. Avg. St. Dev. Avg. St. Dev.
Actual 3.8 2.1 5.2 2.9 7.9 5.3
Predicted 3.9 1.1 4.3 0.3 8.7 2.2

 

 

 

 

 

 

 

 

Table 8. Actual and Predicted Drops to Failure, Freefall Testing, Sheet Metal.

 

 

 

 

30 inches 48 inches
Method Average St. Dev. Average St. Dev.
Actual 9.2 3.6 5.3 1.2
Predicted l .8 .5 1 .2 .1

 

 

 

 

 

The results show good agreement for the PMMA beams.

 

The actual and predicted

 

 

number of drops to failure was always within one drop for all drop heights. This suggests
the material property values, though maybe not physically descriptive, are useful for
predicting damage from different shock pulses. The standard deviation from the

predicted number of drops is about half from the actual observed, a reflection of the

material variability since the PROGRAM’s only variation is the shock input itself.

56

The results for the printer sheet metal show the predicted number of drops to be
too low compared to the actual. The large standard deviation in the observed number of
draps compared to the predicted indicates material variability plays a significant role in
these results. Material variability is particularly important since relatively few samples
were tested at each drop height, both for the plastic programmer testing and the fieefall
testing. This is a practical limitation since it was prohibitively expensive to test many
products. Trade-offs may need to be made between cost and accurate data. It is also
possible the sheet metal is not a single degree of freedom spring/mass system, an
assumption of Equation 24. The sheet metal has a PCB board with many electronic
components affixed to it, making it a complex arrangement of many spring/mass systems.
The model in this study may not accurately represent this type of system. The location of
the accelerometer to capture the input shock may also have affected predicted results. As
shown in Figure 17, the accelerometer was mounted to the wood test fixture, which itself
is a spring/mass system, able to flex during a freefall drop. The true input shock to the
sheet metal may in fact have been something different than the shock captured by the

accelerometer mounted to the base of the wood test fixture.

4.3 RESULTS FOR MATERIAL PROPERTIES FOUND IN FREEFALL DROPS
Using the procedure outlined in 3.2.5, material properties A, B, x0 and x1 for the
PMMA beams and the printer sheet metal were again found from the freefall drop testing,
and are summarized in Table 9. The results for x0 and x1 show good agreement with
those in Table 6 obtained from plastic programmer testing. The agreement is within 3%

for the PMMA beams, and within 12% for the printer sheet metal. As noted in the

57

previous section, it is possible the accelerometer location on the wood test fixture did not
capture the true input shock to the sheet metal, and may explain the difference in the
calculated material property values. This points out one benefit for using the plastic
programmers; the rigid, heavy shock table eliminates the flexing of the wood test fixture
during the drop event, and may provide a more consistent and true input shock.

For the PMMA beams, the good agreement for x0 and x1 between the freefall
method and the plastic programmer method eliminates the possibility of calculation or
data gathering error for the material properties. This suggests that the model does not
accurately describe the PMMA beams. Despite the model’s inability to predict
physically meaningful material properties for the PMMA beams, it still fits values that
are able to correctly predict number of drops to failure from very different shock pulses
when used in the fatigue shock response PROGRAM. This is an acceptable situation,
since the goal is to predict number of drops to failure, not isolate actual material

properties, which have little independent value.

Table 9. Summary of Calculated Material Properties From F reefall Testing.

 

 

 

 

 

 

 

 

Drop F reefall Drop Test Data Calculated Material Properties
Height, Observed Drops to Failure, Fit Results, Material Properties,
inches Mean 3: St. Dev. Equations 4 - 8 inches
24 7.9 :1: 5.3 A = 24.2 _
10% 26 5.2 s 2.9 B = 130.0 :0 ; £3
30 3.8 s 2.1 R2 = 0.99 1 -
Sheet 30 9.2 s 3.6 A f 23.5 M = 0.65
B — 32.5 _
M6131 48 5.3112 R2=1.0* X1 -1-10

 

 

 

 

*Only two data points gives perfect fit for straight line

58

 

4.4 RESULTS FOR VALIDATING MATERIAL PROPERTIES FOUND FROM
FREEFALL DROP TESTDIG
Table 10 summarizes the comparison between the calculated AVClr and Ger
(Equations 4 and 5) using x0 and x1 values found from freefall testing and those found in
the plastic programmer testing, both at the same number of drops to failure, N. Results
show less than 3% difference between the two methods for the PMMA beams, and about

13% for the sheet metal.

Table 10. Comparison of Calculated AVcr and Gcr.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PMMA Beams Sheet Metal
N Plastrc FreeFall N Plastic FreeFall
Programmers Programmers
AVcr, in: 147.2 151.1 124.8 144.6
24. 11.5
Ger: 3 13.1 13.5 33.4 38.7
AVcr, in: 9 8 166.2 170.1 4 8 134.8 151.3
Gcr: ° 14.5 14.9 ° 35.8 40.3
AVcr in: 191.5 195.3 N/A N/A
’ 5.2 N/A
Gcr: 16.0 16.4 1 N/A N/A

 

 

 

 

 

 

 

4.5 RESULTS FOR GENERATING FATIGUE DAMAGE BOUNDARY CURVES

To generate fatigue DBCs, only x0 and x1 are needed, either fi'om the plastic
programmer or freefall test method. AVcf and GClr are then calculated from Equations 4
through 7 for any N. To demonstrate this, Table 11 was constructed using x0 and x1 from
the plastic programmer testing method and three different values of N. This information
is plotted in Figures 19 through 22. The rounded knee has been squared off for

simplicity, and the axis formatted to make the curves easy to read on the pages.

59

Table 11. Data For Constructing Fatigue DBCs.

 

 

 

 

 

 

 

 

 

 

 

 

 

N = 1 N = 5 N = 10
AVcr, in Gcr AVcr, in Gcr AVcr, in Ger
PMMA Beam 340.8 20.6 193.2 16.1 165.8 14.5
Metal Beam 253.9 206.7 153.4 159.3 135.7 144.8
119 Spring 158.9 105.1 131.0 89.3 127.1 86.8
Sheet Metal 186.9 45.2 134.1 35.7 126.0 33.7

 

6O

 

35s

 

 

 

 

 

 

 

 

30 G l I
25 .
AVcr = 340.8
Ger = 20.6
1/ N = 1
20 I
AVcr = 193.2
Ger = 16.1
/ N = 5
15 1
\ AVcr = 165.8
Gcr = 14.5
N = 10
10 a
5 a
O U V ‘ U V V ‘
1 00 1 50 200 250 300 350 400 450
Velocity Change, in/sec

Figure 19. DBCs for PMMA Beams.

61

260

240

220

200

G180

160

140

120

100

1

1

q

 

 

 

AVcr = 253.9

Gcr = 206.7
/ N = 1

AVcr = 153.4

Gcr=159.3

 

AVcr = 1135.7

Gcr= 144.8
{ N= 10

 

 

100

1 50 200 250

Velocity Change, in/sec

Figure 20. DBCs for Metal Beams.

62

300 350

1151

110‘ 1 1

AVcr = 158.9

Gcr= 105.1
/
1051

 

v

1004

 

 

 

 

 

 

95 d
AVcr= 131.0
Gcr= 89.3
90 ‘ / N = 5
AVcr= 127.1
Gcr= 86.8
/ N =10
85 d
80 .. . . u u -
100 120 140 160 180 200 220

Velocity Change, in/sec

Figure 21. DBCs for 119 Spring.

63

 

 

 

 

 

 

 

 

55 .
50 I
AVcr = 186.9
Gcr = 45.2
/ N = 1
45 - -
40 -
AVcr = 134.]
Gcr = 35.7
i/ N = 5
35 -
AVcr = 126.0
Gcr = 33.7
N = IO
30 J
25 -
20 I’ I V I I I
120 140 160 180 200 220 240

Velocity Change, in/sec

Figure 22. DBCs for Sheet Metal.

64

4.6 DISCUSSION OF ERRORS AND IMPACT ON RESULTS

Observing Tables 7 and 8 points out some discrepancy between the actual and
predicted number of drops to failure. There are several possible explanations for these
discrepancies. First and foremost is material variability. Number of drops to failure from
the plastic programmer or freefall test (to determine A and B) will affect x0 and x1. One
approach to handle this variability is to vary N within experimentally observed ranges
and generate the corresponding x0 and x1 values. The PROGRAM can then use these
values to predict a range for number of drops to failure.

Another approach is to directly observe the variation in x0 and x1 by conducting a
static compression test. As pointed out earlier, this is not a recommended method for
determining the material properties. Nevertheless, this test may prove useful for
establishing upper and lower limits on x0 and x1. Testing of this sort was done using the
PMMA beams (Figure 18), and the average and standard deviation for x0 and x1 were
established. The percent standard deviation was then used with the average x0 and x1
values calculated from Equations 19 and 20 to describe ranges for x0 and x1 values.
Table 12 shows the results for the PMMA beams, 0 representing standard deviation. The
results in Table 7 suggest no adjustment is necessary in x0 and x1 to correctly predict
average number of drops to failure for the PMMA beams. However, the smaller standard
deviation from the predicted number of drops suggests the PROGRAM might not account
for variations in number of drops to failure when working with smaller sample sizes.
Conversely, varying x0 and x1 might be beneficial for the sheet metal, based on the results

in Table 8, and a table could be constructed for the sheet metal similar to Table 12.

65

Table 12. Variation of x0 and x1 From Static Deflection Test.

 

 

 

 

 

 

x0, in x], in

Average 1 .92 7 .29
1 o 1.75 — 2.09 5.85 - 8.73
2 o 1.57 — 2.27 4.42 - 10.16
3 o 1.40 — 2.44 2.99 - 11.59

 

 

 

 

Another potential cause of error is the choice of N during the procedure to find A
and B. As previously discussed, the true value of N occurs somewhere between the drop
causing observed failure and the previous drop. Table 13 shows how N from plastic
programmer testing would vary depending on how N is chosen. “High” refers to N when
damage is observed, “Low” for the drop before observed failure, and “Average” for the
average of the “High” and “Low” N values. x0 and x1 are subsequently affected by the
choice of N as shown in Table 14. Eventually number of drops to failure predicted by the
PROGRAM would be affected. Similar tables could be constructed for the freefall test

method showing how x0 and x1 are affected by the choice of N.

Table 13. Values For Number of Drops to Failure.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of Number of
Number of
D Drops to Drops to
rops to . .
Failur Hi h Failure, Failure,
e, g Average Low
1 24.27 23.77 23.27
P113223 2 9.83 9.33 8.83
3 5.17 4.67 4.17
1 17.73 17.23 16.73
11:13:: 2 9.53 9.03 8.53
3 5.77 5.27 4.77
119 1 5.30 4.80 4.30
Spring 2 2.03 1.53 1.03
Sheet 1 11.60 11.10 10.60
Metal 2 4.78 4.28 3.78

 

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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67

A third but less important influence on x0 and x1 is the variation of AV during the
determination of A and B. The AV from the plastic programmers can usually be held to
within +/- 5%, which should minimize its effect on determining A and B. This is
demonstrated in Table 5, where the R2 value (curve fit) is very high, indicating drop
height (AV) does not vary in a way that strongly influences the calculation of x0 and x1
from A and B. However, the AV in freefall drops may be more difficult to control, since
factors such as cushion and corrugated degeneration can occur over multiple drops.
Since the package falls without restraint, it is also more difficult to precisely control drop
orientations, which may lead to small changes in the shock pulse shape, as discussed in
2.1.3. Thus, one might expect less accuracy for the material properties generated from
freefall testing.

Another possible influence on results is natural frequency determination.
Resonance usually occurs within a range of frequencies, which is in contrast to the single
number used by the PROGRAM for shock response analysis.

Other errors introduced include such things as accelerometer calibration,
electromagnetic noise captured by accelerometer cables, and computer round off error.
Care was taken to prove the accelerometers were calibrated properly, and normal
precautions were taken during data gathering so that these errors should be regarded as

incidental.

68

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

5.1 CONCLUSIONS

Material properties are easily obtained from the plastic programmer test method
because the velocity change and number of drops to failure are used directly in a simple
regression scheme. This is made possible because the shock generated by the plastic
programmers is a "spike”, for which the response of the component has a single closed
form solution. The material properties found from the freefall test method require a
computer program because the shock is no longer a spike. Still, they showed good
agreement with those obtained in the plastic programmer test method. Thus, even with
shock pulse and cushion variability, the freefall method appears to be sufficient for
obtaining the characteristic material properties needed for the model. The procedure
outlined in Section 2.1.3 and Appendix B provide a straightforward and easy method for
determining material properties without the need for a shock table.

For both the PMMA beams and printer sheet metal, the freefall method predicts
material properties similar to those obtained from the plastic programmer testing, but the
plastic programmer method is recommended for obtaining them. The reason is two-fold.
First, the plastic programmer method eliminates potential complications when deciding
accelerometer location, and secondly there is no concern with cushion material
performance from drop to drop. The shock response PROGRAM in Appendix C can

then be used with these material properties to predict number of drops to failure for any

69

input shock. Fatigue DBCs can also be constructed using Equations 4 through 7. All of
this can be done without an expensive shock table.

As demonstrated with the PMMA beams, the SRS PROGRAM utilizing the elastic,
perfectly plastic model and dynamic material properties accurately predicts number of
drops to failure for input shocks which are very different from the ones used to find the
material properties, even when the material properties appear to have no physical
meaning. Thus, the model appears to correctly extrapolate material property values that
describe behavior during freefalls from plastic programmer data. For the sheet metal,
using material properties obtained from plastic programmer testing with the PROGRAM
predicts responses in freefall drops to be more severe than actual, most likely because the
sheet metal is not a single degree of fieedom system. It is also possible flexing of the
wood test fixture during freefall drops complicates the input shock to the component,
adversely affecting the predictions of the SRS PROGRAM.

There are several sources of experimental error that affect the calculations of
material properties, and subsequently the predictions made by the SRS PROGRAM.
Small changes in velocity change from drop to drop have some effect on calculating
material properties. Choosing the N corresponding to failure also affects calculated
material properties. The largest contributor to experimental error is material variability,
evidenced by the large variations in the number of drops to failure during plastic
programmer and freefall testing. These experimental errors are unavoidable, and need

consideration during fatigue fragility testing.

70

5.2 FUTURE WORK

The most beneficial outcome from this new fragility model is that once material
properties are defined, they can be used to predict percent damage caused by any type of
input shock, including dissimilar multiple shocks to the same component. Shock pulses
from different drop heights or onto different surfaces could be analyzed for their
cumulative damage effect on the component. This is done by simply entering each shock
pulse into the PROGRAM to predict percent damage, and adding the results. Future
work could concentrate on verifying this use by subjecting a test specimen to several
different shock pulses, and comparing the predicted drops to failure with actual number
of drops to failure.

Other areas for future work include the following:

OSRS is sensitive to the input shock used for analysis. More work should be done
to analyze the effects of SRS predictions from differing accelerometer locations and
mountings.

OThe elastic/perfectly plastic model assumes natural frequency of the critical
component remains the same from drop to drop — the stretching and releasing of the
spring does not alter its natural frequency. Checking the component’s natural frequency
after each drop could serve as an additional validation for this assumption.

OThis study focused on the ability of the PROGRAM to predict only the total
number of drops to failure given identical multiple shocks. The permanent deformation

from each drop should be checked to see if it is constant as the model predicts.

71

OMuch has been said about obtaining the material properties x0 and x1. Another
possible method for obtaining them is to use high-speed video during dynamic testing. A
method similar to the ones described in earlier works [20] might prove useful.

OAS noted in the Introduction, it was observed from earlier work that the fatigue
model might satisfactorily describe crack propagation failure, and more investigation of
this may be warranted.

OThe fatigue shock response PROGRAM in Appendix C can easily be modified
to calculate shock response spectra over any desired range of frequencies. The program
for finding material properties in freefall testing outlined in Appendix B could be
modified to include a print-out of the calculated DBCs.

OInvestigation of weighting methods mentioned in Section 2.1 may prove useful
for particular materials.

OSince S-N curve data already exists for many materials, it may prove worthwhile
to investigate developing a method for combining the elastic, perfectly plastic model in
this study with the existing S-N data. This may reduce or eliminate lab testing for
generating material properties.

OAnd finally, useful testing can be done on other models of interest to validate the

elastic, perfectly plastic model used with SRS.

72

APPENDIX A

73

APPENDIX A

RECOVERING OTHER MODELS

The rigid, perfectly plastic model [7] can be recovered from the elastic, perfectly

plastic model by first incorporating Equation 8 into Equations 6 and 7 to get A and B in

terms of XO before the limit is taken:

A=——-—=—— 35
XOW 2g 2W ( )
Fg x2 x F

B=——0—'—0[—L—1]=-‘0‘ _ 3
xOW g x0 W(x' x0) (6)

Substituting into Equation 5 and simplifying yields

_ 5,— Nxo +2x, —2xO

 

 

G — 37
°’ W 2Nx0 + 2xl — 2x0 ( )
Taking the limit of XO as it approaches zero gives:
1' G F0 ACI’ 38
1m = —— 2
X040 Cf W g ( )

74

which corresponds with the result in [7]. Similarly for AV substituting into Equation 4

cr’

 

and simplifying yields
1' AV — 2 F0)“ — 2(5) 5 39
.23}. °' ' g WN m N ( )

which is the same expressions found in [7].
The brittle model, where x; = x0, can also be recovered from the elastic, perfectly

plastic model by observing that B = 0 in Equation 7, so that Equations 4 and 5 read

 

 

 

 

2 2
AV” = 2g“ x0 =24”, (40)
g
0)sz
2g _ (27%!ny (41)
2 2 _ 2
20) x0 g
28

75

APPENDIX B

76

APPENDIX B

MATERIAL PROPERTIES FROM FREEF ALL TESTING

The following is a summary outlining the steps for find material properties x0 and

x1 from freefall drop testing.

1.

Construct a test package for the test specimen. Mount an accelerometer in a location
to capture the input shock to the specimen.

Drop the test package from a known height, recording the shock pulse from the
accelerometer.

Repeat dropping fiom the same height until failure occurs, recording number of drOps
to failure.

Repeat Steps 2 and 3 for a different drop height.

Calculate starting values for x0 and x1 using the same regression equations (Equations
16 through 18) as for the plastic programmer method.

Using the material properties from Step 5 above, and the Program in Appendix C,
calculate the predicted number of drops to failure for each drop height in Steps 2
through 4.

Calculate SSE for actual and predicted number of drops to failure.

Systematically vary x0 and x1 and repeat Steps 6 and 7 until the SSE is minimized.

77

APPENDIX C

78

APPENDIX C

SRS PROGRAM CODE

The following is Visual Basic 5.0 code for the SRS algorithm incorporating the
elastic, perfectly plastic stress-strain curve to accommodate fatigue failure. The code is
written to demonstrate the algorithm described in Chapter 2, not to necessarily represent a

commercial software application.

 

 

'Modified: 16 December 1998

,1 I . . E l . l :

Option Explicit

Dim lcv, intNumRec, intCounterForArray As Integer 'counter variables

Dim sngSamplan, sngXO, sngXl, sngDuration, sngDT, sngZ3, snan As Single
Dim sngPRIG, sngZMax, sngKM, sngZ, sngZDot, sngZZ, sngFO As Single
Dim sngT, sngOA, sngZDDot, sngIA, sngFZ2, sngPercent, sngDrops As Single
Dim strFileName As String

Dim ShockData As ShockData 'udt for input shock

Dim strTemp As String

Dim strTemp2(2000) As String

' Dim sngDR, sngCM

 

79

Private Sub Form_Load()

frmMain.Top = (Screen.Height - frmMain.Height) / 2 'center form in screen of any size

frmMain.Lefi = (Screen.Width - frmMain.Width) / 2
fraResults.Visible = False
frmMain.Show 'form must be loaded or shown before SetFocus will work

txtX0.SetFocus 'must show form before any objects in the form can have SetFocus

With graShockData
.Visible = False
.DataGrid.RowCount = 5000
.ColumnCount = 4

End With

With graForceDeflection 'set up graph for force/deflection curve
.Visible = False
.DataGrid.RowCount = 5000
.DataGrid.ColumnCount = 2
.Plot.Axis(VtChAxisIdX).AxisTitle = "Deflection (in)"
.Plot.Axis(VtChAxisIdY).AxisTitle = "Spring Force (lbs)"

End With

End Sub

 

80

Private Sub Form_Unload(Cancel As Integer)
Const conBtns = vbYesNoCancel + vixclamation + vbDefaultButton3 +
vbApplicationModal
'Const conmsg = "Do you want to save the current information?"
Dim intUserResponse As Integer
'If blnChange = True Then
intUserResponse = MsgBox("Are you sure you want to exit?", conBtns, "SRS")
If intUserResponse = vbYes Then
End
Else
Cancel = 1
End If
'add dialog to ask if want to save?

End Sub

 

 

Private Sub mnuAboutAbout_Click()
fi'mSplashShow

End Sub

 

 

 

Private Sub mnuCalculateSR__Click()

frmMain.MousePointer = vaourglass 'mouse pointer turns to hourglass

'Open "e:\def_data" For Output As #2 'check values calculated by program

81

'reset variables to 0, allowing repeated opening of files and calc’s w/o quitting application
sngZ = 0
lcv = 0
sngDuration = 0
sngDT = 0
sngZB = 0
sngPRIG = O
sngZMax = 0
sngKM = 0
sngZ = 0
sngZDot = 0
sngZ2 = 0
sngFO = 0
sngT = O
sngOA = 0
sngZDDot = 0
sngIA = 0
sngF Z2 = 0
sngPercent = 0
sngDrops = 0

intCounterForArray = 0

For lcv = 1 To 5000 'clear out arrays for force vs deflection graph

82

graForceDeflectionDataGrid.SetData lcv, l, 0, 1 'row, column, value, not visible
graForceDeflectionDataGrid.SetData lcv, 2, 0, 1

Next lcv

'read values from form into variables
sngSamplan = Val(txtSamplan.Text)
'sngDR = Val(txtDampRatio.Text)
sngXO = Val(txtX0.Text)

sngXl = Val(txtX1.Text)

snan = Val(txtFn.Text)

sngDuration = ShockData.sngMS(intNumRec) 'duration will be equal to last ms
reading from the inported pulse

sngDT = l / sngSamplan 'set time step based on sampling frequency supplied by user

sngKM = (2 * 3.14159 * snan) A 2

For sngT = sngDT To sngDuration Step sngDT 'run program to end of input pulse

intCounterForArray = intCounterForArray + 1 'keep track of iterations

'interpolate to get input G level from shock pulse
For lcv = 1 To intNurnRec - 1
If sngT >= ShockData.sngMS(lcv) And sngT <= ShockData.sngMS(lcv + 1) Then
sngIA = ShockData.sngG(lcv) + ((ShockData.sngG(lcv + 1)-

ShockData.sngG(lcv)) * (sngT - ShockData.sngMS(lcv)) _

83

/ (ShockData.sngMS(lcv + l) - ShockData.sngMS(lcv)))
End If

Next lcv

'place value into datagrid to be plotted
graForceDeflectionDataGrid.SetData intCounterForArray, 1, sngZ, 0

graForceDeflectionDataGrid.SetData intCounterForArray, 2, sngFO, 0

sngZ = sngZ + sngZDot * sngDT + sngZDDot * sngDT A 2 / 2 'set value for

position

'Write #2, "sngz= " & sngZ & "= " & sngFO

If sngZ <= -sngX0 Then Exit For 'exit loop if in elastic region, at elastic limit

deflection and negative

If Abs(sngZ) > sngZMax Then sngZMax = sngZ 'keep track of maximum position

'check for position exceeding x1

If Abs(sngZMax) > sngXl Then
lblCachrops.Caption = "This shock breaks the element"
1blCachamage.Caption = "100%"

Exit For

84

End If

'Elastic Loading Criteria
If (Abs(sngZ) <= sngX0 And sngFZZ = 0) Then 'use sngFZ2 as flag; 0
indicates not in unloading
sngZDot = sngZDot + sngZDDot * sngDT
sngZDDot = sngIA - (sngKM * sngZ) '- (sngCM * sngZDot)
sngFO = sngKM * sngZ
'Write #2, "Elastic." & "F= " & sngFO & "z= " & sngZ & "zdot= " & sngZDot

End If

'Plastic Loading Criteria
If (sngZ > sngXO And sngZDot > 0) Then
sngZDot = sngZDot + sngZDDot * sngDT
sngZDDot = sngIA - (sngKM * sngXO) '- (sngCM * sngZDot)
sngFO = sngKM * sngXO
sngF Z2 = 0.1 'flag to keep program from going back into elastic routine
'Write #2, "Plastic Loading." & "F= " & sngFO & "z= " & sngZ & "zdot= " &
sngZDot

End If

'Plastic Unloading Criteria

'sngFZ2 is flag to keep from elastic routine

85

If (sngZDot < 0 And Abs(sngFO) <= (sngKM * sngX0)) And sngFZ2 <> 0 Then

sngZDot = sngZDot + sngZDDot * sngDT

sngZDDot = sngIA - (sngKM * (sngXO - sngZMax + sngZ)) '- (sngCM *
sngZDot)

sngFO = sngKM * (sngXO - (sngZMax - sngZ))

sngFZZ = sngFO 'set sngFZ2, and hold for plastic loading

’Write #2, "Plastic Unloading." & "F= " & sngFO & "z= " & sngZ & "zdot= " &
sngZDot

If (sngZDot >= 0 And Abs(sngFO) <= (sngKM * sngX0)) Then Exit For

End If

'Plastic Loading, Tension Criteria
If (sngZDot < 0 And Abs(sngFO) >= (sngKM * sngX0)) Then
sngZDot = sngZDot - sngZDDot * sngDT
sngZDDot = sngIA - (sngKM * sngZ2) '+ (sngCM * sngZDot)
If sngZ < sngZZ Then sngZ3 = sngZ
sngFO = sngFZZ 'get sngFZ2 from end of plastic unloading; should be
(sngkm*sngx0)
‘Write #2, "Plastic Loading, tension." & "F= " & sngFO & "z= " & sngZ &

"zdot= " & sngZDot

'exit loop if velocity changes to positive, or when position equals snng — this

forces conformity to Bauschinger model

86

If sngZDot >= 0 Or Abs(sngZ) >= sngXO Then Exit For
End If
'calculate output acceleration, place result into datagrid
sngOA = sngKM * sngZ '+ sngCM * sngZDot
graShockData.DataGrid.SetData intCounterForArray, 3, sngT, 0

graShockData.DataGrid.SetData intCounterForArray, 4, sngOA / 386.4, 0

'keep track of peak output G
If Abs(sngOA) > Abs(sngPRIG) Then sngPRIG = sngOA

Next sngT

 

'check for not reaching elastic limit

If Abs(sngZMax) < sngXO Then
lblCachamage.Caption = "No Damage"
lblCachrops.Caption = "Below Elastic Limit"

End If

'calculate percent damage, etc.

If Abs(sngZMax) > sngXO And Abs(sngZMax) < sngXl Then
sngPercent = Abs((Abs(sngZMax) - sngXO) / (sngXl - sngXO))
sngDrops = Int(1 / sngPercent)

lblCachamage.Caption = Format(sngPercent, "percent")

87

lblCachrops.Caption = Fonnat(sngDrops, "##.#0")

End If

lblCalcPeakG.Caption = Format(sngPRIG / 386.4, "####.#0")
lblCachisplacement.Caption = Format(sngZMax, "##.#0")

fraResults.Visible = True

graShockData.Plot.UniformAxis = False

graShockData.Title.Text = "Shock Input and Shock Response"
graForceDeflection.Plot.UniformAxis = False
graForceDeflection.Visible = True

frmMain.MousePointer = vbDefault 'retum mouse to regular pointer
'Close #2

End Sub

 

Private Sub mnuF i1eExit_C1ick()
Unload Me

End Sub

 

 

Private Sub mnuF i1ePrintAll_Click()
PrintForm

End Sub

 

 

88

Private Sub mnuFilePrintData_Click()
PrintForm

End Sub

 

 

Private Sub mnuFileRetrieveSP__Click()
graShockData.Visible = False

fimMain.MousePointer = vaourglass

With dlgCommon
.DialogTitle = "Retrieve Input Shock Pulse"
.Flags = cleFNHideReadOnly + cleFNFileMustExist 'hide read only box
.Filter = "Text Files(*.txt)|*.txt|CSV Files(*.csv)|*.csv"
.filename = ""
.ShowOpen

End With

'if cancel is selected, exit sub without loading shock pulse data
On Error Resume Next

If Err.Number = 75 Then Exit Sub

'clear out the array from any previous shock pulse data
For lcv = 0 To 2000

strTemp2(lcv) = 0

89

ShockData.sngMS(lcv) = 0
ShockData.sngG(lcv) = 0

Next lcv

For lcv = 1 To 5000
graShockData.DataGrid.SetData lcv, 1, 0, 1
graShockData.DataGrid.SetData lcv, 2, 0, 1
graShockData.DataGrid.SetData lcv, 3, 0, 1
graShockData.DataGrid.SetData lcv, 4, 0, 1

Next lcv

Open dlgCommon.filename For Input As #1

intNumRec = 0

If dlgCommon.filename <> "" And UCase(Right(dlgCommon.filename, 3)) = "TXT"
Then
'input text file, formatted with MS and G, seperated by commas
Do While Not EOF(1)
intNumRec = intNumRec + 1
Input #1, ShockData.sngMS(intNumRec), ShockData.sngG(intNumRec)

ShockData.sngMS(intNumRec) = ShockData.sngMS(intNumRec)/ 1000

'set graph array data to time (sec) and G data

90

With graShockData.DataGrid
.SetData intNumRec, 1, ShockData.sngMS(intNumRec), 0
.SetData intNumRec, 2, ShockData.sngG(intNumRec), 0
.SetData intNumRec, 3, O, l
.SetData intNumRec, 4, 0, 1

End With

ShockData.sngG(intNumRec) = ShockData.sngG(intNumRec) * 386.4
Loop

Close #1

frmMain.Caption = dlgCommon.filename
End If
'open CSV file, directly from TP2
If dlgCommon.filename <> "" And UCase(Right(dlgCommon.filename, 3)) = "CSV"
Then
Line Input #1, strTemp 'get rid of first line
Do While Not EOF(1)
intNumRec = intNumRec + 1 'update record counter
Input #1, ShockData.sngMS(intNumRec), ShockData.sngG(intNumRec),
strTemp2(intNumRec)

ShockData.sngMS(intNumRec) = ShockData.sngMS(intNumRec)/ 1000

91

'fill graph array with time and G data

With graShockData.DataGrid
.SetData intNumRec, 1, ShockData.sngMS(intNumRec), 0
.SetData intNumRec, 2, ShockData.sngG(intNumRec), 0
.SetData intNumRec, 3, 0, 1
.SetData intNumRec, 4, 0, 1

End With

ShockData.sngG(intNumRec) = ShockData.sngG(intNumRec) * 386.4

Loop

Close #1
fimMain.Caption = dlgCommon.filename

End If

With graShockData
.Visible = True
With .Plot
.Axis(VtChAxisIdX).ValueScale.Minimum = 0
.Axis(VtChAxisIdX).ValueScale.Maximum = (graShockData.DataGrid.RowCount)
* 0.05 / 1000
.Axis(VtChAxisIdX).ValueScale.MajorDivision = 0.001

.Axis(VtChAxisIdX).AxisTitle = "Time (ms)"

92

.Axis(VtChAxisIdY).AxisTitle = "Acceleration (G's)"

.UniformAxis = False

.LocationRect.Min.Set 0, 0

.LocationRect.MaxSet graShockData.Width - 450, graShockData.Height - 450
End With

End With

frmMain.MousePointer = vbDefault

End Sub

 

 

Private Sub txtFn_GotFocus()
txtFn.SelStart = 0
txtFn.SelLength = Len(txtFn.Text)

End Sub

 

 

Private Sub txtFn_KeyPress(KeyAscii As Integer)
If KeyAscii = 8 Then 'allow backspacing
Exit Sub

End If

'only allow digits to be entered
If KeyAscii < 48 Or KeyAscii > 57 Then

If KeyAscii = 13 Or KeyAscii = 9 Then txtSamplan.SetFocus '13 is Return, 9 is

93

Tab
KeyAscii = 0 'ignore all other keystrokes
Beep

End If

'criteria for moving focus to next text box
If Len(txtFn.Text) >= txtFn.MaxLength And txtFn.SelLength <> Len(txtFn.Text)
Then
'txtDampRatio.SetFocus
txtSamplan.SetFocus
End If

End Sub

 

Private Sub txtSamplan_GotFocus()
txtSamplan.SelStart = 0
txtSamplan.SelLength = Len(txtSamplan.Text)

End Sub

 

 

 

Private Sub txtSamplan_KeyPress(KeyAscii As Integer)
If KeyAscii = 8 Then 'allow backspacing
Exit Sub
End If

'only allow digits to be entered

94

If KeyAscii < 48 Or KeyAscii > 57 Then
If KeyAscii = 13 Or KeyAscii = 9 Then txtXO.SetFocus '13 is Return, 9 is Tab
KeyAscii = 0 'ignore all other keystrokes
Beep

End If

If Len(txtSamplanText) >= txtSamplanMaxLength And txtSamplanSelLength
<> Len(txtSamplan.Text) Then
txtXO.SetFocus
End If

End Sub

 

 

Private Sub txtX0__GotFocus()
txtXO.SelStart = 0
txtXO.SelLength = Len(txtX0.Text)

End Sub

 

 

 

Private Sub txtX0_KeyPress(KeyAscii As Integer)
If KeyAscii = 8 Then 'allow backspacing
Exit Sub
End If
'only allow digits to be entered

If KeyAscii < 48 Or KeyAscii > 57 Then

95

If KeyAscii = 13 Or KeyAscii = 9 Then txtXl .SetFocus '13 is Return, 9 is Tab
If KeyAscii = 46 Then Exit Sub 'decimal
KeyAscii = 0 'ignore all other keystrokes
Beep
End If
If Len(txtXO.Text) >= txtX0.MaxLength And txtXO.SelLength <> Len(txtXO.Text)
Then
txtX1.SetFocus
End If

End Sub

 

 

 

 

Private Sub txtX1_GotFocus()
txtX1.SelStart = 0
txtX1.SelLength = Len(txtX1.Text)

End Sub

 

 

Private Sub txtXl_KeyPress(KeyAscii As Integer)
If KeyAscii = 8 Then 'allow backspacing
Exit Sub
End If
'only allow digits to be entered
If KeyAscii < 48 Or KeyAscii > 57 Then

If KeyAscii = 13 Or KeyAscii = 9 Then txtFn.SetFocus '13 is Return, 9 is Tab

96

If KeyAscii = 46 Then Exit Sub 'decimal
KeyAscii = 0 'ignore all other keystrokes
Beep
End If
If Len(txtXl .Text) >= txtX1.MaxLength And txtXl .SelLength <> Len(txtX1.Text) Then
txtFn.SetFocus
End If

End Sub

SnlashScrcen;
Option Explicit
Private Sub Form_KeyPress(KeyAscii As Integer)
If KeyAscii = 13 Then
frmMain.Show
Unload Me
End If

End Sub

Private Sub Form_Load()
lblVersion.Caption = "Version " & App.Major & "." & App.Minor & "." &

App.Revision

End Sub

97

Private Sub Frame1_Click()
fi'mMain.Show
Unload Me

End Sub

Private Sub tmrTimer_Timer()
fi'mMainShow
Unload Me

End Sub

Module;

Option Explicit

Type ShockData
sngMS(2000) As Single
sngG(2000) As Single

End Type

98

REFERENCES

99

10.

11.

12.

13.

REFERENCES

. Newton, Robert E. “Fragility Assessment Theory and Test Procedure.” US. Naval

Postgraduate School.

Selected ASTM Standards on Packaging. Fourth Edition. Philadelphia, PA: ASTM,
1994.

. Henderson, George. “Advanced Techniques For Shock and Vibration Analysis As

Applied To Distribution Engineering.” Fall 1991 IOPP P2C2 Boston Meeting, IBM
Corporation Shock and Vibration Seminars, 1987-91, Distribution Dynamics Labs
Advanced Shock and Vibration Seminars, 1991.

Schwartz, HE. and J. Santos. “Shock Response Spectrum (SRS): Analysis,
Interpretation and Application.” Poughkeepsie, NY: IBM Technical Report, 1989.

. Henderson, George. “Improved Protective Package Drop Test Data Analysis.” IBM

Annual Shock and Vibration Seminar. Boca Raton, FL. IBM, 19 October 1990.

Daum, Matthew P. “Application of the Shock Response Spectrum to Product
Fragility Testing.” Masters Thesis. Michigan State University, E. Lansing, MI,
1994.

Burgess, Gary J. “Product Fragility and Damage Boundary Theory.” Packaging
Wm Vol.15-10, 1988-

Burgess, Gary J. “Effects of Fatigue on F ragrlrty Testing and the Damage Boundary
Curve” MW Vol 24, No 6,1996.

Mendelson, Alexander. Wren. New York, NY:
Macmillan, 1968.

Callister, William D. Jr. Wm Fourth
Edition. New York, NY: John Wiley and Sons, 1997.

Collins, Jack A. EailnmflMaten'alsjnMeghamgaLflesign, Second Edition. New
York, NY: John Wiley and Sons, 1993.

Burgess, Gary J Lecture notes, Advanced Packaging Dynamics. Michigan State
University, E. Lansing, MI.

Wiebull, W. Warm; New York, NY: Pergmon
Press, 1961.

100

14. Singh, S. “Effect of Multiple Drops on the Damage Boundary Curve.” Masters
Thesis, Michigan State University, E. Lansing, MI, 1983.

15. Flugge, Wilhelm. Warm, Waltham, MA: Blaisdell Publishing Company,
1987.

16. Bhattacharyya, G. and Richard Johnson. W New
York, NY: John Wiley and Sons, 1977.

17. Ostrem, Fred E. and Godshall, W.D. “An Assessment of the Common Carrier
Shipping Environment.” Techincal Report F PL 22. Forest Products Laboratory,
Forest Service, USDA, Madison, WI, 1979.

18. Giacin, Jack. Lecture notes, Advanced Polymer Materials. Michigan State
University, E. Lansing, MI.

19. Hibbeler, R.C. EngmeenngMechamgsDynamms. New Jersey: Prentice-Hall,1nc.,
1995.

20. Marcondes, J., Waldeck, J, Burgess, G., and Singh, S. “Application of High- speed

Motion Analysis to Measure Shock 1n Cushioned Drops.” WW
Science Vol. 3 51-55,1990.

101