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This is to certify that the

thesis entitled

APPLICATION OF THE SHOCK RESPONSE SPECTRUM
TO PRODUCT FRAGILITY TESTING

presented by

Matthew Paul Daum

has been accepted towards fulfillment
of the requirements for

M . S . degree in Packag ing

   

ajér prefessor'

Date 3.9.94

 

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

 

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LIBRARY
Michigan State
University

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TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

[1:3 2 3’” J

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Wm M l

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution
chlmmpma-DJ

APPLICATION OF THE SHOCK RESPONSE SPECTRUM
TO PRODUCT FRAGTLITY TESTING

By

Matthew Paul Daum

A THESIS

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

MASTER OF SCIENCE

School of Packaging

1994

ABSTRACT

APPLICATION OF THE SHOCK RESPONSE SPECTRUM
TO PRODUCT FRAGILITY TESTING

By

Matthew Paul Daum

The purpose of this study was to apply the concept of the Shock
Response Spectrum (SRS) to fragility assessment. Specifically, the study
addressed the appropriate procedure for the testing of products using SRS,
and using SRS as an alternative to the Damage Boundary Curve (DBC) for
determining product fragility. Using SRS for deflection-type failure criteria as
opposed to G-level failure was also investigated, as well as the accuracy of a
commercial SRS software package.

Results clearly show SRS can accurately model single degree of
freedom spring / mass systems, and that the same information from a damage
boundary curve can be extracted easily from a SRS plot. The lengthy and
costly investment of a DBC testing procedure can be reduced to damaging
only one unit in an inexpensive free fall drop test. SRS is also an accurate

tool for predicting when deflection failure will occur.

Copyright by

Matthew Daum
Gary Burgess

1994

Dedicated to my loving and supportive wife, my best friend, April. Thanks
for your patience these last two years.

To my Lord and Savior Jesus Christ, in whom rests all knowledge and truth.

...yv0mpiov r013 6806 Kai narpos Kat
r013 xptoroi}, sv a) SLOL 1:01va or.
Gnoavpm rfis ao¢ias Kai 1:718 vaosws
anoxpv¢ou

"...the mystery of God, namely, Christ, in whom are hidden all the treasures
of wisdom and knowledge." —Paul of Tarsus, circa 64 AD.

iv

ACKNOWLEDGEMENTS

A big thanks to Dr. Gary Burgess for his genius, patience and humor with this
project.

Thank-you also to my committee members, Dr. S. Paul Singh and Dr. Bob
Ofoli.

Regards to the Center for Distribution Packaging for encouraging new insights
and applications in the area of packaging dynamics.

To the excellent professors at MSU who have helped shape and illuminate
the field of Packaging, much thanks and appreciation.

Finally, to the many special friends in the School of Packaging who have
made my graduate years enjoyable.

TABLE OF CONTENTS

LIST OF TABLES .................................................................................................... viii
LIST OF FIGURES .................................................................................................. ix
CHAPTER 1 ............................................................................................................. 1
INTRODUCTION AND LITERATURE REVIEW .............................. 1
1.1 INTRODUCTION .................................................................... 1
1.2 LITERATURE REVIEW ......................................................... 6
CHAPTER 2 ............................................................................................................. 7
THEORETICAL DEVELOPIVIENT .......................................................... 7
2.1 SR5 THEORY ............................................................................ 7
2.2 RELATIONSHIP BETWEEN DAMAGE BOUNDARY
AND SRS ................................................................................... 18
2.3 SR5 AND DAMPING RATIO ............................................... 25
CHAPTER 3 ............................................................................................................. 30
MATERIALS AND TEST METHODS ................................................... 30
3.1 SOFTWARE .............................................................................. 30
3.2 EQUIPMENT ............................................................................. 31
3.3 TEST METHOD FOR SRS EVALUATION ......................... 31
3.4 TEST METHOD FOR DEFLECIION FAILURE ................. 35
3.5 TEST METHOD FOR FRAGTLITY TESTING ..................... 35
CHAPTER 4 ............................................................................................................. 40

vi

A

RESULTS ..................................................................................................... 40

4.1 RESULTS ................................................................................... 40

4.1.1 Results From Damping Ratio ................................ 40

4.1.2 Results From Plastic Programmers ....................... 40

4.1.3 Results From Gas Programmers ............................ 45

4.1.4 Results From Deflection Failure ........................... 47

4.1.5 Results From Fragility Testing ............................... 47

CHAPTER 5 ............................................................................................................. 53

CONCLUSIONS AND LIMITATIONS ................................................. 53

5.1 CONCLUSIONS ....................................................................... 53

5.2 LIMITATIONS OF SRS ........................................................... 53

5.3 FUTURE WORK ...................................................................... 54
APPENDICES

APPENDIX A .............................................................................................. 55

APPENDIX B ............................................................................................... 57

APPENDIX C ............................................................................................... 59

APPENDIX D ............................................................................................... 63

APPENDD( E ............................................................................................... 78

APPENDD( F ............................................................................................... 87

APPENDIX C ............................................................................................... 92

REFERENCES ............................................................................................. 93

vii

LIST OF TABLES

Table 1. Damping Ratio Using Equation 24 ..................................................... 41
Table 2. Damping Ratio Using Equation 25 ..................................................... 41
Table 3. SRS Evaluation Results and Comparisons ...................................... 43
Table 4. Damping Ratio Calculations Using Equation 25; Peak

Heights Read From Test Partner ........................................................................ 65
Table 5. Damping Ratio Calculations Using Equation 25; Peak

Heights Read From Oscilloscope ........................................................................ 66
Table 6. Damping Ratio Calculations Using Equation 24 ............................. 67
Table 7. Damping Ratio Calculation Comparison For Plastic Beam ......... 68
Table 8. Effect of Damping Ratio On Shock Response .................................. 69

viii

LIST OF FIGURES

Figure 1. Response Of A Spring / Mass System With A Particular Natural

Frequency To A Single Shock Input .................................................................. 2
Figure 2. N ewton's Law Of Motion Can Be Used To Determine The
Response Of A Mass To A Particular Excitation ............................................. 8
Figure 3. Input Shock And Response Of A Spring/ Mass System .............. 11
Figure 4. Typical SRS Plot For A Spike ............................................................ 13
Figure 5. Ideal Square Wave Input ................................................................... 14
Figure 6. SRS Plot For An Ideal Square Wave Input .................................... 17
Figure 7. Relationship Between Damage Boundary And Results From
SRS ............................................................................................................................ 20
Figure 8. Ideal Model Of A Product With A Critical Element .................... 22
Figure 9. Damped And Undarnped Systems ................................................... 27
Figure 10. Determining Damping Ratio Using Successive Peak Heights
After Setting Beam Into Motion ......................................................................... 29
Figure 11. Linear Damped Spring / Mass System With Accelerometer
Attached ................................................................................................................... 33
Figure 12. Pulse Reflecting Deflection Failure ................................................ 36
Figure 13. Test Unit Used For DBC ................................................................... 37
Figure 14. Damping Ratio Comparison Between Equation 24

And Equation 25 ..................................................................................................... 42
Figure 15. SRS Plot From 30" Gas Programmer Drop ................................... 49
Figure 16. SRS Plot From 23" Free-Fall Drop Test Onto Ethafoam 220 ..... 50

ix

Figure 17. SRS Plot From 17" Plastic Programmer Drop .............................. 51

Figure 18. SRS Plot From 33" Free—Fall Dr0p Test Onto Polyurethane

Foam ......................................................................................................................... 52
Figure 19. Damping Ratio Comparison Between Test Partner

and Oscilloscope, Using Equation 25 ................................................................ 70
Figure 20. Oscillation of Variable Beam, 9.5 Hz ............................................. 71
Figure 21. Oscillation of Fixed Beam, 14.3 Hz ................................................. 72
Figure 22. Oscillation of Variable Beam, 20.2 Hz ........................................... 73
Figure 23. Oscillation of Variable Beam, 50 Hz .............................................. 74
Figure 24. Oscillation of Plastic Beam, 16 Hz .................................................. 75

Figure 25. Comparison Between Actual and Predicted Response,
0 Damping; Plastic Programmers ....................................................................... 76

Figure 26. Comparison Between Actual and Predicted Response,
0 Damping; Gas Programmers ............................................................................ 77

Figure 27. Actual and Predicted Response To a Plastic Programmers
Drop, Variable Beam, 9.5 Hz ................................................................................ 79

Figure 28. Actual and Predicted Response To a Gas Programmers
Drop, Variable Beam, 9.5 Hz ................................................................................ 80

Figure 29. Actual and Predicted Response To a Plastic Programmers
Drop, Fixed Beam, 14.3 Hz ................................................................................... 81

Figure 30. Actual and Predicted Response To a Gas Programmers
Drop, Fixed Beam, 14.3 Hz ................................................................................... 82

Figure 31. Actual and Predicted Response To a Plastic Programmers
Drop, Variable Beam, 20.2 Hz .............................................................................. 83

Figure 32. Actual and Predicted Response To a Gas Programmers
Drop, Variable Beam, 20.2 Hz .............................................................................. 84

Figure 33. Actual and Predicted Response To a Plastic Programmers
Drop, Variable Beam, 50 Hz ................................................................................. 85

A

Figure 34. Actual and Predicted Response To a Gas Programmers

DrOp, Variable Beam, 50 Hz ................................................................................. 86
Figure 35. 30" Gas Programmers Drop, Damage Boundary Test ............... 88
Figure 36. 23" Free Fall Drop Test Onto Ethafoam 220 ................................. 89
Figure 37. 17" Plastic Programmers Drop, Damage Boundary Test ........... 90
Figure 38. 33" Free Fall Drop Test Onto Polyurethane Foam ...................... 91

xi

CHAPTER 1
INTRODUCTION AND LITERATURE REVIEW

1.1 INTRODUCTION

The purpose of this study was to apply the concept of the Shock Response
Spectrum to fragility assessment. Specifically, the study addressed the
appropriate procedure for the testing of products using SRS, and using SRS as an
alternative to the Damage Boundary Curve for determining product fragility.
Using SRS for deflection-type failure criteria as opposed to G-level failure was
also investigated, as well as the accuracy of a commercial SRS software package.

The term shock response refers to the reaction of a single degree of
freedom spring / mass system with a particular natural frequency to a single
shock input. (See Figure 1). The reaction of the system is measured by the peak
deceleration, G. A Shock Response Spectrum is a plot of the peak responses of
different single degree of freedom systems to an input transient. Unless
otherwise specified, the systems are considered to be undamped [1]. Thus, SRS is
primarily concerned with the response of a system, not the characteristics of the
input shock.

The fragility question to be answered is: what is it about the nature of the
input shock to the product that causes it to break? To quantify this, the current
accepted method is to collect experimental data and construct a damage
boundary curve. This is done by subjecting the product to both short and long
duration shocks, and recording the velocity change and acceleration levels that
cause damage. This method can be costly and time consuming since at least two
products must be damaged. Using SRS, the same information can be obtained

without the need for a shock machine and damaging several units.

Accelerometer captures peak G ram:

Mass

 

M

Spring/ mass
system with a ' .
natural frequency SPEHS

I J

 

 

 

 

Input shock

Figure 1. Response Of A Spring/Mass System With A Particular Natural
Frequency To A Single Shock Input.

3
Traditionally for packaged products, an accelerometer is mounted to the

product, usually somewhere on the base structure [2], to capture the product's
response to a shock input. However, the component of a product that is the most
fragile during shipping and handling is generally not what the accelerometer is
mounted to. It is something else, called the "critical element". It is this
component of the product that must be protected against input shocks to keep it
from being damaged. The critical element must satisfy the condition of being a
single degree of freedom spring/ mass system in order that SRS be applicable, as
will be discussed later. The traditional method begins by taking the product and
its cushion and dropping it on a free-fall drop tester, recording the shock pulse.
The acceleration level is then checked against the G level from a Damage
Boundary Curve to predict if there is damage. The DBC test on the product
really gives a picture of the fragility of the component that fails - the critical
element, not the whole product. The DBC does not predict or even monitor the
actual response of the element, it only describes the velocity change and G level
of the input shock that caused the critical element to fail. Furthermore, DBC is
specific as to what velocity change and G levels break the critical element: they
are measured from a spike (produced by plastic programmers) and a square
wave (produced by gas programmers). In real world testing of a product and its
cushion, neither of these waveforms is reproduced. The square wave is generally
considered the most damaging of waveforms [2] so the DBC can become very
conservative in its fragility assessment.

The two biggest misapplications resulting from the above traditional
approach are mounting the accelerometer to the product and recording the shock
pulse, and trying to mount the accelerometer to the critical element itself.
Mounting the accelerometer to the base of the product for example, capturing the
input pulse, and then using the G level and DBC to predict damage does not take

4
into account the actual response of the critical element. Fragility of the critical

element is defined by a square wave in the DBC procedure, but actual drops are
not normally square in shape, so serious overestimations of its fragility are
possible. Also, without knowing the response of the critical element, using the
DBC is a very indirect way of measuring fragility, since it is not the product's
shell (or whatever else the accelerometer is mounted to) that is damaged, but a
component inside it. The actual response of the critical element is completely
ignored.

Mounting the accelerometer to the critical element to measure its response
is also ineffective for several reasons. First, the critical element may be too small
for an accelerometer, like the filament of a light bulb. Another problem may be
the location of the critical element; for instance, how does one attach an
accelerometer to the magnetic head of a hard drive? Finally, by attaching the
accelerometer to the critical element, the mass will necessarily change, causing a
change in the behavior of the critical element. Even if the response can be
accurately measured, comparing it to the DBC is not a correct application, since
the DBC describes the shock input, not the critical element's response.

A problem with capturing response pulses in general is the presence of
other components in the product that were not meant to be monitored. And
since there are many other components present in a product besides the critical
element, each vibrating at different natural frequencies which the accelerometer
can detect, filtering the pulse to eliminate "ringing" is necessary. Without
filtering, ringing (undesirable motion superimposed on the original waveform)
may cause the data to appear to have false peak accelerations, inaccurate velocity
changes and false coefficients of restitution. Using SRS would eliminate the need
for filtering, since filtering has virtually no effect on SRS plots, except when the

natural frequency of the element matches the ringing [3,4]. Using SRS would

5
also eliminate the ambiguity in the DBC, since capturing the input shock and

knowing the natural frequency of the critical element is all that is necessary to
predict whether there will be failure or not. There is no need to mount an
accelerometer to the critical element, since its response can be accurately
calculated by SRS. In other words, SRS eliminates the need to capture the
element's response, since the response can be calculated from the input shock.
This significantly reduces the worries of filtering. SRS also gives a much truer
picture of fragility of the critical element, since it is not dependent on or derived
from a spike or square wave.

Since SRS is concerned with a critical element's response to a shock, not
the input shock itself, any shape input shock pulse can be used so long as it
damages the critical element. This is not true with damage boundary curves: the
input shock used to damage the product must be a square wave, or a half sine
wave. With SRS, a free fall drop tester could be used, moving the height up in
some reasonable increments until damage has occurred, and analyzing that input
pulse using SRS to predict the peak G of the critical element. From the peak G
predicted by SRS, a damage boundary curve could also be constructed. This
would eliminate the need for an expensive shock table, and reduce the number of
units required to construct a damage boundary curve, saving money and time.

There are times when failure will occur due to the deflection of a
component, not the shock it receives. A component Within a product may fail,
for example, because it deflects to the point Where it bumps into something else.
SRS could be used to predict the maximum allowable deflection, and the G level
that causes the critical deflection.

Finally, since damage boundary is a very conservative approach to

assessing fragility, SRS could reduce the amount of cushioning for packages [5].

1.2 LITERATURE REVIEW

The Shock Response Spectrum (SRS) has been known and used in the
scientific community for many decades [6]. SRS was first used as a tool for
determining the resistance of buildings to earthquakes [1,6]. Since then, methods
to calculate SRS have been deve10ped and applications sought in several fields
[7,8,9,10,11]. Uses have been identified for the standardization of shock
machines, and making direct response-based correlations between product
fragility and a package drop test [3,12]. Several authors have outlined the basics
behind SRS analysis [7,11,12,13], while pointing out its virtues as a design tool
and as a shock test control parameter [3,6]. Little application has been made
directly to the field of packaging, however, particularly its use as a testing
alternative to the Damage Boundary Curve (DBC). A study by Robert E. Newton
has recognized and outlined the relationship between the DBC and Shock
Response Spectrum, namely how the DBC is derived from the basic premises of
the Shock Response Spectrum [14].

CHAPTER 2
THEORETICAL DEVELOPMENT

2.1 SRS THEORY

For any particular input pulse, the theoretical response of an undamped,
single degree of freedom spring / mass system with a particular natural frequency
can be calculated using Newton's laws of motion. The peak acceleration
response of a spring / mass system is a function of its natural frequency. A plot of
peak deceleration versus natural frequency is called the Shock Response
Spectrum. For an ideal undamped spring/ mass system, the SRS plot is

determined by applying Newton's laws of motion to the situation in Figure 2.

From Newton's law, Force = Mass x Acceleration
d2
k(x-y)=m-d?g- Eq.1
where: k = spring constant for a linear spring
' m = mass

y(t) = response position at any instant measured from
the static equilibrium position
x(t) =prescribed base position at any instant measured
t = time
(x - y) = relative displacement
This is a second order ordinary linear differential equation in the unknown
function of time y(t). For no motion of the base (x = O), the system executes free
vibration and the solution is [16]:
y = C1 sin wt+ C2 coswt Eq. 2.
where C1 and C2 are constants determined by the starting positions and

velocities, and a) is related to the natural frequency of vibration through [9]

 

y(t)

 

 

 

 

 

/////7

x(t)

 

........................................
.......................................
........................................
.......................................
........................................
---------------------------------------
........................................
.......................................
uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu
.......................................
........................................
.......................................

 

 

 

 

Driven Base

Figure 2. Newton's Law Of Motion Can Be Used To Determine The
Response Of A Mass To A Particular Excitation.

w=2mz
_ 1 kg

fn—Zn' W

W=mg

g = acceleration due to gravity (386.4 in / secz)
For a prescribed motion of the base, the system executes forced vibration and the
solution is [16]:

t

y= Clsinwt+C2coswt+wfx(s)sinw(t-s)ds Eq. 3.
0

Applying the starting conditions that y and ‘31—}: are both zero at t = 0 (mass is at

rest initially) gives C1 = C2 = O.
This solution is not very useful in its present form because it relates
positions to each other: what is needed for SRS is to relate accelerations to each

other. This is easily accomplished by first differentiating this equation twice to
2

obtain fig- and then using integration by parts [16] along with the initial

conditions x = 4%: O at t = 0. The result is:

2 t 2 t 2
d
‘63—; = a) sir(a)t) fit—t: cos(wt)dt - cos(wt) ~fd—gsin(a)t)dt Eq. 4.
O 0
2
where: édT-g— = the response acceleration at any instant
2
Tin = the input acceleration at any instant (known)

Equation 4 is the governing equation for finding the peak acceleration

response of an ideal undamped spring/ mass system to any input shock pulse.

10
There are two important parts to the response acceleration which can be

obtained using this equation. These two parts are for t < T and for t > T, where T
is the duration of the input shock. The response during the input shock (t < T) is
known as the primary response, and the response after the input shock (t > T) is
called the residual response. The peak response acceleration during the primary
and residual responses will be evaluated and compared: the larger of the two is
used for the SRS.

There are two important types of input shock pulses for which the
response of a single degree of freedom spring/ mass system can be evaluated
exactly. The first is a short duration pulse, (or spike), such as that obtained using
the plastic programmers on a shock table. See Figure 3. If the input shock is very

short in duration compared to the natural period of vibration (i) of the

I!

2

x
spring/ mass system, then :1?- in the integral is non-zero only for a very short

time period around t = O. This means that cos wt and sin cot are essentially 1 and

0 respectively. Therefore:

dzy I . ‘de ’de l
Et—z=w[smwt{E2—°1dt-coswt-{F OdtJ Eq.5.
=w - sin wt°AV Eq. 6.

tdzx . .
The term fog? ~1dt, Wthh represents the area under the acceleration versus

time curve, is just the velocity change, AV, for the input shock. The peak

response occurs when sin (2cot) is equal to 1. Using (0 = 2rrfn and reporting the
2 .

peak response in G's = Liz};- / g,

peak response G = 23f" AV . Eq. 7.

8

 

11

4

acceleration

input shock, spike

response of spring/ mass system

//

 

 

/
,_ T —[ U time

I
in 1

 

_._1_._

Figure 3. Input Shock And Response Of A Spring/ Mass System

12
This says the response of any spring/ mass system to a spike does not depend on

the peak acceleration or the duration of the spike, but only on the velocity change
of the spike. This also says that the response G is proportional to the natural
frequency of the spring / mass system for a spike. Since an SRS plot is a plot of
peak response G versus natural frequency of a spring/ mass system, the SRS plot
for a spike begins as a straight line with a slope of (2 J'EAV)/ g, as shown in Figure
4. In fact, all SRS plots must start out as straight lines. This is true because as the

SRS plot starts out, the natural frequencies are low and so the period -1- will

always be large compared to the duration of the input shock. When the natural
period of vibration is not sufficiently larger than the duration of the input shock,
the linear relationship between G and natural frequency does not hold true. The
response must then be determined by evaluating Equation 4. The linearity of
SRS plots as they start out is one feature which can be used to check the accuracy
of commercial software.

The second important type of input shock pulse for which the response
can be evaluated exactly is a square wave pulse, such as that obtained using the

2
gas programmers on a shock table. In this case, 2%: A = constant for the

duration T of the pulse as shown in Figure 5. Two separate times of the input
shock must be evaluated, the input shock during duration T, and the input shock
after T. The response to the input shock during duration T is known as the
primary response, and the response to the input shock after T is known as the
residual response.

2
x . .
Let A 2 E7 = input acceleratron.

The primary response then is:

sin(wt) - ésin wt — cos wt ° 5(1 — coscot)1 Eq. 8.

—=w
a) a) J

dzy I
dt2 [

13

 

 

 

 

 

Evaluation of SRS Software " ,///’/l 10:30:11 PM
C:\IP2\DRTA\MPD\B i 1.PL2 04-29-1993
Channel I Channel 2

 

 

SRS Tspe:Conposite

Resolution: 20 PPO
Damping: 0.00
Peak too: 195.91 G’s
Peak Fn: 278.58 Hz
"go too: -193.36 G’s
Min In: 251.0? Hz

 

   
  

 

2 . 5 Frequency (Hz) 640. 0

 

Notes: Tag line is naxinun response
Bo ton line 15 max Hlnlflufl response

 

6
. TEST PA, RTNER

\ I
§ fl Michigan State University

 

   

 

Figure 4. Typical SRS Plot For a Spike.

14

dzx/df

 

Figure 5. Ideal Square Wave Input.

 

 

15

2
or: %Z— = A[1— cos wt] Eq. 9.

Dividing by g to obtain G's:

Gresponse = Gmput[l — cos annt] for t < T Eq. 10.
It is important to note that this is only the primary response. This is valid only
when the time t chosen is less than the duration T of the square wave. The
residual response occurs after the duration T of the input shock. Since the input

acceleration is discontinuous, two separate integrals must be evaluated. From

Equation 4,
t T t
f0 =1; +1; Eq. 11.
T A
Part 1. f Acos(wt)dt = —sin wT Eq. 12.
0 a)
T A
f A sin(a)t)dt = —(1 — cos (01") Eq. 13.
a)
0
t dzx
Part 2. f = 0 because E7 = 0 for t> T Eq. 14.
T
The response then becomes:
2
L32,— = a) rsin(a)t) ~ésin(wT) — cos wt . £0 - cos can]. Eq. 15.
dt [ co a) l
or:
Gresponse = Ginput[ COS (0(i - T) - COS cut] when t > T. Eq.16.

This is the residual response.

16

 

Therefore, if T < 21 , the peak primary response according to Equation 10

fl

 

 

 

will be:
peak response G = G1,,[1 — cos(279‘nT)] Eq. 17.
And if T 2 21 , the peak response G will be:
peak response G = ZGinput- Eq. 18.
. . 1 1
The peak resrdual response for either T < 2f or T 2 2 f can be

determined using Equation 16 (see Appendix A). By differentiating with respect

to t to find the maximum, the result is:

 

peak residual response G = G... J2(1 - cos (01") Eq. 19.

1

 

In summary, for square waves, if T .2 , then:

I!

1) The peak G will be 2Ginput-

2) The peak G will occur during the input shock.
3) The peak residual response will be G... J2(1 — cos 2.1721") < 2G,".

 

 

AndifT< 21 ,then:

H

 

1) Peak (3 will be G,,,J2(1— cos 21rfnT) < 2G,,,.

2) Peak G will occur after the input shock.
3) The peak (3 during the input shock will be c,,,(1 — cosZJrfnT).

Therefore, all SRS plots for ideal square waves will be as shown in Figure 6.

17

peak acceleration
response

 

2Gin

Gin

natural frequency

Figure 6. SRS Plot For An Ideal Square Wave Input.

18
To relate displacement failure to SRS, simply rearrange Equation 1, and divide by

acceleration, g, to obtain:

G 42315,)? Eq. 20.

where: G = maximum acceleration from an SRS plot at fn

d = distance between the mass and base structure.
If the maximum allowable displacement is known, Equation 20 gives the
acceleration response for when the critical deflection is reached. Therefore,
performing SRS on an input shock will tell the user how far the component will

deflect in response to that shock.

2.2 RELATIONSI-ITP BETWEEN DAMAGE BOUNDARY AND SRS

The conventional approach to fragility testing is a method called the
Damage Boundary Curve (DBC). The DBC is essentially a two dimensional
index of fragility that takes into account both the amplitude and duration of the
shock. The DBC presumes that only two aspects of an input shock pulse to a
product are related to damage: the peak G and the velocity change (area under
the acceleration versus time waveform). The DBC relates peak G on the vertical
axis to the total velocity change (AV) on the horizontal axis to product damage
[15]. The development of a damage boundary curve for a product is presented in
full in ASTM D 3332-77 (1983) "Standard Test Methods for Mechanical Shock
Fragility of Products Using Shock Machines" [2]. The following is a brief
summary of some of the important steps in conducting a damage boundary test.

The first step is to determine the critical velocity change (AVcr) for a
product. This is done using the plastic programmers on a shock machine, which
produce short duration (about two milliseconds) half-sine shocks. Affixing the
product to the table, the table is raised and dropped from a series of heights in

19
slightly increasing increments until the product is damaged. The last drop that

causes damage is used to determine the product's critical velocity change.

The second step is to determine the critical acceleration for the product.
This is accomplished using the gas programmers on the shock table. Gas
programmers use gas compressed in a piston as the contact surface to generate a
trapezoidal pulse. High gas pressures produce short trapezoidal pulses with
high acceleration levels, and low gas pressures produce long duration, low
acceleration pulses. It is important to note the desired pulse is a perfectly square
waveform, but an instantaneous rise time is impossible to produce in a
commercial setting. The trapezoid pulse has a very fast (not instantaneous) rise
time, and is a close approximation. The table drop height used in this part of the
test must be chosen to produce a table velocity change of at least: [(n/ 2) x
(critical velocity change obtained using the plastic programmers)]. To begin this
phase of the testing, a second product is affixed to the table and dropped onto
the gas programmers. The gas pressure in the cylinder is then increased while
the drop height is held constant: this increases the acceleration level while
keeping velocity change the same. This pressure is gradually increased until
damage occurs. The last drop that causes damage is the product's critical
acceleration.

The fragility picture for this particular product is now constructed using
the product's critical velocity change and critical acceleration as shown in Figure
7. The resulting plot is able to show the relationship between the critical velocity
change and critical acceleration of an input pulse, and damage to the product for
which that DBC was constructed. Any input shock pulse with AV > AVCI- and G
> Ger will result in damage. A pulse with AV > AVcr but G < Ger will in theory
not cause damage. The same can be said of a pulse that has AV < AVG- and G >

Ger.

20

 

 

 

Acceleration
(G'S)
AVcr
y Damage
A
N o Damage

 

 

Velocity Change (in / s)

Figure 7. Relationship Between Damage Boundary And Results From SRS.

21
An important point to the above process is the fact that the DBC generated

is valid only for one orientation of the product. A product the shape of a cube
would require six DBC's to represent the overall fragility of the product. This can
obviously become a time consuming and expensive undertaking.

The reason that the DBC in Figure 7 has the shape it does is that it is based
on an idealized model of the product. See Figure 8. Note the component called
the "critical element." The critical element can be any component which must not
be damaged, and that meets the following criteria:

1) The critical element must behave like a spring / mass system.

2) The critical element in question must be the thing that breaks during
both the plastic and gas programmer shocks.

Using the results from the spike and square wave shocks from SRS, a
correlation can now be made to the damage boundary curve. Recall that the
damage boundary curve attempts to answer what it is about the nature of the
input shock to the product that causes damage to the critical element. It answers
the question by identifying a critical velocity change and a critical acceleration,
both of which must be satisfied for failure to occur. To relate the experimental
results, AVcr and Ger, to the nature of the critical element, two results from SRS
are applicable. The first is the input shock from the plastic programmers which
is essentially a spike, for which the peak response of the critical element is given
by Equation 7. Since by assumption the critical element breaks when G = Gee
where Gee is the fragility of the critical element and f n is its natural frequency,

this predicts that

AV =§fl£ Eq. 21.

22

/\/ Product

Mass
M

 

 

Critical Element

 

 

 

Spring

 

 

 

////////A/////////

 

Input Shock

Figure 8. Ideal Model Of A Product With A Critical Element.

 

23
where AVcr is the critical velocity change required for damage. In other words

the product's critical velocity change obtained in the plastic programmer part of
the test is related to two properties of the critical element alone: Gee and f n. In
the second part of the test, the gas programmer essentially produces a long
duration square wave shock. For the square wave, the response is G = 2Ginput,

from Equation 18. Again the critical element fails when the response G = Gee, so
Gce = 2Ginput,

01‘: Ger = — Gee. Eq. 22.

This says that the product's critical acceleration is half the fragility of the critical
element

Therefore, based on SRS, the results from damage boundary test relate
directly to the natural frequency and Gee of the critical element Conversely, SRS
can provide the needed information to construct a DBC without the need for
going through the shock machine test procedure, provided that the natural
frequency for the critical element is known beforehand. Equations 21 and 22 give
the desired information. One way to obtain the required information about the
critical element is to perform the gas programmer part of the test exactly as
outlined for the DBC and obtain Gcr- Then Gce can be obtained from this
through Equation 22. The natural frequency fn can then be determined through
a non-destructive test by vibrating a new product on a vibration table and
recording the table frequency at which the critical element resonates: this is fn.
Now the products critical velocity change can be obtained through Equation 21.
The utility of this approach is that only one product need be destroyed compared

to the conventional DBC test procedure.

24
The SRS relations in Equations 21 and 22 can also be used to check the

validity of the entire model behind the DBC. Eliminating Gee between Equations

21 and 22 gives a prediction for the natural frequency of the critical element
based on the results of the DBC test:

G g
=-—-—-“ . E . 23.
f" 2:1:AVCr q

A quick check of the natural frequency on a vibration table can confirm this
relationship.

There is an inherent problem with the damage boundary curve results
since damage occurs before the last drop but after the next-to-last drop because
of the way the drop tests are carried out. For example, damage may have
occurred at a ten inch drop on the plastic programmers, but not at nine inches.
So the damage just occurs somewhere between nine and ten inches. In a 10" drop
on the plastic programmers, the velocity change may be 150 in/ sec, and at 9" it
may be 140 in / sec. Likewise, on gas programmers, damage may occur at 200 psi
but not at 175 psi, corresponding to 25 and 20 G's respectively. This then means
that the natural frequency of the critical element is somewhere between
20 x 386.4 25 x 386.4

= 16Hz and —— = 22Hz. The assumed model of the product is
It x 150 Jr x 140

therefore considered to be correct if the resonance test on the vibration table
produces a resonant (natural) frequency between 16.and 22 Hz.

To avoid ambiguity, the last drop is often used to determine the critical
acceleration and critical velocity change, and it is obvious now that the damage
boundary curve becomes a very conservative tool for identifying fragility. There
is always the potential for significantly overshooting the true fragility of an
element while at the same time not being sure of its exact fragility. So how does

one know if the product can be accurately modeled as in Figure 8 and at the same

25
time obtain accurate fragility levels? This can be answered either by making the

check described in Equation 23 with more accurate (more refined) values for Ger
and AV cr, or by applying SRS to the two waveforms that just cause damage in
the DBC test. The steps are:

1) Drop the product on the plastic programmers from a height that just
causes damage, and then do an SRS plot on that pulse.

2) Drop a new product on gas programmers as described earlier, and do
an SRS plot on that pulse.

3) Overlay the plots from 1 and 2. The intersection point identifies the
natural frequency and critical G level for the critical element

4) Get the natural frequency of the critical element on a vibration table. If
the frequencies in Steps 3 and 4 do not match, then the system does not behave
like an ideal spring/ mass system. If the product is modeled accurately, then any
two input shocks that just cause damage will intersect at the natural frequency
and Ger of the critical element. The implications of this are far reaching, since it
means that determining the fragility of a product containing a critical element
could be done simply by using a free fall drop tester. The shape of the input
shock pulse is not the determining factor: the response to the input is all that
matters. The SRS approach requires only the natural frequency of the critical

element and any input waveform that just damages it to determine the complete

fragility picture.

2.3 SRS AND DAMPING RATIO

One crucial assumption for the aforementioned model is that it does in
fact behave as a linear nondamped spring/ mass system. However, there are no
real ideal undamped systems, so all models should incorporate some degree of

damping. Most commercial software packages for SRS allow for damping.

26
Damping simply means that after a system is set into oscillating motion, the

movement will gradually die out.

The degree of damping is usually specified by a "damping ratio", which
has a value between 0 and 1. A damping ratio of 0 means the system vibrates
indefinitely. A damping ratio of 1 corresponds to a system with no cycle of
motion (the mass returns .to equilibrium position immediately after being
disturbed). A damping ratio in-between corresponds to a system which
gradually dies out as shown in Figure 9. It is possible to incorporate damping
into the model and evaluate the shock response. This is done in commercial
software packages for SRS. The approach is the same as the ideal model, except
that a dashpot is included in the damped model. It is usually the case that the
peak response of the mass is smaller with damping than without. Using an SRS
plot with no damping to assess the effect of an input shock on a spring / mass
system therefore overestimates the effect. With damping included in the
analysis, there are now two properties of the system which determine the
response to a shock: its natural frequency and its damping ratio. For a real
system, both can usually be determined by mounting an accelerometer on the
mass, disturbing it (by tapping it or dropping it), and analyzing the response.
The natural frequency is easily determined by simply counting the number of
cycles per second. The damping ratio can be calculated by measuring the decay
in the peak heights of the recorded waveform.

The first method used in this study to estimate the damping ratio of a
beam was to count the number of cycles, N, the beam oscillated through before

returning to rest. The damping ratio then is:

 

Damping Rafio = Eq. 24.

1+N2

27

 

Mass

 

 

 

spring

f///////

 

The ideal system vibrates indefitenely

 

Mass

 

 

 

spring g E? dashpot
f///////

 

A real system dies out gradually

Figure 9. Damped And Undamped Systems.

28
where: N = the approximate number of peaks before the system dies out.

The second method uses the successive peak heights of the decaying waveform
to determine damping ratio. In Figure 10 the peak heights are labeled H1, H2,
etc. This method is described by the equation:

 

1n Er
Damping Ratio = H2 Eq. 25.
H 2
Jan —-1-) + 41:2
H 2
Where: H1 = acceleration in GS of the first peak
H2 = acceleration in G's of the second peak
Since the damping ratio is a property of the system, i = _Iiz_ = £3- = etc.
H 2 H 3 H 4

Experimentally determined damping ratios will be used later to check the ability
of a commercial software to do SRS plots on systems with damping. The

derivations of the above equations can be found in Appendix C.

29

.5322 85 88m macaw as?
me83 v—aon— 333850. “3.63 03mm wCEEmD wEEEuBoQ .3 0.5me

 

VI

 

 

 

 

 

 

 

 

 

m..-—

 

__-_

 

\\\\ \\\

CHAPTER 3
MATERIALS AND TEST IVIETHODS

3.1 SOFTWARE

Computer software and hardware from the Lansmont Corporation was
used to evaluate the responses to input shocks during testing. Specifically, Test
Partner 2 (TT’2) was used. Test Partner captures analog acceleration versus time
signals from shocks. The data is digitized and processed in a number of different
ways. The system features include the following:

°Four channel data capture

OHigh-resolution color displays

0 lMHz 12 bit analog-to-digital conversion

Support for dot, color and laser printers

OSupport for keyboard, mouse and touchscreen input devices

OAutomatic non-destructive digital filtering

OAutomatic or manual analysis

OAnalysis modes tailored to specific waveforms

0Peak and minimum acceleration, durations

OVelocity change calculation

OTrace shift and zoom

OTriaxial resultant vector magnitude

ODeflection calculation and display

°Extended transient computation

OMil-spec tolerance bands

OShock response spectrum (SRS) analysis

OShock response time domain (SR) analysis

30

31
3.2 EQUIPMENT

Accelerometers: 1. Dytran, Model 301UA5, S / N 612
2. Kistler, Model 818, S/ N 2834

Charge Amps: Kistler Piezotron Coupler, #5112

Cables: Coaxial Belden 8263 Shielded.

Shock Table: Lansmont Model 65/ 81, Serial Number 57-681-0016

Vibration Table: Lansmont Touchtest Vibration System
Model 10000-10, Table size 152 cm, Hydraulic Power Supply

Oscilloscope: Kikusui Electronics Corporation
CO SSOZO-ST Storage Oscilloscope 20 MHz

Drop Tester: Lansmont Model PDT-56E Free Fall Drop Tester

3.3 TEST METHOD FOR SRS EVALUATION

The objective of this test procedure was to evaluate the ability of SRS to
accurately predict the response of a product to an input shock. By necessity, this
procedure also tests the validity of the model of a product as a linear damped
spring / mass system. This was done by evaluating the accuracy of the SRS
analysis of two different input pulses, using the known damping ratio and
natural frequency of the system. The models used were two different metal
beams, each with a mass attached to one end. One beam (referred to as the fixed

beam) was tested at only one length, corresponding to a natural frequency of 14.3

 

 

32
Hz. The second beam (referred to as the variable beam) was affixed in varying

lengths so as to vary the natural frequency of the system. See Figure 11 (variable
beam setup in the 20.2 Hz position). The beams were held in place with
clamping plates fastened to a wood mounting.

An accelerometer was mounted on the mass in order to measure the
response. For purposes of modeling, the accelerometer was assumed to be a part
of the mass, and natural frequencies were determined with the accelerometer
attached. The steps involved in the evaluation process were as follows:

Step 1. Determine the damping ratio of the beam. Two methods were
used. The beam was set into motion by hand and the decaying output
acceleration signal was captured on a storage oscilloscope. The storage scope
was equipped with an output channel, so the signal was also fed into TP2. The
oscillation was captured by both the oscilloscope and Test Partner
simultaneously. Equation 24 determined damping ratio by counting the
approximate number of cycles of oscillation before the beam returned to rest
from the oscilloscope. Equation 25 was used by taking the peak G's from
successive peak heights using TP2. The results using Equation 24 and the
oscilloscope are found in Appendix D.

Step 2. Determine natural frequency of the beam. The method used to
determine the natural frequency of each spring/ mass system was to place the
system on a vibration table and perform a sine sweep test as outlined in ASTM
D999 [17]. A .5 G input and 3 to 100 Hz sweep was used, to look for resonance.
The resonant frequency was then taken to be the natural frequency. At this
point, we now know the two properties of the spring/ mass system needed to do
an SRS plot (fn and damping ratio).

Step 3. The third step was to determine if the predicted SRS from a

particular input shock matched the actual response of the beam. The unit was

 

n '1
JD

 

Figure 11. Linear Damped Spring / Mass System With Accelerometer
Attached.

34
mounted on the shock machine. An accelerometer attached on the shock table (to

measure the input acceleration) was routed to channel one on TP2. Another
accelerometer was mounted to the mass on the beam and routed to channel two
on TP2. This recorded the output acceleration. A low level drop onto plastic
programmers was then performed, using a height of two inches. Both the short
duration half-sine input pulse and the output response of the beam were
recorded in the same time domain. Refer to Figure 3. When evaluating each
input shock pulse, a filtering frequency of zero was chosen. This unfiltered spike
was then used by TPZ to calculate the SRS. Acceleration levels were affected very
little by choosing not to filter the input pulse, and no filtering gave the worst case
scenario and provided consistency in evaluation. The peak acceleration at each
of the beams' natural frequency was then compared to the actual response of the
beam. This comparison was done using no damping (damping ratio = O), and
using the calculated damping ratios from experimentation. Using the shock
response (SR) time domain function of TP2, the natural frequency of the beam
was entered and the theoretical shock response to the beam was calculated and
compared to the actual response of the beam.

Step 4. The procedure in Step 3 was duplicated using gas programmers.
A table height of eight inches and a pressure of 150 psi was used for each drop.
The filtering frequency used for the input shocks was the default frequency,
which was usually about 370 Hz. Too much filtering destroyed the integrity of
the trapezoid pulse, and there were negligible differences of G levels in the SRS
when filtered at 200 Hz, so the shock response was calculated using the pulse
filtered at the default frequency. Theoretically, the damping ratio is a property of
the spring/ mass system, so acceleration levels were compared using the same

damping ratios as in Step 3 for each beam.

 

35
3.4 TEST METHOD FOR DEFLECTION FAILURE

To test the prediction for deflection failure, (Equation 20), a new beam
with a natural frequency of 17 Hz was mounted to the setup in Figure 11. The
height from the bottom of the mass to the surface below was measured. The
height of the beam above the surface was adjusted to give two different heights
from which to test. The first height was 15 / 16" (.9375"), and the second was 1
3 / 8" (1.375"). The drops were done using the plastic programmers, and the table
height was raised incrementally until the bottom of the mass made contact with
the surface below. Contact was evident from looking at the indentation on the
wooden surface, and the small spike present in the captured waveform. See
Figure 12. TP2 was then used to generate an SRS on the pulse, and the peak G
was used with Equation 19 to predict the deflection. This predicted deflection
was then compared to the known measured distance. Since damping ratios for
the beam were so low (< .1), they were not considered in the calculation, as it
would have little effect. See Appendix E for using damping ratio with deflection
failure.

3.5 TEST METHOD FOR FRAGILTIY TESTING

The objective of this part was to test the widely used assumption that an
element of a product fails when its peak acceleration reaches a certain level. In
other words, a critical element inside a product fails when the product gets any
kind of shock which produces a prescribed G-level to the element. A second
objective is to show that a DBC can be constructed from damage information
obtained from drops done on equipment other than the shock machine. The
wooden box in Figure 13 was used as the model. For this thesis, the product
used was a wooden box with no lid containing a plastic rectilinear piece attached

to the box on one end, and weighted with a metal mass on the other end.

 

 

36

 

—i

 

e ection Failure 2:04:36 PM
BEIi 09-03-1993
Channel 1 Channel 2
Paired too: 6’
Peak noc: 220.62 72.30 G’s
Min goo: -13.22 -22.41 G’s
Duration: 1.80 14.80 nseo
[ta 0: 92.68 149.34 ln/S
Filter: 500 676 H

 

 

20.0 nsec/Dio

 

 

Notes: Deflection failure testing

4 inch drop onto plastic programmers
Plastic Bean urth natural frequeno of 17 Hz
Maxrnun allowable deflection: 15/1 inch
Shock response predicts nax G of 27.51

  

 

   

O

\ ’
§ { Michigan State University

TEST PAN RTN ER

      

Figure 12. Plastic Programmers Drop Showing Deflection Failure.

 

 

 

 

 

 

 

Figure 13. Test Unit Used For DBC.

 

38
Damage was defined as the breaking of the plastic beam. Since the beam is the

object of interest, we call it the critical element, designating it the most important
piece to observe during testing.

Step 1. Find the natural frequency of the critical element. This was done
using a vibration table, as in Step 2 of the previous test. The plot of the sine
sweep indicated a natural frequency of about 16 Hz.

Step 2. A standard damage boundary test was performed, as outlined in
ASTM D3332-77(1983). One accelerometer was mounted on the shock table, and
routed to TP2 to capture the pulse of the waveform. The pulses captured were
the drop just before failure and the drop that caused failure. Using the natural
frequency of the critical element from Step 1 in an SRS plot with no damping,
(since damping was quite small), the critical peak acceleration of the plastic beam
was predicted. A range of natural frequencies within which the beam should fall
into can be found using the above information since damage occurs somewhere
between the drop height increments. This is also a good way to test the accuracy
of the model as a linear spring/ mass system. The critical G is between the two
drops, since failure occurs at one and not the other, and it is not known if the last
drop height that caused failure is the borderline failure.

Step 3. Continue the damage boundary using gas programmers. Again,
do SRS plots on the pulses just before failure and the drop causing failure. This
SRS plot was compared to the one generated in Step 2. The two plots should
intersect at the same G level corresponding to the same natural frequency. If
they are somewhat close, the probable reason is that the spring/ mass system is
slightly damped. If they are not close, the system is probably not modeled
accurately.

Step 4. Perform a free fall drop test onto two different cushions. This step

was done to change the shape of the input shock. The two previous steps

 

 

 

‘__——————‘

39
generated a square wave input and a half-sine spike. This step produced input

shocks of considerable shape difference. A corrugated box was constructed to
house the test unit, with space left for the insertion of a cushion. The first
cushion used was a high density polyethylene 220 foam. Two planks were cut,
each 2 1/ 8 inches high by 2 1 / 2 inches wide by 15 inches long. They were placed
on opposite edges under the test unit's bottom side. This set—up attempts to
simulate a product resting on its cushion in a normal drop test situation. The
second foam was low density polyurethane. This material was two inches deep,
and covered the entire bottom surface of the test unit. The test procedure began
by dropping the unit from 17 inches using a Lansmont free fall drop test
machine. If no failure occurred at the drop height, the height was raised by
increments of two inches until the plastic beam broke. Each pulse was captured
for analysis. This procedure was done for the polyethylene foam and for the
polyurethane foam. For each pulse causing damage, an SRS plot was generated
and compared to those from Steps 2 and 3. Again, it is expected that they should
intersect close to the same G level and natural frequency of the beam. Using the
known natural frequency and critical element fragility from the above drops, the
same damage boundary curve generated from the traditional method was
generated using the information obtained from the intersection points on the SRS

plots, and Equations 21 and 22.

 

 

CHAPTER 4
RESULTS

4.1 RESULTS
4.1.1 Results From Damping Ratio

Table 1 shows the results using Equation 24 and the decaying waveform
read from the oscilloscope. Table 2 shows the results using Equation 25 and the
peak heights of the decaying waveform captured by T'P2. There is no significant
difference between the two methods in terms of the overall damping ratio
average for each natural frequency. The damping ratio of .04 for the 50 Hz beam
using Equation 24 appears to be a bit high. A probable reason is the stiffness of
the beam, which produces very small oscillations making them very difficult to
read from the oscillosc0pe. Also, since the beam's length was shortened to obtain
50 Hz, disturbing the mass produces higher strain than for a longer beam. Thus
we would expect the damping ratio to be slightly higher for the higher natural
frequency. However, the damping ratio for all four natural frequencies is below
.1, which is very low, and hence not significantly different than a damping ratio
of 0, as will be shown shortly with the results from SRS. (Figure 14).

4.1.2 Results From Plastic Programmers.

Table 3 shows the results from Step 3 of the first procedure. The values
reflect the average of five trials performed at the same conditions for each set-up.
The four different natural frequencies were subjected to the same two inch drop
onto the plastic programmers, and the actual response was captured. Both the

input and output pulses were not filtered for evaluation.

40

41

Table 1. Damping Ratio Using Equation 24.

 

 

 

 

 

 

 

 

 

 

Beam Frequency I 9.5 Hz 14.3 Hz 20.2 Hz 50 Hz

Average Damping

Ratio .02 .03 .02 .04
Table 2. Damping Ratio Using Equation 25.

Beam Frequency I 9.5 Hz I 14.3 Hz I 20.2 Hz 50 Hz

Avera e Dam in

Ratio g p g I .02 I .06 I .02 .02

 

 

 

 

 

 

 

42

 

—Cl— Equation 24

........ 0..."... EquallOn 25

Damping Ratio
0
to
U
l

 

 

 

Beam Frequency, Hz

Figure 14. Damping Ratio Comparison Between Equation 24 and Equation 25.

 

Table 3. SRS Evaluation Results and Comparisons.

43

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PLASTIC PROGRAMMERS
9.5 Hz 14.3 Hz 20.2 Hz 50 Hz
Actual Output

Acceleration, G's 12.35 11.50 19.02 48.08

Damping Ratio 0.00 0.00 0.00 0.00
SRS Peak

Acceleration G's 11.10 13.06 20.89 49.63
Damping Ratio 0.00 0.00 0.00 0.00
Peak Acceleration,

G's D.R. Eq. 24 9.17 11.34 17.00 43.77
Damping Ratio 0.02 0.03 0.02 0.04
Peak Acceleration,

G's 11R, Eq. 25 9.76 11.19 17.00 45.78
Damping Ratio 0.01 0.04 0.02 0.01
GAS PROGRAMMERS

9.5 Hz 14.3 Hz 20.2 Hz 50 Hz

Actual Output

Acceleration, G's N / A 24.89 21.76 31.24
Damping Ratio 0.00 0.00 0.00
‘—s—:Rs Peak
Acceleration G's N/ A 26.06 28.54 29.31
Damping Ratio 0.00 0.00 0.00
Peak Acceleration,

G's D.R. Eq. 24 N / A 24.91 27.66 27.50
Damping Ratio 0.03 0.02 0.04
Peak Acceleration,

G's D.R. Eq. 25 N / A 24.56 27.66 28.77
Damping Ratio 0.04 0.02 0.01

 

 

 

 

44
9.5 Hz Beam. In all five drops, the actual output G's was slightly higher

than the theoretical G level predicted by SRS with zero damping. The damping
ratios calculated by Equations 24 and 25 were also used to calculate SRS by TP2,
as Table 3 shows. A damping ratio of 0 gives the most accurate results, but even
with a damping ratio of .02 the actual and predicted G's differed by less than
three and one-half G's. In this case, for such a low natural frequency, 0 damping
is most accurate. To verify the prediction of TP2 of the shock response to a spike,
we can check it against Equation 7. The natural frequency is known, and the
value used for AV was calculated by TP2 (done digitally by integration). The
input pulse is a short duration spike (from the pulse BRESK, the third trial), and
has a peak G of 121 and a duration of 2.4 ms. So from Equation 7: peak G =
2rtanV/ g = 8.80 when using the AV from TP2. The actual G level was 11.87 , and
SRS predicted 10.75 G, with no damping.

14.3 Hz Beam. SRS predicted results very similar for all damping ratios
compared to the actual G level. In most cases the G's differ only by one or two.
The check on pulse YRESl, the first trial, using Equation 7 reveals: Peak G =
13.77 when using the AV from TP2 analysis. These values compare with the
actual of 11.53 G and the SRS value calculated as 13.60 G, with no damping.

20.2 Hz Beam. The damping ratio calculated for this frequency was about
.02. For plastic programmers, no damping produced higher G's than the actual
and damping of .02 produced slightly lower G's suggesting there is some
damping. It should be stressed this small level of damping is negligible. Using
Equation 7 on pulse BRES4, the second trial: Peak G = 18.62 using AV from TP2.
The actual response was 19.04, and TP2 calculated a peak G of 20.59 with no
damping. There are very small differences between these calculations,

suggesting TP2 and Equation 7 give accurate results.

 

45
50 Hz Beam. Using no damping TP2 predicted G's slightly higher than

the actual, and damping ratios of .01 and .04 predicted G's slightly lower. Again,
there are very small differences, and using no damping gives very accurate
results. From Equation 7 and AV from TP2: Peak G = 48.58 for pulse BRESE,
trial five. Compare that to the actual of 49.80 G's, and TP2 calculating 49.95 G's
using SRS.

4.1.3 Results From Gas Programmers.

For the gas programmers, the beams with different natural frequencies
were subjected to the same eight inch drop height, 150 psi square wave shock,
and the response was recorded. The results are also shown in Table 3. Again,
the values reflect an average from five trials for each set-up.

9.5 Hz Beam. Only one drop was completed, since the shock was too
severe for this setup. The beam bent, so testing was stopped. However, based

on the input square wave, the predicted G from TP2 was 22.46 G's. Since T <
1

2f.

the peak occurring sometime after the duration of the square wave. The input

 

 

, we would expect the peak response of the beam to be Ge. J2(1 — cos (DI) and

acceleration reaches a peak of 14.89, and the duration of the pulse is about 27 ms.
The predicted G is therefore 22.32 G's. This is essentially the same as calculated
by TP2.

14.3 Hz Beam. The actual G's and G's predicted by TP2 with 0, .03, and .06
damping ratio were all very similar. TP2 predicts a peak G response of 26.02 G's,
and from Equation 19, G is predicted to be 28.06. The small discrepancy can be
attributed to the peak acceleration of the input shock (14.98), which is not really

constant over the pulse's duration.

46

 

20.2 Hz Beam. For this case, I is about equal to so the peak G

1
2f.
response should occur almost exactly at the end of the input shock (where t = T).

Since T 2 2%, the SRS G's should be twice the input shock acceleration. This set.

up produced the largest discrepancy between the actual and predicted G level.
The actual output averaged 21.76 G's and SRS predicted an average G of 28.54.
Although the G level is different, the predicted is still slightly larger, so a worst
case scenario situation is produced. A possible explanation for the difference
could be that the beam is not really 20.2 Hz. More likely, the duration of the
pulse is very near the breakpoint between Equation 17 and Equation 18, so TP2
may have chosen to treat the square wave as being long enough to double its

acceleration level.

50 Hz Beam. Here, T > 21

 

, so the peak response G will occur sometime

If

during the duration of the input pulse. This is in fact what does occur. The peak
G using the peak from the input shock BRESF and Equation 19 should be 29.78 G,
and TP2 predicts 29.01. Again, the small difference can be attributed to the fact
that the acceleration of the "square wave" produced by the gas programmer is
not perfectly constant over its entire duration. The small damping ratios do not
differ significantly from a damping ratio of 0. The expected peak G reached
during the duration T should be almost exactly 2Gin, since there is very little
deflection of the beam. Less deflection of the beam will mean the path the beam
travels in during motion will be more linear, not angular, and will more closely

model a true spring/ mass system.

 

 

 

47
4.1.4 Results From Deflection Failure.

Using the peak G predicted by SRS from the pulses obtained just at the
critical deflection yielded very accurate results. For the 15/ 16" setup, peak G was
27.52 G's. Using Equation 20, the predicted deflection is then .9320 inches.
Similarly, for the 1 3/ 8" setup, peak G was 40.91 G's, so the predicted deflection
becomes 1.385 inches. Obviously SRS is an accurate tool to predict when

deflection failure will occur.

4.1.5 Results From Fragility Testing.

A new plastic beam was used for this testing. The natural frequency was
determined by finding the resonance on a vibration table. A very small
accelerometer with negligible mass was mounted on the weight attached to the
end of the beam. The approximate natural frequency was 16 Hz.

To show the system behaves as a linear spring / mass system we can use
the results from the damage boundary testing. Performing a damage boundary
test revealed that on plastic programmer drops there was no damage at a AV of
165.03 in / s, but there was damage at 171.56 in / 3. Similarly, for gas
programmers, there was no damage at 20.56 G but damage at 23.76 G's. Since we
are dealing with a range, the natural frequency can be found within the upper

and lower limits of the above results.

f" __g&_ = 20.56 x3864 =14.7Hz at least, and 23.76x386.4 = 17.7Hz at

’ 2mm, x x 171. 56 n x 165.03

 

 

most. The known natural frequency of 16 Hz falls within this range.

The drops onto the plastic programmers, gas programmers and cushion
that broke the beam were all used to construct their respective SRS plots. If SRS
is valid in predicting the critical G for an element despite the shape of the input

pulse which caused damage, the plots should all intersect at the same natural

48
frequency and critical G level. Overlaying the plots from testing reveals that this

is in fact what happens. See Figures 15, 16, 17 and 18. All four plots are
composite plots, meaning the minimum and maximum G levels of both the
primary and residual responses were plotted. The upper most curve is the
maximum response, and the one of interest. All four SRS plots intersect at
approximately 16 Hz and roughly 46 G's. The implication is that by using SRS
one may obtain the same information for DBC by simply dropping a product
from a normal free fall drop tester. This is shown by comparing the SRS data to
the damage boundary test results.

From the damage boundary test, AVcr was found to be 171.56 in / sec, and
Gcr was 23.76 G's. These are results taken from the drop that caused damage, so
they are conservative and probably represent a slight over-estimation. For
consistency, the drops causing damage from the other inputs were also used.
From the 21 inch free fall drop onto the polyethylene cushion, (pulse DTB), the
SRS plot shows a peak G of 43 G's at 16 Hz. So, using Equations 21 and 22, the
same DBC curve can be constructed, since from the SRS plot:

43 ° 386.4

AV =———=165.27in s
Gr 2-Jr-16 /

Gcr = icce = 21.5 G'S

A check with the SRS plot from a dr0p onto the polyurethane cushion (pulse K2)

yields the same results.

 

 

49

 

 

Evaluation of SRS Software

12:43:28 Pli
05-25-1993

C:\IP2\DMA\D84.PI.21

srs r cimit
: I 051 e
Resolutiii: pro pro

Damping: 0.00
Peak soc: 59.10 G's
Peak Fn: 109.28 Hz

itin ticc: -65.43 G's
itin Fn: 144.20 ilz

 

12.
100
.0
S.

 

4.22.5 “0.0

Frequency (Hz)

 

Notes: Plastio Bean, :6 Hz

1
rarrisuei ht or 39-
Bean he on thrs drop

 

 

\.r TEST PARTNER
§ { Michigan State University

Figure 15. SRS Plot From 30" Gas Programmer Drop.

 

 

 

50

 

 

E luation of SRS Software 3:34:36 Pm
ciiremrtmrzmrz 08-18-1993
Channel 1

SRS Igpe:Composite
Resolution: PPO
Damping: 0.00
Peak Rec: 1
Peak In: 5
Hrn too: -1
urn Po: 6

 

12.
100
O.
6.

 

-12
2. Frequency III) 640. 0

 

Notes: Free {all drop test, onto Ethafoam 220 cushion
23 inches
Damage
Plastic beam, 16 Hz

 

 

\'9 TEST P RTNER
§ a Michigan State University V

Figure 16. SRS Plot From 23" Free Fall Drop Test Onto Ethafoam 220.

 

 

51

 

 

SRS d Dama e Boun 12:08:09 PM
enri'z'onmxinz. 91.2“” 05-25-1993

SRS Type: Composite1

Resolution: 20 PPO
amping: 0.00
Pe ncc: 803.11 G’s
e In: 367.58 Hz
1n Ace: -804.76 CPS
to En: 380.55 Hz

 

 

-.I.2
g. 5 Frequency (Hz) 640. O

 

Notes: Plastic Beam. 16 112
Plastic Programmers, 17' drop height
Beam breaks on this pulse

 

\|’ TEST PARTNER
§ { Michigan State University

Figure 17. SRS Plot From 17" Plastic Programmers Dr0p. ‘

 

 

52

 

 

Evaluation of 311: Software. 9 36 15 em
carrzumcvheqafi P12 1 08-18-1993

ne
SRS Type:Composite

Resolution: 20 PPO
Damprng: 0.00
Peak ace: 414.09 C’s
Peak Pn: 618.20 Hz
Hrn ncc: -458.17 G's
Irn In: 618.20 Hz

 

13.
100
.0
6.
4.
'20

 

-12 ‘
2.5 Frequency (Hz) 640.0

 

Notes: Free fall drop test
33 1nches
Damage
Drop onto Polyurethane foam
Plastic beam, 16 Hz

 

 

 

 

\ ’ TEST PARTNER
§ { Michigan State University

Figure 18. SRS Plot From 33" Free Fall Drop Test Onto Polyurethane Foam.

 

CHAPTER 5
CONCLUSIONS AND LIMITATIONS

5.1 CONCLUSIONS

TestPartner and Equation 7 accurately predict the actual peak acceleration
level with no damping. This suggests the model does in fact behave as a linear
spring/ mass system. Also, it appears that for very low damping ratios there is
not a significant difference from zero damping. Using a damping ratio of 0 for
short duration, high acceleration impacts gives very accurate results compared to
the actual output. The proposed model for a spring/ mass system is accurate, and
Equations 18, 19, and TP2 are in agreement with the actual output C levels. SRS
can be used confidently to predict the response of a spring / mass system to an
input shock.

The results in Figures 15-18 show the predicted response of the critical
element was the same for the different input shocks at the time of failure. This
result supports the assumption the component fails at a prescribed G level
regardless of the source of the shock. This means the same information
contained on a damage boundary curve can be extracted easily from any SRS
plot that just damages the critical element. The shape of the input pulse does not
matter, eliminating the need for a shock table. Thus, the lengthy and costly
investment of a DBC testing procedure is reduced to damaging one unit in an

inexpensive free fall drop test.

5.2 LIlVflTATIONS OF SRS
SRS is designed to predict the response of a spring/ mass system to a

shock, and should therefore not be expected to predict failure in cases Where

53

 

54
damage occurs to a component that does not behave as such. Products that have

no identifiable critical element, specifically the "spring" component, such as glass,
or boxes of agricultural products are some examples.

SRS is also limited to products that fail due to some critical G—level, not a
critical velocity change. A general rule of thumb is that SRS applies to brittle or
stiff products, like the plastic beam that was modeled, or a magnetic head in a
VCR. Components that are soft, or ductile, and fail because of velocity change or
fatigue will not be treated satisfactorily with SRS.

5.3 FUTURE WORK

Though current commercial SRS software packages accurately calculate
SRS plots, there are some useful additions that could be made. In addition to
predicting the peak G response of a spring / mass system, the velocity change
could also be given, since the required information is already provided. The
software could also be improved to allow the user to enter hypothetical shocks.
There are instances Where the input shock may be known beforehand, so the
response could be calculated without the effort of instrumenting a product and
performing drop tests. This would eliminate the need for dependence on analog

signal hardware.

 

APPENDIX A

APPENDD( A

To show the peak residual response to a square wave input is

 

 

 

1 , recall Equations 16 and 17.

G... fill — cos ml") for either T < 21 or T 2
From Equation 17, when t > T,
Gmpons, = G... [cos w( t — T) - cos wt] Eq. 17.

Expand by using the cosine difference identity, and simplify to obtain:
Gm = Gin[asin wt + bcos wt] Eq. 26.

where a = sin (OT
and b = (cosz — 1)
The amplitude of asin wt + bcos cut is:

Jaz + 192 Eq. 27.

So the peak residual response is:

 

Gm = G... (sian)2 +(cos LOT —1)2 Eq. 28.

01':

 

G," J2(1 — cos cor) Eq. 29.

This is the peak residual response to a square wave input when t > T. It is also

the peak residual response when t < T, since
55

56
G,,,(1 — cos wT) < G,,,J2(1 — cos wT) < 2G,, Eq. 30.

 

This is shown by first eliminating Gin from each expression. To show that

 

1 — cos wT < J20 — cos wT) , Eq. 31.

square both sides and simplify to obtain:

1+ cos2 (01" < 2 Eq. 32.

And this inequality is true because cos _<. l.

 

APPENDIX B

APPENDIX B

The SRS plot for a square wave will start out as a straight line with the

same relationship as that of the spike. See Equation 7. To show this, recall the

 

peak primary response when T < . 2; :

 

G... J2(1- cos (07") Eq. 29.

For a small natural frequency, estimate the expression cos wT with a power

 

 

 

 

 

series:
w2T2\ w4T4\
cosz==1—( 2! }+( 4! l ...... Eq.33.
so:
2 2 4 4
cm. zG,,\/2(1—{1— ‘0 T + w T +..}) Eq. 34.
2! 4!
2 2
Since is small compared to l by assumption (i.e. w T is small), simplify to
obtain:
GmwT Eq. 35.
Therefore:
Gpeak = (2minT)fn Eq' 36'

57

 

58
Since (G mg)T = AV for a square wave, divide by g and simplify to obtain:

fonAV
8

G

 

Eq. 37.

peak =

which is equal to Equation 7.

APPENDD( C

APPENDD( C
The following is the derivation for each of the damping ratios used in this
thesis. Begin with the differential equation of Newton's law of motion, with c

representing the dashpot, or damping constant:

 

mk+cx+kx=0 Eq.38.
The solution yields:
x = Aexp(2_—n:t) ~ [cos wt + mw sin wt] Eq. 39.

where: m = mass = W / g
c = dashpot constant
k 2 spring constant
A = amplitude released from
w = 23f" = J; . all—35
C2

4mk'

 

and where R = damping ratio =

If you knew m, k and c, then you would know R. But they are never really
known, especially c, so R must be obtained by looking at the response of the
beam after it is set into motion.

Since the model is an accelerometer mounted on the mass, the output is >2
versus t (acceleration versus time), not x versus t (position versus time), so take

two derivatives of Equation 39, divide by g, and obtain:

= G = %exp< it) (J—ié—E-Z—sin wt — cos wt) Eq. 40.

OQIH:

59

 

60
All of the above damping ratio equations are based on looking at successive

periods of the oscillation:

wt = 2N7r Eq. 41.
where: N = 1, 2, 3, .......
The value of (3 after the Nth cycle (i.e. the G at the Nth peak) is:

 

Eq. 42.

Using the starting value of G at time 0 as ~kA/ w = Go, substituting variables and

rearranging gives:

 

 

 

—2NJrR
G = G ex ( ) E . 43.
\/ 0 P JT-—Rz q
Solving for R yields the damping ratio:
1
R = 2 Eq. 44.
J1+ Q
Where: Q = an .
1n(.§u.)
GN

Equation 44 is the basis for Equation 24. Recall Equation 24 substitutes N for Q,
where N is the number of cycles before the system dies out. Making the
substitution with N in place of Q uses an arbitrary argument that says the motion
of the system appears to die out when GN reaches about 2% of Go. The 2% is

purely arbitrary also, but it is used a lot for many other kinds of exponentially

61
decaying phenomena. So if the motion of the system dies out when (30/ GN =

1 / .002, then:

 

 

In—
.002
50:
1
R z E . 24.
1+ N2 q

where: R is the approximate damping ratio

N is the number of cycles before the system dies out.

The second method of determining damping ratio, Equation 25, follows
from Equation 43. Equation 25 uses the ratio of two successive peak heights. If
the first peak height is noted as ON, and the next peak height as GN+1, then:

 

 

 

—2(N+1)JrR
Gm =GO exp( J1 -R2 ) Eq. 46.
or:
—2NJrR -275R
G ex ex E .47.
0 p( 1—R2) p( 1-R2] q

Therefore:

 

62

 

 

GV+1 _ ( -2'7tR)
GM - exp\ 1 _ R2 Eq. 48.
Solving for R gives:
1,1111
H2 Eq. 49.

2
(In-Ii- +41:2
J H2

where: H1 is the peak height, in G's, of the first height
H2 is the peak height, in GS, of the second height.

 

APPENDIX D

 

 

APPENDIX D

The following is the data collected during the determination of the
, damping ratio.

In both Table 4 and Table 5, H1, H2, etc., indicate the peak height of the
first and second peaks, respectively, after the beam was put into motion. Table 4
reports the peak heights in G's, and Table 5 uses the number of grid divisions
read from the oscilloscope. DR followed by two successive numbers indicates
the damping ratio between those two peaks. The average under each column is
the average damping ratio between all the peaks, and Table 2 reports the average
of these five averages. Table 7 is similar to Table 5, except that it applies only to
the plastic beam.

Table 6 reports the damping ratio calculated using Equation 24, so N is the
number of peaks (cycles) the system oscillated before appearing to die out. The
procedure was repeated five times for each natural frequency, so the average
from the five trials is reported in Table 1.

Table 8 shows the effect damping ratio has on the prediction of the shock
response for each pulse. Each set-up was performed five times for each natural
frequency, except for the gas programmers at 9.5 Hz. The column named
"Equation 24 Peak G, SRS" reports the peak G's calculated by SRS on TP2 using
the damping ratio calculated by Equation 24 for that particular natural
frequency. The calculated damping ratio is found in the same column. The
column named "Equation 25, TP2 Peak G's" reports the peak G's calculated by
SRS on TP2 using the damping ratio that Equation 25 calculated. Similarly, the
next column, (with "O" replacing "TP2"), is the peak G's calculated by SRS using
the damping ratio obtained from the readings from the oscilloscope and

Equation 25. The last column is SRS G's calculated using zero damping. PP

63

64
represents the results from the plastic programmer drops, and GP represents the

results from the gas programmer drops.

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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69

Table 8. Effect of Damping Ratio On Shock Response.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9.5 Hz Equation 24 . Equation 25, TP2 Equation 25, 0 Peak G's
Actual Output G's Peak G, SRS DR Peak G's DR Peak G, SRS DR DR = 0
PP 1 11.87 8.95 0.02 8.95 0.02 8.42 0.03 10.75
2 12.16 9.25 0.02 9.25 0.02 8.70 0.03 10.82
3 13.13 9.10 0.02 9.10 0.02 8.56 0.03 11.80
4 12.60 9.34 0.02 9.34 0.02 8.79 0.03 10.90
5 12.01 9.22 0.02 9.22 0.02 8.67 0.03 11.23
AVG 12.35 9.17 0.02 9.17 0.02 8.63 0.03 11.10
GP 1 13.56 19.46 0.02 19.46 0.02 18.64 0.03 23.17
14.3 Hz
PP 1 10.72 11.48 0.03 11.06 0.06 10.92 0.07 13.08
2 12.21 10.91 0.03 10.51 0.06 10.40 0.07 12.90
3 11.53 11.93 0.03 11.48 0.06 11.35 0.07 13.60
4 11.38 11.26 0.03 10.83 0.06 10.69 0.07 12.80
5 11.64 11.11 0.03 10.68 0.06 10.56 0.07 12.94
AVG 11.50 11.34 0.03 10.91 0.06 10.78 0.07 13.06
GP 1 24.90 24.90 0.03 23.91 0.06 23.63 0.07 26.06
2 24.41 24.96 0.03 23.99 0.06 23.70 0.07 26.11
3 25.39 24.91 0.03 23.93 0.06 23.64 0.07 26.07
4 24.90 24.89 0.03 23.91 0.06 23.62 0.07' 26.05
5 24.85 24.87 0.03 23.91 0.06 23.62 0.07 26.02
AVG 24.89 24.91 0.03 23.93 0.06 23.64 0.07 26.06
20.2 Hz
PP 1 17.30 16.53 0.02 16.53 0.02 16.53 0.02 20.47
2 19.04 16.86 0.02 16.86 0.02 16.86 0.02 20.85
3 20.07 17.80 0.02 17.80 0.02 17.80 0.02 21.33
4 19.04 17.00 0.02 17.00 0.02 17.00 0.02 20.59
5 19.63 16.79 0.02 16.79 0.02 16.79 0.02 21.23
AVG 19.02 17.00 0.02 17.00 0.02 17.00 0.02 20.89
GP . 1 21.64 27.45 0.02 27.45 0.02 27.45 0.02 28.81
2 21.68 27.67 0.02 27.67 0.02 27.67 0.02 28.32
3 21.82 2/.82 0.02 27.82 0.02 27.82 0.02 28.54
4 21.97 7.42 0.02 27.42 0.02 27.42 0.02 28.70
5 21.71 27.93 0.02 27.93 0.02 27.93 0.02 28.31
AVG 21.76 27.66 0.02 27.66 0.02 27.66 0.02 28.54
50 Hz
PP 1 42.77 43.56 0.04 44.83 0.02 44.83 0.02 50.05
2 47.85 43.82 0.04 45.20 0.02 45.20 0.02 49.85
3 50.57 43.66 0.04 44.97 0.02 44.97 0.02 48.80
4 49.41 43.39 0.04 44.70 0.02 44.70 0.02 49.51
5 49.80 44.34 0.04 45.76 0.02 45.76 0.02 49.95
AVG 48.08 43.75 0.04 45.09 0.02 45.09 0.02 49.63
GP 1 28.96 27.27 0.04 28.12 0.02 28.12 0.02 29.01
2 31.30 27.28 0.04 28.10 0.02 28.10 0.02 29.06
3 31.74 27.61 0.04 28.43 0.02 28.43 0.02 29.39
4 31.88 27.68 0.04 28.53 0.02 28.53 0.02 29.48
5 32.32 27.65 0.04 28.49 0.02 28.49 0.02 29.59
AVG 31.24 27.50 0.04 28.33 0.02 28.33 0.02 29.31

 

 

 

 

 

 

 

 

 

 

 

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Dan in Ratio 11:20:56 AM
: 1 \MPDWRBS. 11.2 05-24-199
C\P2 “mun e11 3

Paired nco:

G’s
an : 23.78 G’s
iinfiis use
neit§°3i 155273 In/s
Filter: 0 Hz

 

3O
25
20
15

10

 

50. O moo/Div

 

Notes: Bean has natural ire?uencg of 9. 5 Hz
Results iron 1P2 ana 951$
Damping ratio using successive peak heights, Eq. 25

 

Q
S ? 11:51 PA. RTNER

Michigan State University

Figure 20. Oscillation of Variable Beam, 9.5 Hz.

 

 

72

 

 

Danf1ngDRatio 3: 6:93 0H
c:\ P2 AIA\HPD\DR91.PL2 34-29-1993
Channel 1
Faired Ace: G’s
Peak Rec: 33.11 ’s
H1n ecc: -25.49 G's
Durat1on: 15.03 asec
Delta 0: 98.99 ln/S
Filter: 0 Hz

 

‘0
50
40
3O
20

10

 

50 . O moo/Di 9

 

Notes: Bean has natural freguencg of 14. 3 Hz
Results iron 1P2 ana gs1s
Danping ratio using successive peak heights, Eq. 25

 

Q
TEST P RTNER
S? __J_,\/__\_

Michigan State University

Figure 21. Oscillation of Fixed Beam, 14.3 Hz.

 

 

73

 

 

3:23:22 AH

 

33333 33333112961121.1212 94-29-1993
Cohan e11
Paired Acc: 6’5
1 : 21.83
33133133: -21.63 G’s
Duration: 14.00 msec
Delta 0: 53.63 In/S
Filter: 9 Hz
3.
as
3.
15

10

 

50 . O moo/Di v

 

Notes: Beam has natural {re encg of 20. 2 Hz
Results from 1P2 ana gsis
Damping ratio using successive peak heights, Eq. 25

 

   

TEST PA. RTNER

Figure 22. Oscillation of Variable Beam, 20.2 Hz.

\II
§ { Michigan State University

 

 

 

74

 

 

D in Ratio :
c3? 92 IAWDWRBAJLZ “953
Channel 1 _

Paired Rec: 6' 5
Peak flcc: 26.27 G’s
Nin ncc: -23.05 G’s
Duration: 5.00 msec
Delta 0: 25.41 ln/S
Filter: 0 Hz

 

3O
25
20
15

1.

 

50. O moo/Div

 

Notes: Beam has: natural 2freguencg of 50 Hz
Result lts Hrom ags1s
Damping ratio using successive peak heights, Eq. 25

 

o
TEST PAM RTN ER

\ ’
§ { Michigan State University

Figure 23. Oscillation of Variable Beam, 50 Hz.

 

 

 

75

 

 

SRS Evaluation, Damping Ratio 3:39:46 Pl
D:DRP2.P12 02-23-1993
Channel 1
Paired ncc:

C’s
Peak Acc: 8.15 C’s
Nin ecc: -6.01 C’s
Duration: 12.50 msec
Delta 0: 17.56 In/S
P11ter: 0 Hz

 

12

1.

N O O 0

6'8

 

30.0 nsoc/Dio

 

 

Notes: Dampin ratio determined from successive peak heights using
at ion2
Results from 1P2 analgsis
Plastic beam vith hnatural {reguencg at 16 02
Peak E £50m first 6 heights: 15, 5. 71, 4.25. 3. 03, 2.29

 

  

O

\ I
§ { Michigan State University

Figure 24. Oscillation of Plastic Beam, 16 Hz.

   

 

 

76

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APPENDIX E

 

APPENDIX E

The following are Figures showing the input shock, the predicted
response and the actual response of the beam from the sections 4.1.2 "Results

From Plastic Programmers," and 4.1.3 "Results From Gas Programmers."

78

 

 

79

 

 

Evaluation of SRS Software 13%34:29 nn’

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:\IP2\DRIR\HPD\B f K.PL2 -25-1993
C Channel 1 Channel 2
Hodel.Pn: 9.50 . Hz
Danp1ng: 0.00
Peak Acc: 110.26 G’s
Model Peak: 10.16 0’5
.11 :12
e 1n: " e
Filter: ‘ 0.00 Hz
130 i ‘
100
.0 a
“input shock
60
6'8
40
2° (Actual response
a a. Predicted response
° 4 g . 53‘5zs:r---ff:::::s7"’zg=:::::~~E==I~J
-20 ’
-40

 

 

 

 

 

 

 

 

 

 

 

 

20.0 useo/Div

 

Notes: Channel 1: Plastic Programmers, 2' drop height
Channel 2: Response of the bean
Bean has natrual frequency of 9.5 Hz

 

\ ’
§ 1’ Michigan State University

 

TEST PAM RTNER

Figure 27. Actual and Predicted Response To a Plastic Programmers Drop,
Variable Beam, 9.5 Hz.

 

 

 

8O

 

 

 

 

 

 

 

 

Evaluation of SR8 Software 11:95:91 DH
: \DHTH\HPD\1 V‘P.PL2 95-25-1993
C \IP2 hanne; 1 Channel 2
Hodel,Fn: 9.h0 Hz
Danp1ng: 0.00 ’
Peak Hcc: 17.54 C’s
Hodel Peak: 22.46 C,
1.17%?“ iii: 8:
e 1n: - .
Filter: 387.60 Hz

 

Input hock

as
ual r sponse

10

 

-3O 20 . O “cc/Div

 

Notes: Channel 1: Gas Programmers, 8' drop, 150 psi
Channel 2: Response of bean
Bean has natural frequency of 9.5 Hz
Bean bent on thls drop

 

   

- TEST PA, RTNER

\ ’
§ { Michigan State University

 

 

Figure 28. Actual and Predicted Response To a Gas Programmers Drop,
Variable Beam, 9.5 Hz.

81

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Evaluation of SRS Softm 10:21:56 PH
C: \IP2\DAIR\NPD\7RES1. PLZ 04-29-1993
Channel 1 Channel 2
Ngdel Pn: 14. .38 Hz
Pe 371335 77212 G’s
Nodel Peak: 13.03 . C’s
Nin ncc: -7.48 6?
Model Nin: -12.96 C’s
Filter: 156.00 Hz
120
100
80 l
31 /.-lnput shock
c' ‘60 l/
40
1 /-Predicted response
20 /
o 3.1 _ jctual re:¢?”\ ['4
rm ‘1, / V: :V
-20
-“ 20.0 nsoc/Div
Notes: Channel 1: Plastic Progranners, 2' drop
Channel 2: Response of
Bean has natural frefluencghof 14. 3 Hz
Channel 1 is filtere to show nore clearly channel 2;
channel Moriginal peak G of 118.59? which was used to
calculate1 the shoe response
\., TEST PARTNER
§ { Michigan State University V

Fi

 

 

gure 29. Actual and Predicted Response To a Plastic Programmers Drop,

Fixed Beam, 14.3 Hz.

 

82

 

 

Evaluation of SRS Software 10:47:35 PM
C:\IP2\DAIA\NPD\PRESO.P12 04-29-1993
Channel 1 Channel 2
Model Fn: 14.30 Hz
Danping: 0. 0
Peak Rec: 18.28 C's
Model Peak: 26. 02 C’s
M1n 99c: -9.31 . C’s
Model M1n: -27. 43 0’5
P1lter: 384. 62 Hz

 

ict res se
ua r pons

 

 

Notes: Fixed Bean withP natural freguencgP of 14. 3 Hz
Channel 1:C Cas Progranner
Channel 2: Response of C. t.

150 p51

 

. TEST PARTNER
S? -—’\r-—

Michigan State University

Figure 30. Actual and Predicted Response To a Gas Programmers Drop,
Fixed Beam, 14.3 Hz.

 

 

 

83

 

 

 

 

 

 

 

 

 

 

Evaluation of SRS Software. 19:39:11 p"
:\IP2\DRIR\MPD\BRE81.P1.2 - -
C Channel 1 Channel 2 04 29 1993
Mgdel,Fn: 23.33 Hz
p.383; 115247 C's
Model Peak: 20.45 C’s
Min Rcc: -7.92 C’s
Model Min: -19.42 C’s
Filter: 4545. Hz
120
no ' 4,_lnput shock
1 /
I“
00 H
60
6'3
‘. ///rprqdicted_LeQPnn:p

. Actual re51onse

/ '/ ._//

-° $7 __\

20.0 asec/Dio

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notes: Uariahle bean vith natural frequency of 20.2 Hz
Channel 1: Plast1c Progranner, 2' drop
Channel 2: Response of the bean

 

   

Q '
TEST PA. RTN ER

\ V ‘
§ { Michigan State University

 

 

Figure 31. Actual and Predicted Response To a Plastic Programmers Drop,
Variable Beam, 20.2 Hz.

 

 

8828371838183” 3‘33““ P"
' Channel'l Channel 2 09-81-1993
Model Flt: 20.20 ' Hz
Danping: 0.00 ,
Peak hcc: 19.33 C,
Model Peak: 28.32 c,5
‘ ...:‘1 es -531: t:
iiltigi 367:65 Hz

 

Predi ted re ponse

 

20.0 usec/Div

 

Notes: Variable bean with natural fre ency of 20.2 Hz
Channel 1: Gas Progranner, 8' op
Channel 2: Response of the bean

150 p51

 

\I’ TEST PARTNER
§ { Michigan State University

Figure 32. Actual and Predicted Response To a Gas Programmers Drop,
Variable Beam, 20.2 Hz.

 

 

85

 

 

Evaluatlon of SRS Software 10:29:50 RM
C:\IP2\DRIR\MPD\BRESB.PL2 05-25-1993
Channel 1 Channel 2
Model Fn: 50.00 Hz
Danping: 0.00
Peak Acc: 114.19 C's
MhIPuL 4%N ' 85
Min acc' c. .-5.35 C‘s
Model Min: -47.81 C's
Filter: 4166.67 Hz

 

 

 

 

 

Notes: Uariable bean with natural frequency of 50 Hz
Chm nel 1: Plastic Progranner r, 2' drop
Channel 2: Response of the bean

 

 

 

Q

\ ’
§ { Michigan State University

TEST PA. RTNER

 

Figure 33. Actual and Predicted Response To a Plastic Programmers Drop,
Variable Beam, 50 Hz.

 

 

 

 

86

 

 

Evaluation of SR8 Softm 10: 59:17 ON
C: \IP2\DRIH\MPD\BRESF. PL2 05-25-199 93
Channel 1 Channel 2
Model Fn: 50. 00 Hz
Danping: 0. 00 ’
Peak Hcc: 17.75 G’s
Model Peak: 29.04 C s
1.1": i°°= 5213 21:
e 1n: - .
Filter: 373.13 Hz

 

30
25
30
15
10

Act al re s

 

20.0 nsec/Div

 

Notes: Uariable bean with natural freflgency150 of 50 Hz
Channel 1: Cubogranner,8 op,10psi
Channel 2: Response of the bean

 

. ’
TEST PAR RTN ER

\ I
§ { Michigan State University

 

 

Figure 34. Actual and Predicted Response To a Gas Programmers Drop,
Variable Beam, 50 Hz.

APPENDD( F

 

 

APPENDIX F

The following are original pulses that broke the plastic beam from the
section 4.1.5 "Results From Fragility Testing." The corresponding SRS plots are
Figures 15 to 18.

87

 

88

 

 

 

I TEST P RTNER
% a? Michigan State University

 

ation of SRS Software 12:43:28 PM
Eiiiiz\nnrn\npn\ncs.rLz 05-25-1993
Channel 1
Paired Acc: C's
Pe Acc: 23.76 C’s
Min acc: -11.83 C's
Duration: 38.60 nsec
Delta 0: 285.34 ln/S
F11ter: 262 Hz

 

 

20 . 0 nsec/Di o

 

Notes: Plastic Bean

200 p51 .
Table He1 ht of 30'
Bean bre on th1s drop

 

Figure 35. 30" Gas Programmers Drop, Damage Boundary Test.

 

 

89

 

 

1 to
has.
00-

Evaluation of SRS Software

PM
C:\TP2\DHIR\MPD\D121.PL2 1993
Channel 1

8w

Fa1red ncc: 74.56 st
Peak ncc: 157.57 G's
N1n ecc: -139.31 C s
Durat1on: 15.20 nsec
Delta U: 199.54 In/S
F1lter: 0 Hz

 

 

120
100
80
69

4O

 

20.0 nsec/Div

 

Notes: Free fall drop test
23 1nches

Daagge
Et can 220
Plast1c bean has natural frequency of 16 Hz

 

TEST PAM RTNER

-
\r
§f

  

Michigan State University

   

Figure 36. 23" Free Fall Drop Test Onto Ethafoam 220.

 

 

 

9O

 

 

9 . TEST PARTNER
$ { Michigan State University

 

SRS and Dana e Bound 12:08:09 PM

C:\TP2\DAln D\D82.P 05-25-1993
Channel 1

Paired Acc:

C’s
Peak Rcc: 419.04 C's
M1n occ: -60.94 C’s
Durat1on: 1.80 nsec
Delta 0: 171.56 In/S
F1lter: 0 Hz ,

 

 

‘0.

 

 

 

 

300
6'8

 

 

100

A_L L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20.0 nsec/Div

 

Notes: Plastic Bean #1
Plast1c Progranners, 17' drop height
Bean breaks on th1s pulse

Figure 37. 17" Plastic Programmers Drop, Damage Boundary Test.

 

 

 

91

 

 

Evaluation of SRS Software 9:36:15 PM
C:\IP2\DATA\MPD\H2.PL2 08-18-1993
Channel 1
Faired Hcc: 48.58 G‘s
Peak flcc: 323.97 C's
M1n Qcc: -270.75 C's
Durat1on: 25.00 nsec
Delta U: 176.32 In/S
F1lter: 0 Hz

 

 

 

 

 

Notes: Free fall drop test
33 inches
an age
Polyurethm fom
Bean has natural frequency of 16 Hz

 

 

 

I TEST PA. RTNER

\ ’
§ { Michigan State University

 

Figure 38. 33" Free Fall Drop Test Onto Polyurethane Foam.

 

 

 

 

APPENDIX G

APPENDD( G

The following is the derivation for incorporating damping ratio into the

deflection failure criteria.
Recall Equation 38, which can be rewritten as
Eq. 48.

W dzy (dx dt\

_.__ ___ k _

g dtz det dt/ + (x y)
where: x-y = deflection of mass relative to its base

locity of the mass relative to the base.

 

and d_x — fl 2 ve
dt dt
At maximum relative deflection, where (x-y) = d, the relative velocity is zero, and
the peak deceleration of the mass becomes:
dzy kgd
—— =— E . 49.
dt2 W q
Using the relations
2
G = 1.31.— Eq. 50.
g dt’
kg (an )2
and — = " 2 Eq. 51.
W 1 — DR
where DR = damping ratio.
Substituting into Equation 49 yields:
(28 Yd
ak G = " E . 52.
pe (1 _ DRZ) q

92

 

REFERENCES

....1

\0

REFERENCES

. Greenfield, J.F. "Dealing With the Shock Environment Using the Shock

Response Spectrum Analysis." Iournal of the Society of
Environmental Engineers. Sep. 1977 pp 3-15.

. ASTM D332-77 (1983). "Standard Test Methods for Mechanical Shock

Shock Fragility of Products Using Shock Machines."

. Henderson, George. "Improved Protective Package Drop Test Data

Analysis." IBM Annual Shock and Vibration Seminar. Boca Raton,
Florida. IBM, 19 Oct. 1990.

. Burgess, Gary. Lecture notes, Advanced Packaging Dynamics. East

Lansing, Michigan State University.

. Leinberger, David A. "Study of SRS for Product/ Package Testing."

Unpublished Report. East Lansing, Michigan State University. June,
1992.

. 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, '88, '89, '90, and '91, Distribution Dynamics Labs Advanced Shock
and Vibration Seminars, 1991.

. Yang, Robert C. and Herbert R. Saffill. Development Of A Waveform

Synthesis Technique - A Supplement To Response Spectrum As A
Definition Of Shock Environment. Shock and Vibration Bulletin No.

42, Part 2, NRL Jan. 1972.

. Smallwood, David O. "An Improved Recursive Formula For Calculating

Shock Response Spectra." Shock and Vibration Bulletin No. 51, Part 2,
pp211—216, May 1981.

. Harris, Cyril M. Shock and Vibration Handbook. 3rd ed. New York:

McGraw-Hill, 1988.

10. Keller, A.C., B.G. Pollard and EA. Andress. Shock Spectrum Analysis

And Structural Design. San Diego: Spectral Dynamics Corporation,
1973.

93

11.

12.

13.

14.

15.

16.

17.

94
Schwartz, H.B.. and ]. Santos. Shock Response Spectrum (SRS): AnalysisI

Interpretation and Application. Poughkeepsie: IBM Technical Report,
1989.

Henderson, George. Standardization of Shock Machines Using SRS
Analysis. San Pedro: GHI Systems, Incorporated, 1988.

Tustin, Wayne, George M. Heiber, et a1. Understanding And Measuring
The Shock Response Spectrum. San Diego: Spectral Dynamics
Corporation, 1975.

Newton, Robert E. "Fragility Assessment Theory and Test Procedure."
US. Naval Postgraduate School.

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