HEALTH MONITORING FOR DAMAGE INITIATION & PROGRESSION
DURING MECHANICAL SHOCK IN ELECTRONIC ASSEMBLIES
Except where reference is made to the work of others, the work described in this thesis is
my own work or was done in collaboration with my advisory committee. This thesis does
not include proprietary or classified information.
_____________________________________________
Prakriti Choudhary
Certificate of Approval:
______________________________ ______________________________
Robert L. Jackson Pradeep Lall, Chair
Assistant Professor Thomas Walter Professor
Mechanical Engineering Mechanical Engineering
______________________________ ______________________________
John L. Evans Joe F. Pittman
Associate Professor Interim Dean
Industrial & Systems Engineering Graduate School
HEALTH MONITORING FOR DAMAGE INITIATION & PROGRESSION
DURING MECHANICAL SHOCK IN ELECTRONIC ASSEMBLIES
Prakriti Choudhary
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
May 10, 2007
iii
HEALTH MONITORING FOR DAMAGE INITIATION & PROGRESSION
DURING MECHANICAL SHOCK IN ELECTRONIC ASSEMBLIES
Prakriti Choudhary
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of the individuals or institutions and at their expense. The author
reserves all publication rights.
______________________________
Signature of Author
______________________________
Date of Graduation
iv
VITA
Prakriti Choudhary, daughter of Mr. Subhash and Dr. Veena Choudhary, was
born on May 15, 1983 in New Delhi, India. Prakriti graduated with her Bachelors in
Electrical Engineering from Delhi College of Engineering, New Delhi, India. During her
bachelors she did internships at Lund University, Lund, Sweden and at the Dresden
Design Centre (DDC), the R&D centre for Advanced Micro Devices (AMD) at Dresden,
Germany. In the pursuit of enhancing her academic qualification she joined the M.S.
Program at Auburn University in the Department of Mechanical Engineering in Fall
2004. During the M.S. program at Auburn University, she has worked under the guidance
of Professor Pradeep Lall, in the Department of Mechanical Engineering and the Center
for Advanced Vehicle Electronics (CAVE), as a Graduate Research Assistant in the area
of reliability of electronic packages in a drop and shock environment.
v
THESIS ABSTRACT
HEALTH MONITORING FOR DAMAGE INITIATION & PROGRESSION
DURING MECHANICAL SHOCK IN ELECTRONIC ASSEMBLIES
Prakriti Choudhary
Master of Science, May 5, 2007
(B.E.E., Delhi College of Engineering, Delhi, India, 2004)
205 Typed Pages
Directed by Pradeep Lall
Electronic products may be subjected to shock and vibration during shipping,
normal usage and accidental drop. Highstrain rate transient bending produced by such
loads may result in failure of fine-pitch electronics. Current experimental techniques rely
on electrical resistance for determination of failure. Significant advantage can be gained
by prior knowledge of impending failure for applications where the consequences of
system-failure may be catastrophic.
This thesis focuses on an alternate approach to damage-quantification in
electronic assemblies subjected to shock and vibration, without testing for electrical
continuity. The proposed approach can be extended to monitor product-level damage.
Statistical pattern recognition and leading indicators of shock-damage have been used to
study the damage initiation and progression in shock and drop of electronic assemblies.
vi
Closed-form models have been developed for the eigen-frequencies and mode-shapes of
electronic assemblies with various boundary conditions and component placement
configurations. Model predictions have been validated with experimental data from
modal analysis. Pristine configurations have been perturbed to quantify the degradation in
confidence values with progression of damage. Sensitivity of leading indicators of shock-
damage to subtle changes in boundary conditions, effective flexural rigidity, and transient
strain response have been quantified. A damage index for Experimental Damage
Monitoring has been developed using the failure indicators.
The above damage monitoring approach is not based on electrical continuity and
hence can be applied to any electronic assembly structure irrespective of the
interconnections. The damage index developed provides parametric damage progression
data, thus removing the limitation of current failure testing, where the damage
progression can not be monitored. Hence the proposed method does not require the
assumption that the failure occurs abruptly after some number of drops and can be
extended to product level drops.
vii
ACKNOWLEDGEMENTS
The author acknowledges and extends gratitude for financial support received
from the National Science Foundation. Many thanks are due to the author?s advisor Prof.
Pradeep Lall, and other committee members for their invaluable guidance and help
during the course of this study. Deepest gratitude are also due to the author?s parents, Mr.
Subhash Chowdhury, Dr. Veena Choudhary and brother Ankush Chowdhury for being
constant source of inspiration and motivation, and to friends, Sameep Gupte, Dhananjay
Panchagade and all other colleagues and friends whose names are not mentioned, for
their priceless love and support.
viii
Style manual or journal used Guide to Preparation and Submission of Theses and
Dissertations
Computer software used Microsoft Office 2003
ix
TABLE OF CONTENTS
LIST OF FIGURES xiii
LIST OF TABLES xviii
CHAPTER 1 INTRODUCTION 1
1.1 Statistical Pattern Recognition 1
1.2 Health Monitoring 2
1.3 Current Testing Techniques 4
1.4 Closed Form Models 4
CHAPTER 2 LITERATURE REVIEW 6
2.1 Experimental Techniques 7
2.2 Statistical Pattern Recognition 9
2.3 Closed- Form Analytical Models 10
CHAPTER 3 STATISTICAL PATTERN RECOGNITION 13
3.1 Wavelet Transforms 14
3.1.1 Daubechies Wavelet 16
3.2 Wavelet Packet Approach 22
x
3.3 Distance Based Similarity 30
3.3.1 Euclidean Distance 30
3.3.2 Mahalanobis Distance Approach 31
CHAPTER 4 FAST FOURIER TRANSFORM & TIME FREQUENCY ANALYSIS 33
4.1 Fourier Tansforms 33
4.1.1 Discrete Fourier Transform 34
4.1.2 The Radix-2 FFT Algorithm 35
4.1.3 FFT Frequency Bands 41
4.2 Time Frequency Analysis 45
4.3 Linear Time Frequency Transforms 45
4.3.1 Short Time Fourier Transform 46
4.3.2 Continuous Wavelet Transform 47
4.3.3 Gabor Expansion 48
4.4 Quadratic Time Frequency Transforms 49
4.4.1 Wigner-Ville Distribution 49
4.4.2 Cohen Class of Transforms 52
4.4.3 Reduced Interference Distributions 53
4.5 Time Frequency Moments 57
4.6 Confidence Value Computation 61
4.6.1 Testing Hypothesis 61
CHAPTER 5 CLOSED FORM ANALYTICAL MODELS 63
xi
5.1 Derivation of the Lagrangian Functional 63
5.1.1 Development of the Virtual Strain energy 65
5.1.2 Development of the Virtual Kinetic Energy 67
5.1.3 Development of the Virtual Potential Energy 68
5.2 Development of Governing Differential Equation 68
5.2.1 Isotropic plates 70
5.2.2 Orthotropic plates 73
5.3 Plate Functional Derivation using Plate Strips 74
5.3.1 Plate Strip Displacement Function 74
5.3.2 For Simple-Simple plate strip 76
5.3.3 Free-Free plate strip 78
5.3.4 For a Clamped-Free Strip 81
5.3.5 For Clamped-Clamped plate strip 84
5.4 Application of Ritz Method 86
5.4.1 Completely Free (FFFF) Plate 87
5.5 Point Mass Components on the PCA 95
5.5.1 Eigenvalue Equation of a Constrained Plate 96
CHAPTER 6 APPLICATION AND VALIDATION OF PREDICTIVE MODEL 100
6.1 Development of Training Signal and High-Speed Measurement Transient
Dynamic Response 100
6.2 Training of the Predictive Model 108
xii
6.3 Closed Form Model Results 119
6.3.1 CFFF to FFFF Boundary Condition change with change in aspect ratio 119
6.3.2 Point Mass Fall off from Assembly corresponding to Package Falloff 123
6.4 Model Based Correlation of Damage 128
6.4.1 Solder Ball Cracking and Failure 131
6.4.2 Chip Failure 138
6.4.3 Chip Delamination 138
6.4.4 Package Fall Off 145
6.5 Experimental Validation 145
6.6 Solder Joint Built in Reliability Test 157
CHAPTER 7 SUMMARY & CONCLUSIONS 162
BIBLIOGRAPHY 165
xiii
LIST OF FIGURES
Figure 1 A N
th
Level Wavelet Decomposition Structure. 16
Figure 2 The Daubechies-6 Wavelet. 20
Figure 3 The Scaling Function of a Daubechies 6 Wavelet. 20
Figure 4 Frequency Response of the Low-Pass Filter. 21
Figure 5 Frequency Response of the High-Pass Filter. 21
Figure 6 Wavelet Packet decomposition structure for Level three decomposition. 23
Figure 7 First six wavelet packets for a DB6 filter packet decomposition. 26
Figure 8 Transient Strain-History at Location of CSP during Drop-Event. 28
Figure 9 Wavelet Packet Energy Feature Vector. 29
Figure 10 Mahalanobis distance Feature Vector. 32
Figure 11 Fast Fourier Transform decimation based on Decimation in time algorithm. 37
Figure 12 A basic FFT butterfly structure used to combine the decimated signal to
obtain the frequency spectrum. 40
Figure 13 Example of the Structural Combination of the decimated signal to produce
the frequency spectrum. 40
Figure 14 : Transient Strain-History at Location of CSP during Drop-Event. 42
Figure 15 : FFT Frequency Band Energy Feature Vector. 43
Figure 16 The Receptance Plot obtained by the Modal Analysis of the TABGA Board. 44
Figure 17 The Mode shapes and natural frequencies of vibration of the Board. 44
xiv
Figure 18 Time Frequency Analysis Techniques. 45
Figure 19 Time Frequency Distribution for a Transient Strain signal. 59
Figure 20 Time Moment Feature Vector for a Transient Strain Signal. 60
Figure 21 Frequency Moment Feature Vector for a Transient Strain Signal. 60
Figure 22 Modeshape Correlation of a Completely Free plate with [Leissa 1969]. 94
Figure 23 Point Mass representation of the Electronic Assembly. 99
Figure 24 Interconnect array configuration for Test Vehicles. 101
Figure 25: Interconnect array configuration for 95.5Sn4.0Ag0.5Cu and 63Sn37Pb
Test Vehicles. 103
Figure 26 Measurement of Velocity, Acceleration, and Relative Displacement During
Impact. 106
Figure 27 Relative Displacement and Strain Measurement in Horizontal Orientation. 106
Figure 28 Transient Strain-History at Location of CSP during Drop-Event. 107
Figure 29 Strain data for Repeatable Drops of an electronic Assembly. 109
Figure 30 Repeatable Feature Signatures obtained using Wavelet Packet Energy
Vectors. 110
Figure 31 Confidence Values obtained by applying Wavelet Packet Energy Approach
to Repeatable Drops (No Failure). 111
Figure 32 Repeatable Feature Signatures obtained using Mahalanobis Distance
Vectors. 112
Figure 33 Confidence Values obtained by applying Mahalanobis Distance
computation to Repeatable Drops (No Failure). 113
xv
Figure 34 Repeatable Feature Signatures obtained using FFT Frequency Bands
Energy Vectors. 114
Figure 35 Confidence Values obtained by applying FFT Frequency Band Energy
computation to Repeatable Drops (No Failure). 115
Figure 36 Repeatable Feature Signatures obtained using Time Moment Vectors. 116
Figure 37 Repeatable Feature Signatures obtained using Frequency Moment Vectors. 117
Figure 38 Confidence Values obtained by applying Time Frequency Analysis to
Repeatable Drops (No Failure). 118
Figure 39 Confidence Value Degradation with Change in Aspect Ratio for Mode 1. 121
Figure 40 Confidence Value Degradation with Change in Aspect Ratio for Mode 2. 122
Figure 41 Point Mass Closed Form Model and Numbering of Location of Packages. 124
Figure 42 Degradation in Confidence Value with respect to Location of Package Fall
off. 125
Figure 43 Degradation in Confidence Value with Package Fall off from Location 1. 126
Figure 44 Effect of Package Fall off on Modeshape of Assembly. 127
Figure 45 Package with Solder Beam Array. 129
Figure 46 Solder Bam Array modeled to represent Solder Balls. 129
Figure 47 Vertical Drop Model developed for the Study. 130
Figure 48 Horizontal Drop Model developed for the Study. 130
Figure 49 Model Configurations for Correlation of Interconnect Failure to Confidence
Value Degradation. 132
Figure 50 Model Configurations for Correlation of Interconnect Damage to
Confidence Value Degradation. 133
xvi
Figure 51 Confidence Value degradation in Transient PCB Strain with Solder Ball
Failure for Vertical Drop. 134
Figure 52 Confidence Value degradation in Transient PCB Strain with Solder Ball
Failure for Horizontal Drop. 135
Figure 53 Confidence Value degradation in Transient PCB Strain with Solder Ball
Damage for Vertical Drop. 136
Figure 54 Confidence Value degradation in Transient PCB Strain with Solder Ball
Damage for Horizontal Drop. 137
Figure 55 Model configuration for Chip Fracture (cracking). 139
Figure 56 Confidence Value degradation in Transient PCB Strain with Chip Failure
for Vertical drop orientation. 140
Figure 57 Confidence Value degradation in Transient PCB Strain with Chip Failure
for Horizontal drop orientation. 141
Figure 58 Model configuration for Chip Delamination. 142
Figure 59 Confidence Value degradation in Transient PCB Strain with Chip
Delamination for Vertical drop orientation. 143
Figure 60 Confidence Value degradation in Transient PCB Strain with Chip
Delamination for Horizontal drop orientation. 144
Figure 61 Model configuration for loss of package from assembly (a) Vertical Drop
(b) Horizontal Drop. 146
Figure 62 Confidence Value degradation in Transient PCB Strain with Package Loss
for Vertical drop orientation. 147
xvii
Figure 63 Confidence Value degradation in Transient PCB Strain with Package Loss
for Horizontal drop orientation. 148
Figure 64 Wavelet Packet Energy Feature Vector used in Failure Classification. 150
Figure 65 Mahalanobis Distance Feature Vector used for Failure Classification. 151
Figure 66 FFT Frequency Band Energy Feature Vector used for Failure
Classification. 152
Figure 67 Time Moment Feature Vector used for Failure Classification. 153
Figure 68 Frequency Moment Feature Vector used for Failure Classification. 154
Figure 69 Confidence value degradation showing progressive damage with the Drops. 155
Figure 70 Confidence value degradation showing progressive damage with the Drops. 156
Figure 71 Solder Joint Built in Reliability test Circuit Design. 159
Figure 72 Voltage Characteristics obtained due to variation in Solder interconnect
resistance due to heath degradation. 160
Figure 73 Damage Detection using a Solder Joint Built in Reliability Test (SJ-BIRT). 161
Figure 74 The Degradation in Confidence Value relative to Damage occurrence in
Assembly. 164
xviii
LIST OF TABLES
Table 1-1 Statistical Pattern Recognition Techniques and Applications. 3
Table 4-1 Kernels applied during Time Frequency analysis [Cohen 1995]. 54
Table 4-2 List of Common Correspondence Functions. 58
Table 5-1 Roots of the transcendental equation for a simply supported-simply
supported plate. 78
Table 5-2 Roots of the transcendental equation of a Free-Free plate. 80
Table 5-3 Values of ?
r
for a free-free plate. 81
Table 5-4 Roots of the transcendental equation of a Clamped-Free plate. 83
Table 5-5 Values of ?
r
for a Clamped-Free plate. 84
Table 5-6 Roots of the transcendental equation of a Clamped-Clamped plate. 86
Table 5-7 Values of ?
r
for a Clamped-Clamped plate. 86
Table 6-1: Test Vehicles. 102
Table 6-2: Test Vehicles. 104
1
CHAPTER 1
INTRODUCTION
Electronic packaging refers to an electromechanical platform which is both
economical and manufacturable and provides protection to the delicate silicon die [Gilleo
2002]. It provides the geometric translations which is required for the compatible
interface between the electronic device and the next system level. The package provides
protection from the external environment, external loads and stress by enclosing the
silicon die in electrically insulative materials and from moisture by hermetically sealed
packages like metal vacuum-sealed packages and gas-impervious ceramic packages.
Some other major functions of electronic packages are heat dissipation, signal
distribution and power distribution. The advances in semiconductor fabrication and
packaging techniques have caused an increase in the interconnect densities of the
packages on the PCB and also an increase in the reliability considerations for the
electronic devices. Damage due to shock and vibration on portable products may
manifest in solder interconnects and copper traces and cause failure of the packages. The
damage may be due to sudden overstress or from cumulative stress.
1.1 Statistical Pattern Recognition
Statistical Pattern Recognition refers to the study of algorithms that recognize
patterns in data. The above research area also contains various sub disciplines like
2
discriminant analysis, feature extraction, error estimation, and cluster analysis. Some
important application areas of statistical pattern recognition are image analysis, character
recognition, speech analysis, man and machine diagnostics, person identification and
industrial inspection. The various methods applied for statistical pattern recognition and
their applications are summarized in Table 1-1.
Though research and development in the field of statistical pattern recognition has
been going on for the past 50 years its application to reliability studies of electronic
assemblies is new. Currently statistical pattern recognition is being employed in a variety
of engineering and scientific disciplines such as biology, psychology, medicine,
marketing, artificial intelligence, computer vision and remote sensing [Jain et.al.2000].
1.2 Health Monitoring
Structural health monitoring, i.e. the process of establishing knowledge of the
current condition of a structure has found application in various fields, like shaft crack
detection [Lebold, et. al. 2004, Gyekenyesi, et. al. 2003] and aircraft maintenance
[Hedley, et. al. 2004, Hickman, et. al. 1991, Castanien, et. al. 1996]. This method also
finds application in performance assessment of Machinery systems [Lee 1995, Chuang,
et. al. 2004, Wegerich 2003]. While, structural health monitoring is popularly used in
various fields, its application to the field of reliability of electronic structures is new. The
relevant features, like vibration, temperature etc. are extracted from strategically placed
sensors on the machine structure, and the algorithms developed for performance
assessment of a system are applied. Experience in other applications indicates that
structural health monitoring produces gains in the performance and cost-effective
3
Table 1-1 Statistical Pattern Recognition Techniques and Applications.
Method for Statistical Pattern
Recognition
Application Reference
Neural Networks
(Probabilistic and Artificial
combined with fuzzy logic)
Fault in Gas Turbine
engines
[Atlas 1996, Sick 1998, Chuang
2004]
Hidden Markov Model
Speech Recognition
Machine Tool Wear
[Litao 2001, Heck 1991]
Multivariate Similarity
Modeling
Machinery Health
Monitoring
[Wegerich 2003]
Auto Regression models
Machinery Health
Monitoring
[Logan 2001, Shao 2000, Lei 2003,
Casoetto 2003, Yan 2004, Engel
2000]
Wavelet Packet Approach Tool wear [Yan 2003]
FFT based frequency domain
analysis
Machine monitoring [Yuan 2004]
Time series methods
(Time-frequency moments)
Machine Tool
Monitoring
[Zheng 1992, Djurdjanovic 2002]
Statistical Data Comparison
(Kurtosis, Crest factor etc.)
Railway Bearing
Diagnostics
[National Research Council Canada
1999]
4
maintenance of high-value assets such as aircrafts, civil infrastructure and maritime
vessels. Structural health monitoring systems help in reducing down-time and eliminating
component teardown inspections, thus reducing the risk of failure during operation.
1.3 Current Testing Techniques
Currently the main reliability tests performed on electronic assemblies undergoing
drop and shock are the JEDEC drop test [Lall, et. al. 2005], Shear testing [Hanabe, et. al.
2004] and ball pull testing [Newman 2005]. The JEDEC drop test is based on the JEDEC
test standards, and studies the affect of drop and shock experimentally on Test boards.
These tests are limited to board level drops with the packages on the board connected in a
daisy chain as the experimental techniques relies on measurement of electrical resistance
for determination of failure. Ball-pull testing and shear testing has also been applied to
test structures to study the reliability of electronic assemblies in drop and shock
environments. These tests quantitatively study the impact toughness of solder joints by
means of various tests including the Charpy test [Date, et. al. 2004], Shear test, and the
package to board interconnection shear strength (PBISS) technique [Hanabe, et. al.
2004]. The above mentioned test procedures cannot monitor the damage occurring during
shock and drop, hence they cannot be used for health monitoring of electronic
assemblies.
1.4 Closed Form Models
The electronic assembly is modeled as a rectangular board with point masses on it,
representing the PCB with the packages attached to it. Various kinds of boundary
conditions have been studied, determined by the packaging of the assembly at the product
5
level. Hence a press fit edge of a PCB is modeled as a clamped edge condition, while a
configuration with screws attaching the PCB to the casing is modeled as a plate having
point supports. In this paper, the JEDEC drop test assembly has been modeled as a
rectangular plate on rigid point supports, with packages being modeled as attached
masses on the PCB. The vibrational frequency and mode shapes have been correlated
with FEM models developed for JEDEC drop testing and also with experimental data
obtained during the tests. Various case studies of failure occurring due to change in
effective attributes of the assembly have been discussed and damage monitoring done to
show damage progression.
6
CHAPTER 2
LITERATURE REVIEW
Thermal loading is generally considered the major cause of failure in electronic
devices. The mismatch in the coefficient of thermal expansions (CTE) of the various
materials in the package causes various types of failures, some of which are solder joint
cracking, chip delamination, and silicon chip cracking. However many of the electronic
devices are subjected to extreme environments where the devices have to sustain high
amounts of vibrations and shock. The U.S. Air Force estimates that vibration and shock
cause 20 percent of the mechanical failures in airborne electronics [Zhao et.al. 2000]. The
nonlinear stress-strain behavior of solder joints under vibration is still not clear, and the
role of vibration in the life of solder joints has not been studied sufficiently. Electronic
assemblies are susceptible to failure due to shock and drop as the electronic products may
be subjected to drop and shock due to mishandling during transportation or during normal
usage. Some specific electronic products such as portable communication and computing
products which contain fine-pitch ball-grid arrays, and quad-flat no-lead packages, are
very susceptible to shock-related impact damage. Electronic assemblies used in military
applications are repetitively subjected to extreme shock due to various factors such as
artillery fire but require reliable functioning even after such high impacts [Lall et.al
2005].
7
2.1 Experimental Techniques
Several experimental tests are performed to study the reliability of electronic
devices that are designed to operate in specific environmental conditions. These tests are
designed and performed to study the effect of the various environmental parameters such
as temperature, humidity, and vibration on the packages. Researchers [Silverman 1998,
Chengalva 2004] have applied various test conditions such as the accelerated thermal
cycling, thermal shock, HAST (highly accelerated stress test) and vibration tests, to
analyze the reliability of the packages for various applications.
The current drop and shock testing of electronic assemblies fall into mainly two
categories, the constrained drop testing and the unconstrained or free drop testing. One of
the test standard defined for the constrained drop testing is the JEDEC test standard
[2003] which has found vast application in the comparative drop performance assessment
of surface mount electronic components found in compact handheld electronic products.
The JEDEC standard test are generally performed at the component level, thus the
primary goal of the JEDEC test is to provide a reproducible assessment of drop
performance by standardizing the test board and the test methodology. The limitation of
the JEDEC testing lays in the correlation of performance at product level with the test
results. The reliability failures in product assemblies are pertinent on various affecting
factors such as the product casing, and the drop orientation which might not be
perpendicular to the board surface [Lim 2002]. The previous research efforts [Tian 2003,
Lall 2004] on constrained drops for edge-based drop orientation have shown repeatable
drops. Unconstrained or free drop testing has been proposed by the use of high speed
8
photography [Goyal 2000] with the limitation of difficulty in getting repeatable drops due
to edge clattering i.e. one corner of the product touches the ground first and the other
corner rebounds repeatedly.
Both the constrained and unconstrained testing of the electronic assemblies requires
the packages on the test board to be connected in a daisy chain so as to facilitate failure
monitoring. These tests are limited to board level drops with the packages on the board
connected in a daisy chain as the experimental techniques relies on measurement of
electrical resistance for determination of failure
Mechanical tests performed to test solder joints are the shear test, the pull test and
the peel-off test [Nishiura et.al 2002, Jeon et.al. 2002]. These tests are utilized to perform
bulk or bond testing of the joints as compared to impact testing. The test speeds for these
tests are less than 10mm/s, which is very low when compared to the strain rates applied
to solder joints during drop and impact.
Another test suggested and applied to the mechanical testing of solder joints during
shock and drop is the miniature Charpy test [Morita et.al. 2002] where the shear rate is
approximately 1m/s [Date et.al. 2004]. The quantification of the shear strength and the
effect of pad finish in Chip Scale Packages is performed by Shear testing [Canamulla
et.al. 2003] and ball pull testing [Newman 2005] performed at a strain rate of 100?m/sec.
As the strain rate in shear tests are low, the shear strength behavior shown by solders due
to slow deformation is not applicable to the study of electronics in shock and drop
[Hanabe et.al. 2004].
9
The above mentioned test procedures cannot monitor the damage occurring during
shock and drop, hence they cannot be used for health monitoring of electronic
assemblies.
2.2 Statistical Pattern Recognition
Statistical Pattern Recognition refers to the study of algorithms that recognize
patterns in data and contains various sub disciplines like discriminant analysis, feature
extraction, error estimation, and cluster analysis. Some important application areas of
statistical pattern recognition are image analysis, character recognition, speech analysis,
man and machine diagnostics, person identification and industrial inspection.
Statistical pattern recognition has been developed using several methods and
applied to a plethora of applications including neural networks applied to faults in gas
turbine engines [Atlas 1996, Sick 1998, Chuang 2004], Hidden Markov models applied
to speech recognition and machine tool wear [Wang 2002, Heck 1991], multivariate
similarity models applied to machine health monitoring [Wegerich 2003], auto-regression
models applied to machine health monitoring [Logan 2003, Shao 2000, Lei 2003,
Casoetto 2003, Yan 2005, Engel 2000], wavelet packet approach applied to tool wear
[Yan 2004], FFT based frequency-domain analysis applied to machine monitoring [Yuan
2004], time-series methods applied to machine tool monitoring [Zheng 1992,
Djurdjanovic 2002], and statistical data comparison applied to railway bearing
diagnostics [National Research Council Canada 1999]. Application of statistical pattern
recognition to health monitoring of electronic assemblies subjected to shock and
vibration environments is new.
10
2.3 Closed- Form Analytical Models
A closed-form modeling approach has been used to analyze the dynamic behavior
of printed circuit assembly with component masses. The Lagrangian Functional for a
rectangular printed-circuit assembly is,
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defined by, a which is the length of the PCB, and b which is the width of the PCB. The
distributed loads on the PCB is represented by q, ? is the density of the PCB, and I
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Work on the analytical solution for the free vibration of rectangular plates having
various boundary conditions can be found easily in previous publications [Leissa 1969,
Young 1950]. There are various techniques that have been utilized during the solution of
the governing differential equation of a rectangular plate,
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11
The first comprehensive collection of solutions for rectangular plates was
presented by Warburton [Warburton 1954, Leissa 1969, ]. The Rayleigh method was
used to analyze the deflection and frequency data for the free vibration of rectangular
plates. The deflection function was defined as the product of the beam deflection
functions, where the beam
functions represented the fundamental mode shapes of the beams having the boundary
conditions of the plate.
)y(Y)x(X)y,x(W =
The above method satisfies all the boundary conditions for the plate, except the
free edge condition, where the shear condition on the plate is approximately satisfied.
Janich [Leissa 1969] also suggested a comprehensive set of solutions for the free
vibration of rectangular plates having different boundary conditions. He obtained the
fundamental frequencies of vibration for 18 sets of boundary conditions. The method
suggested also used the Rayleigh Ritz solution and utilized simple trigonometric
functions to represent the beam functions. These two methods yield the upper limits of
the frequency and mode shape values, but are not very suitable to study the vibrations of
plates with certain boundary conditions like a completely free plate. The results for
higher mode shapes also decrease in accuracy with the mode number.
Several other studies and methods have been developed for the analytical
solutions of rectangular plates based on the specific boundary conditions of the plate.
Some of the previous solutions specifically solved particular boundary conditions.
Gorman also developed a solution for the complete set of boundary conditions and
12
developed the displacement functionals based on the antisymmetric and the symmetric
modes of the rectangular plate [Gorman 1982].
The Rayleigh-Ritz method was also employed by Young to solve the free
vibration of rectangular plates with various boundary conditions [Young 1950]. The
study also provides the upper limit of the vibration frequencies of a rectangular plate but
yields satisfactory results for studying various problems in equilibrium, buckling and
vibration. This study employs the Rayleigh Ritz method to solve for the vibrational
frequencies and modeshapes of an electronic assembly.
The electronic components placed on the PCB surface have been modeled as
point masses on a rectangular plate while developing the closed form analytical models.
The normal vibrations of a rectangular plate with point masses attached on the surface
has been studied in various works [Das et.al. 1963, Chintakindi 1964, Shah et.al. 1969,
G?rg?ze 1984, and Ingber et.al. 1992] but have focused on simply supported plates and
beams. The representation of electronic assemblies as rectangular plates with point
masses attached is new. The analytical solution developed using the Rayleigh Ritz
method for the free vibration of rectangular plates can be modified to obtain the vibration
frequencies and modeshapes of plates with attached masses [Wu et.al. 1997]. This
method allows the use of previously developed models for the free vibration of free
rectangular plates having various boundary conditions and hence is applicable to all the
developed boundary conditions.
13
CHAPTER 3
STATISTICAL PATTERN RECOGNITION
Statistical Pattern Recognition is defined as a set of algorithms that recognize
patterns in data and has found application in the areas of image analysis, character
recognition, speech analysis, man and machine diagnostics, person identification and
industrial inspection. The application of Statistical pattern recognition to the study of
prognostics and damage monitoring in electronic assemblies undergoing a shock and
drop event is new though several methods have been developed and applied to various
applications.
In this study statistical pattern recognition is used to study the degradation of
reliability in electronic assemblies, due to shock and drop. The health monitoring of
assemblies has been accomplished by monitoring the confidence values computed by
applying statistical pattern recognition techniques to the transient-strain response,
transient displacement-response, vibration mode shapes and frequencies of the electronic
assembly under shock and drop. Correlation of structural response, damage proxies and
underlying damage has been accomplished with closed-form models, explicit finite
element models and validated with high-speed experimental data. In this chapter, two
14
statistical pattern recognition techniques, including the wavelet packet approach and the
Mahalanobis distance approach have been investigated.
3.1 Wavelet Transforms
Wavelets have been used in several areas including data and image processing
[Martin 2001], geophysics [Kumar 1994], power signal studies [Santoso 1996],
meteorological studies [Lau 1995], speech recognition [Favero 1994], medicine [Akay
1997], and motor vibration [Fu 2003, Yen 1999]. Wavelets based time-frequency analysis
is specifically useful to analyze non-stationary signals.
The wavelet transform is defined by
dt
s
ut
*
?f(t)
s
1
su,
?f,s)Wf(u, ?
?
?
?
?
? ?
?
+?
??
==
where the base atom ?* is the complex conjugate of the wavelet function which is a zero
average function, centered around zero with a finite energy. The function f(t) is termed as
the mother wavelet and is decomposed into a set of basis functions called the wavelets
with the variables s and u, representing the scale and translation factors respectively. The
original signal is first passed through a half-band highpass filter g[n] and a lowpass filter
h[n]. After the filtering, half of the samples are eliminated according to the Nyquist?s
rule, since the signal now has a highest frequency of p/2 radians instead of p. The signal
is therefore sub-sampled by 2, simply by discarding every other sample.
15
This constitutes one level of decomposition and can mathematically be expressed as
follows:
?
?
??=
??=
n
low
n
high
n]h[2ksignal[n][k]y
n]g[2ksignal[n][k]y
where y
high
[k] and y
low
[k] are the outputs of the highpass and lowpass filters, respectively,
after subsampling by 2. However, the number of average number of data points out of
the filter bank is the same as the number input, because the number is doubled by having
two filters. Thus, no information is lost in the process and it is possible to completely
recover the original signal. Aliasing occurring in one filter bank can be completely
undone by using signal from the second bank. Further, the time resolution after the
decomposition halves as the sub-sampling occurs. However this sub-sampling doubles
the frequency resolution, as after decomposition the frequency band of the signal spans
half the previous frequency band, effectively reducing the uncertainty in the frequency by
half. At every level, the filtering and subsampling will result in half the number of
samples (and hence half the time resolution) and half the frequency band spanned (and
hence doubles the frequency resolution).The frequencies that are most prominent in the
original signal will appear as high amplitudes in that region of the Wavelet transform
signal that includes those particular frequencies. The time localization will have a
resolution that depends on which level they appear. The wavelet decomposition of the
signal is shown below in Figure 1.
16
Signal
Approximation 1 Detail 1
Approximation 2 Detail 2
Detail N Approximation N
Figure 1 A N
th
Level Wavelet Decomposition Structure.
3.1.1 Daubechies Wavelet
The orthonormal expansion was developed to improve on the performance of the
Fourier expansion and other classical expansions. The Fourier series expansion is not
well localized in space and the Haar series used in the Haar wavelets is very well
localized and hence limits the observation of the signal behavior in a given time interval.
If the mother wavelet used in the wavelet transforms forms an orthonormal basis
in )(L
2
? , then the mother wavelet is capable of generating any function in
)(L
2
? [Benedetto et.al. 1994].
In the wavelet analysis performed in this study, the Daubechies wavelet has been
chosen for analysis of transient dynamic signals mainly based on resemblance of the
wavelet with the true signal. The Daubechies-wavelets are defined two functions, i.e. the
scaling function ?(x), and the wavelet function ?(x). The Daubechies wavelet algorithm
17
uses overlapping windows, so the high frequency spectrum reflects all changes in the
time series. Daubechies wavelet shifts its window by two elements at each step.
However, the average and difference are calculated over four elements, so there are no
"holes" unlike other wavelet transforms such as the Haar transform, which use a window
which is two elements wide. With a two element wide window, if a big change takes
place from an even value to an odd value, the change will not be reflected in the high
frequency coefficients. The scaling function is the solution of the dilation equation,
?
?
=
?(2?=)(?
1L
0u
)ut)u(h2t
where h(u) are a sequence of real or complex numbers called the scaling function
coefficients, 2 represents the normalization of the scaling function with a scale of two,
)t(? is normalized
?
?
??
=? 1dt)t( . The wavelet )t(? ?is defined in terms of the scaling
function,
?
?
=
?(2?=?
1L
0u
)ut)u(g2)t(
where the coefficients g(u) defines the scaling function. Building on the orthonormal
basis from )t(? and )t(? ?by dilating and translating, the following functions are
obtained,
)ut2)t(
s
2
s
s
u
?(2?=?
18
)ut2(2)t(
and
s
2
s
s
u
??=?
where s is the dilation parameter and u is the translation parameter. The variables
)u(gand)u(h are the filter coefficients of the Quadrature mirror filters H and G
respectively [Walnut 2002]. The QMF H and G must satisfy the following properties.
I
*
GG
*
HH
and
0
*
GH
*
HG
==
==
where I is an identity matrix. To obtain QFM, the matrix computed using the filter
coefficients h (u) and g (u) shown below must have a unitary value for a value of R??
[Daubechies 1988].
?
?
?
?
?
?
?
?
)?+??
)?+??
(m)(m
(m)(m
11
00
?
?
?
=
?
?
=
?
=?
=?
1L
0u
in
1
1L
0u
in
0
e]u[g)(m
and
e]u[h)(m
where
The advantages of the Daubechies wavelet are attributed to the property of
vanishing moments exhibited by the Daubechies wavelet basis. The limitation of the Haar
wavelet is attributed to the presence of jump discontinuities in the wavelets which cause
19
the Haar coefficients to decay poorly for smooth functions and also give poor signal
reconstruction. All the wavelet functions satisfy the condition
?
=?
R
0dx)x(
which represent the zeroth moment of )x(? , and hence this states that the zeroth moment
is vanishing for all the wavelets.
The m
th
moment of )x(? is given by
?
<>??1<<)?,?(? for
In this study the binomial time-frequency kernel proposed by [Jeong et.al. 1992 .] has
been applied to study the drop and shock characteristics of an electronic assembly. The
kernel proposed satisfies various kinds of conditions such as:
1 The time frequency distribution obtained is not negative and real in value.
2 The RID kernel does not depend on time and frequency and some of the other
properties satisfied the kernel are
?=)?(?
?=),?(?
)?,????=)?,?(?
?
of valuesall for1,0
of valuesall for10
(
3 The time and frequency marginals are satisfied and the energy of the signal is exactly
represented by the distribution.
The binomial time-frequency Distribution defined by [Jeong et.al. 1992] is defined as
????
?+=?
??=?
?
?
?=?
??=?
??
)???+)?+?+
?
?
?
?
?
?
?
?
?+?
?
?
)?=?
4i
2
en(fn(f
2
2
)(g
(h),n(TFR
where )?)? g( and (h is the frequency smoothing window and the time smoothing
window respectively and )n(f represents the signal where N2,1n L= . The term ??=? ,
and here is used to define the RID kernel as the RID kernel constraint is that 0>>?? .
The frequency smoothing window )?(h and the time smoothing window )?g( used here
is a hamming window of size(N) as outlined in [Jeong et.al. 1992]. The binomial
distribution provides an efficient and fast computation method for computing Time
Frequency distributions. The kernel values in the binomial kernel are based on the
57
binomial expansion and a time saving and recursive application is applied by convolving
kernel with the autocorrelation using shifts and add-ins [Williams 1996].
4.5 Time Frequency Moments
The Time frequency moments are used in our study to represent the time frequency
distribution of the transient signals obtained during the shock and drop testing of
electronic assemblies. The time frequency moments are calculated from
dt)t(f)W,(C)t(ft
nm
*mn
?=?
?
where )W,T(C
nm
represents a correspondence term between the time frequency
moments and the signal. The value of )W,T(C
nm
is obtained from the formula
0=?,?
?+?
+
)?,??
????
?
=
WjTj
m
mn
mn
nm
e(
jj
1
)W,T(C
While calculating the moments some of the common correspondence terms )W,T(C
nm
is
shown below in Table 4-2 [Cohen1995]
58
Table 4-2 List of Common Correspondence Functions.
Type
)W,T(C
mn
Nonmixed signals
0n when W0m when T
mn
==
Normal
WT
mn
Symmetrization
{ }
nmmn
TWWT
2
1
+
where
lnlm
)n,mmin(
0l
lmn
lmln
)n,mmin(
0l
lnm
TW
l
m
l
n
!l)i(WT
WT
l
m
l
n
!l)i(TW
??
=
??
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?=
?
?
In this study the individual time moments and frequency moments of the signal are
computed and used as a feature vector to study the damage progression in electronics in a
shock and drop event. The above approach has also been applied to the transient-strain
response, the transient -displacement response, vibration modeshapes and frequencies of
the electronic assembly under drop and shock. The time frequency distribution obtained
for a JEDEC standard horizontal drop of an electronic assembly is shown in Figure 19.
The moment feature vectors obtained for the transient strain signal are also shown in
Figure 20 and Figure 21 respectively. The transient strain signal (Figure 14) obtained
from the strain sensors placed on the electronic assembly while performing drop and
shock testing, as explained in section 6.2 has been used to obtain the Time Frequency
moment signature.
59
Figure 19 Time Frequency Distribution for a Transient Strain signal.
60
Time Moment Feature Vector
-2000
-1000
0
1000
2000
3000
4000
5000
0.000 0.002 0.004 0.006 0.008 0.010
Time (sec)
Instantaneous Frequency (Hz)
Figure 20 Time Moment Feature Vector for a Transient Strain Signal.
Frequency Moment Feature Vector
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 200 400 600 800 1000
Frequency (Hz)
Instantaneous Time (sec)
Figure 21 Frequency Moment Feature Vector for a Transient Strain Signal.
61
4.6 Confidence Value Computation
The confidence value of an electronic assembly as described in this study defines the
state of reliability of an electronic system in a shock and drop environment. Statistical
pattern recognition has been applied to the transient-strain response, the transient-
displacement response, vibration modeshapes and frequencies of the electronic assembly
under drop and shock. The feature vectors obtained by the various signal processing
techniques, Wavelet Energy signature using the wavelet packet transform, Mahalanobis
Distance vector using the Mahalanobis distance computation, FFT Frequency Band
Energy signature using the Fast Fourier transform and the Time Frequency Moments
using Time Frequency analysis are used to determine the health of an electronic
assembly.
4.6.1 Testing Hypothesis
The statistical hypothesis is defined as an assumption made about a parameter of a
given statistical population. The truth of an assumed hypothesis is verified by performing
a statistical test on the population. The probability of the occurrence of the event assumed
in the hypothesis is calculated and if the probability is above a certain significance level
then the hypothesis is considered to be true. The hypothesis assumed in this study is that
the means of the two populations being compared are identical, i.e.
0
2
?
1
?:
a
H
0
2
?
1
?:
0
H
??
=?
The distribution of the assumed hypothesis, i.e. the Null Hypothesis is studied and
a statistical test is performed to check whether the Null hypothesis might be rejected in
62
favor of the alternative hypothesis. There are two different types of tests that can be
applied while verifying the hypothesis, the one-tailed test and the two-tailed test. The
choice between one sided or two-sided test depends on the alternative hypothesis
assumed to the null hypothesis. The one sided p ?value is the measure of the evidence
against the null hypothesis, 0
2
?
1
? =? . If the alternative hypothesis is limited to only
one direction of possible inequality i.e. either
21
?>? or
21
?