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
?
<<?
R
m
Nm0wheredx)x(x
It has been shown that the greater the number of vanishing moments in a wavelet, the
smoother is the wavelet [Walnut 2002]. The presence of a higher number of vanishing
moments causes the wavelet functions to have fewer wavelet coefficients, as functions
which are smooth require few wavelet coefficients to represent them accurately as
compared to a non smooth function. A Daubechies wavelet of order N has N vanishing
moments and is supported on the interval [0, 2N-1]. As the vanishing moments for the
Daubechies wavelets are large in number hence they can better represent signals and do
not suffer from jump discontinuities. The Daubechies wavelets show regularity which
ensures the smoothness of the wavelet and the wavelet is compactly supported in [0,?N-
1], i.e. )t(? is zero outside [0,?N-1]. The Daubechies Wavelet also satisfies the
admissibility condition which implies that the mean of the wavelet is zero and hence it
represents oscillatory motion. The basic shape of a Daubechies wavelet (DB 6) used in
this study and its scaling function has been shown below in Figure 2 and Figure 3
respectively.
20
Figure 2 The Daubechies-6 Wavelet.
Figure 3 The Scaling Function of a Daubechies 6 Wavelet.
21
The strain signal obtained from the impact drop test has a sampling time of 0.2 to
0.4 s, i.e., a sampling frequency of 2.5 to 5 MHz. For the 2.5 MHz test data, in order to
avoid aliasing during our analysis, we perform our transforms and calculations using the
Nyquist frequency, which is half of the sampling frequency, i.e. 1.25 MHz. The Low-
Pass and High-Pass filters used during the transform have a frequency response shown in
Figure 4 and Figure 5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1500
-1000
-500
0
Normalized Frequency (?? rad/sample)
P
h
ase (
d
egr
ees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-300
-200
-100
0
100
Normalized Frequency (?? rad/sample)
M
a
gni
t
ude (
dB
)
Figure 4 Frequency Response of the Low-Pass Filter.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-600
-400
-200
0
Normalized Frequency (?? rad/sample)
P
h
ase (
d
egr
ees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-300
-200
-100
0
100
Normalized Frequency (?? rad/sample)
M
a
gni
t
ude (
dB
)
Figure 5 Frequency Response of the High-Pass Filter.
22
3.2 Wavelet Packet Approach
The application of a pair of lowpass and highpass filters corresponds to
decimating a signal into a low frequency approximation component and a high frequency
detail component. In wavelet decomposition the low frequency component is further
decimated while the high frequency component is not filtered. This causes some high
frequency data loss and hence wavelet packets have been applied in this study. In wavelet
packet decomposition we can decimate both the high frequency and the low frequency
component. The wavelet filters are applied to the high frequency part at each level and
the component is also decimated into a high frequency part and a low frequency part.
This allows better frequency localization of the signals as the frequency space is divided
into more number of tiles.
The transform has been used to analyze transient strains signals at different
frequency bands with different resolutions by decomposing the transient signal into a
coarse approximation and detail information. The signal is decimated into different
frequency bands by successively filtering the time domain signal using lowpass and
highpass filters. The original stress signal is first passed through a halfband 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 decimation structure of a signal based on a wavelet packet
approach has been outlined in Figure 6.
23
Signal
A
1
D
1
AA
2
DA
2
AD
2
DD
2
AAA
3
DAA
3
ADA
3
DDA
3
AAD
3
DAD
3
ADD
3
DDD
3
Figure 6 Wavelet Packet decomposition structure for Level three decomposition.
24
The signal is therefore subsampled by 2, simply by discarding every other sample.
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. The compression and de-noising in the wavelet packet transform
is same as those for a wavelet transform framework. The only difference is that wavelet
packets offer a more complex and flexible analysis, because in wavelet packet analysis,
the details as well as the approximations are split.
Wavelets have been shown to be useful for texture classification because of their
finite length [Lee, et. al. 1995]. If an orthonormal basis is chosen, the wavelet coefficients
are independent and possess the distinct features of the original signal. Wavelet packets
can be described by the following collection of basis functions,
() ( )
() ()? ??
?
=
?
?
?
?
?
?
?
?
+
? ??
?
=
?
?
?
?
?
?
?
?
m
mx
p
2
n
W
p
22lmg
pl
2lx
1p
2
12n
W
m
mx
p
2
n
W
p
22lmh
pl
2lx
1p
2
2n
W
where p is a scale index, l is a translation index, ?h? is a lowpass filter, and ?g? is a high-
pass filter with k)h(l
k
1)(g(k) ??= . The function (x)
o
W can be identified as the low-
pass scaling function ? and (x)
l
W as the high-pass mother wavelet ?. A 2-D wavelet
packet basis function is given by the product of two 1-D wavelet packet basis along the
horizontal and vertical direction.
25
The corresponding 2-D filter coefficients have four groups,
g(k)g(l)l)gg(k,
g(k)h(l)l)gh(k,
h(k)g(l)l)hg(k,
h(k)h(l)l)hh(k,
=
=
=
=
Wavelet packet basis functions have the properties of smoothness, number of
vanishing moments, symmetry, good time and frequency localization, satisfy the
admissibility condition and are absolutely square integrable functions. The discrete
wavelets can be classified as non-orthogonal, biorthogonal or orthogonal wavelets. Non-
orthogonal wavelets are linearly dependent and redundant frames. Orthogonal wavelets
are linearly independent, complete and orthogonal. In the analysis of reliability of
electronic assemblies, one of the most widely used wavelets constructed by Daubechies
has been applied. The Daubechies wavelets are orthonormal, compactly supported, have
maximum number of vanishing moments, and are reasonably smooth. The low-pass and
band (high)?pass filter coefficients of the Daubechies wavelets satisfy the conditions of
Orthogonality, Normality and Regularity.
An entropy-based criterion is used to select the most suitable decomposition of a
given signal. This implies that at each node of the decomposition tree, the information
gained by performing a split is quantified. Simple and efficient algorithms exist for both
wavelet packet decomposition and optimal decomposition selection. Adaptive filtering
algorithms, allow the Wavelet Packet transform to include the "Best Level" and "Best
Tree" features that optimize the decomposition both globally and with respect to each
node.
26
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
024681012
ti
-1.5
-1
-0.5
0
0.5
1
1.5
024681012
-1.5
-1
-0.5
0
0.5
1
1.5
2
024681012
-1.5
-1
-0.5
0
0.5
1
1.5
024681012
-2
-1.5
-1
-0.5
0
0.5
1
1.5
024681012
-1.5
-1
-0.5
0
0.5
1
1.5
024681012
Figure 7 First six wavelet packets for a DB6 filter packet decomposition.
27
For obtaining the optimal tree, a node is split into two nodes, if and only if the
sum of entropy of the two nodes is lower than the sum of entropy of the initial node.
After the wavelet packet transform, the wavelet packet energy is calculated at each node
of the decomposition tree. An energy signature E
n
for each sequence of wavelet packet
coefficients
p
kn,
C for
1p
....40,1,2.....n
?
= can be computed by using the formula
?
=
=
N
1k
2
p
n
C
2
N
1
n
E
where N is the total number of points in the signal at a given node in the wavelet packet
tree, p is the decomposition depth, and C
i
is the wavelet packet coefficients obtained
during the wavelet packet transform at the particular node where energy is being
calculated. The feature vector of length
1p
4
?
, is formed for the signal.
The packet energies obtained from the wavelet packet decomposition of the
various mode shapes and frequencies of vibration of the electronic assembly are the basis
for the computation of confidence values for health monitoring. The Strain signal
obtained from sensors placed on the electronic assembly as shown in Figure 8 is used to
obtain the wavelet packet energy signature. The wavelet packet energy signature for a
transient strain signal is shown in Figure 9.
28
Figure 8 Transient Strain-History at Location of CSP during Drop-Event.
29
Figure 9 Wavelet Packet Energy Feature Vector.
30
3.3 Distance Based Similarity
The application of algorithms that automatically map data points known as the
feature vectors to compute the degree of similarity between two objects is termed as a
Similarity measure. The application of distance-based similarity utilizes the relative
distance among data objects to perform similarity measures between the two given data
objects. It has found application in various application such as Face recognition [Fraser
et.al. 2003, Kamei 2002], Image analysis [Kato et.al. 1999, Kokare et.al. 2003], and
Medical signals [Babiloni et.al. 2001, Momenan et.al. 1994]. The approach creates a data
structure that allows the scalable retrieval of complex data required for patter recognition.
The distance based similarity approach requires the distance function to be metric, hence
requires the distance to satisfy the metric conditions.
)Z,Y(D)Y,X(D)Z,D(X:Inequality Triangle
)X,Y(D)Y,X(D:lSymmetrica
0)Y,X(D:Value Positive
nnnnnn
nnnn
nn
+?
=
?
3.3.1 Euclidean Distance
The mathematical definition of the Euclidean distance states that the distance
measure between any two points that would be obtained by measuring the length of the
straight line joining the two points is the Euclidean distance. Considering two points in a
n-dimensional metric space, the Euclidean distance is given by:
()() ( ) )yx()yx(yxyxyx)Y,X(D
t2
nn
2
22
2
11nn
??=?+?+?= LL
The Euclidean distance represents a spheroid whose centre is
n
X and has radius
)Y,X(D
nn
and hence while computing the distance measure equal contribution of each
31
coordinate is assumed. However for statistical pattern recognition, considering the
variability of each coordinates is desired while computing the distance between two data
objects. The distance computation desired requires applying less weight to high
variability components and high weight to low variability components.
3.3.2 Mahalanobis Distance Approach
The Mahalanobis Distance classification is similar to the Maximum Likelihood
classification except for the class covariances which are all assumed to be equal, hence
the method is more efficient [Babiloni 2001]. It is based on correlations between
variables by which different patterns can be identified and analyzed. It is a useful way of
determining similarity of an unknown sample set to a known one. It differs from
Euclidean distance in that it takes into account the correlations of the data set. The
Mahalanobis distance from a group of values with mean ( )
n4321
,,,, ?????=? K and
covariance matrix ? for a multivariate vector ( )
n4321
x,x,x,x,xx K= is,
()()?????=
?
xx)x(D
1T
M
Mahalanobis distance can also be defined as dissimilarity measure between two random
vectors x
r
and y
r
of the same distribution with the covariance matrix ? ,
()()yxyx)y,x(d
1T
rrrrrr
???=
?
The Mahalanobis distance approach has been chosen over other classification approaches
as it considers the variance and covariance of the variables as opposed to only the
average value. The distance measure is taken as a basis for the calculation of the
32
confidence values for prognostics. The wavelet packet energy signature for a transient
strain signal is shown in Figure 10.
Mahalanobis Distance Feature
0
5
10
15
20
25
-0.02 0 0.02 0.04 0.06 0.08 0.1
Time (seconds)
Ma
ha
la
no
bis
D
i
s
t
a
n
c
e
Figure 10 Mahalanobis distance Feature Vector.
33
CHAPTER 4
THE FAST FOURIER TRANSFORM & TIME FREQUENCY ANALYSIS
The feature extraction techniques utilized for statistical pattern recognition outlined
in this chapter are the Fast Fourier Transform and the Time Frequency Analysis. The
mathematics and the various advantages and disadvantages of using these methods have
been discussed to help understand the applicability of these methods to perform health
monitoring of the transient shock and drop characteristic of electronic assemblies. The
Fourier transform decomposes the signal by superimposing weighted sinusoidal functions
and measure the similarity of the signal to the sinusoidal basis while a Time Frequency
analysis provides the exact behavior of the frequency content and its variation over time.
The confidence value for damage monitoring and the hypothesis applied for the
computation has been outlined.
4.1 Fourier Tansforms
The Fourier transform is widely used signal-analysis tools in real-time signal analysis and
has found application in several areas such as speech recognition [New et.al. 2003, Polur
et.al. 2005, Prasanthi et.al. 2005, Sakurai et.al. 1984, Tin et.al. 2005, Wang et.al. 1996],
biomedical signals [ Cesarelli et.al. 1990, Clayton et.al. 1993, Cote et. al. 1988, Pannizzo
et.al. 1988] , image processing [Ahlvers et.al. 2003, Feihong 1990, Uzun et.al. 2003] , in
solving differential and integral mathematical equations [Branick 2004, Helms 1967,
34
Olejniczak et.al. 1990, Qing et.al. 2000], geology [Axelsson 1997, Liu et.al. 1988], and
astronomy [Kulkarni 1995] . The basic objective of a Fourier transform is to decompose
the signal by superimposing weighted sinusoidal functions and measure similarity of the
signal to the sinusoidal basis. This causes the frequency attributes of the signal
transformed to be exactly described. The Fast Fourier transform is an efficient algorithm
to compute the Discrete Fourier transform (DFT), and reduces the computation time for
an N size signal from )
2
?(N to (N))
2
?(Nlog [Chu 2000].The FFT algorithms are based on
two classes of decimations techniques, the decimation in time and the decimation in
frequency.
4.1.1 Discrete Fourier Transform
A discrete Fourier transform is given by the matrix-vector product
1N0,1,n
1N
0k
kn
N
?
k
f)
n
F(? ?=?
?
=
= LL
and the inverse Fourier transform is defined as
1N0,1,k
1N
0n
kn
N
)?
n
F(?
N
1
k
f ?=?
?
=
?
= LL
The transient strain signal being processed is represented by
k
f, and
kn
N
? is called the
twiddle factor where
N
? is the Nth root of unity and defined as
)N/2sin(i)N/2cos(
i/N2-
e
N
? ???=
?
=
where 1i ?= .
35
The Discrete Fourier Transform requires the computation of the product of a
matrix, called the DFT matrix where the matrix components are the twiddle factors with
the signal being transformed represented as a vector.
Hence the DFT can be written as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? 1N
f
3
f
2
f
1
f
1)1)(N(N
N
?
1)2(N
N
?
1N
N
?1
1)2(N
N
?
4
N
?
2
N
?1
1N
N
?
2
N
?
1
N
?1
1111
1N
F
2
F
1
F
0
F
M
L
MMMMM
L
L
L
M
Though the DFT can be applied to any kind of real or complex signal, the computation
time required when processing large size signals is a limitation. The various algorithms
available to perform the FFT are the Cooley-Tukey FFT algorithm, Prime-factor FFT
algorithm, Bruun's FFT algorithm, Rader's FFT algorithm, and Bluestein's FFT algorithm
to name a few.
4.1.2 The Radix-2 FFT Algorithm
The Cooley-Tukey algorithm arranges the input data in bit-reversed sorted order
and builds the output transform using decimation in time technique. The Decimation in
time FFT is computed by splitting the data over the odd and even index k of the input
signal.
1N,1,0n
)k2(n
N
12N
ok
1k2
f
n
N
)k2(n
N
12N
ok
k2
f)
n
(F ?=??
?
=
+
?+??
?
=
=? LL
36
The decimation in frequency FFT is obtained by decimating the frequency spectrum data
into even-indexed and odd-indexed sets. The decimation in frequency algorithm is
applied when the data is in the frequency domain. As the data obtained by the drop and
shock testing of electronic assemblies is in the time domain hence the decimation in time
FFT is applied. The basic methodology and the structure for the DIT FFT is shown in
Figure 11.
After splitting the data to the nth level where N=2
n
, the frequency spectra of the 1
sample time domain signal is calculated. As the frequency spectra of a one sample signal
is represented by the signal itself, hence the N signals obtained after decimation now
represent the frequency spectrum of the signal. To compute the FFT the N frequency
spectra obtained is combined in the exact reverse order in which the decimation in time
occurred. The individual spectra obtained are combined using the FFT butterfly as shown
in Figure 12.
37
Signal
Sample Size= N=2
n
First Level Decimation
Even indexed terms(E
1
)
Sample Size =N/2
First Level Decimation
Odd indexed terms(O
1
)
Sample Size =N/2
Second Level Decimation
Even indexed terms(EE
2
)
Sample Size= N/4
Second Level Decimation
Odd indexed terms(OE
2
)
Sample Size =N/4
Second Level Decimation
Even indexed terms(EO
2
)
Sample Size= N/4
Second Level Decimation
Odd indexed terms(OO
2
)
Sample Size =N/4
n Level Decimation
Even indexed term (E?E
n
)
Sample Size= 1
n Level Decimation
Odd indexed term (O?.E
n
)
Sample Size= 1
n Level Decimation
Even indexed term (E?O
n
)
Sample Size= 1
n Level Decimation
Odd indexed term (O?.O
n
)
Sample Size= 1
Figure 11 Fast Fourier Transform decimation based on Decimation in time algorithm.
38
The mathematical formulation of the combination of these individual spectra can be
understood by looking at the basic FFT transform where
1N,1,0n
)k2(n
N
12N
ok
1k2
f
n
N
)k2(n
N
12N
ok
k2
f)
n
(F ?=??
?
=
+
?+??
?
=
=? LL
Using the mathematical properties of the twiddle factors the equation can be represented
as
1N,1,0n
nk
2/N
12N
ok
1k2
f
n
N
nk
2/N
12N
ok
k2
f)
n
(F ?=??
?
=
+
?+??
?
=
=? LL
as
2
N
2
N
?=? .
The computational time for the FFT is further reduced as the DFT sums needed are
computed for only half the values of n, i.e. 12/N,1,0n ?= LL , as
k)
2
N
n(
2
N
nk
2
N
+
?=?
and
n
N
)
2
N
n(
N
??=
+
? .
Representing the two individual DFTs as
12N,1,0nfO
12N,1,0nfE
nk
2/N
12N
ok
1k2n
nk
2/N
12N
ok
k2n
?=?=
?=?=
?
?
?
=
+
?
=
LL
LL
39
The FFT of the signal for 12/N,1,0n ?= LL , is given by
12N,1,0nOE)(F
n
n
Nnn
?=?+=? LL
The FFT for the signal terms from 1,......2,12 ??= NNNn is given by as
12N,1,0nff)(F
k)
2
N
n(
2/N
12N
ok
1k2
)
2
N
n(
N
k)
2
N
n(
2/N
12N
ok
k2
2
N
n
?=??+?=?
+
?
=
+
++
?
=
+
??
LL
Using the twiddle factor properties,
k)
2
N
n(
2
N
nk
2
N
+
?=? and
n
N
)
2
N
n(
N
??=?
+
the transform is given by
12N,1,0nff)(F
nk
2/N
12N
ok
1k2
n
N
nk
2/N
12N
ok
k2
2
N
n
?=????=?
??
?
=
+
?
=
+
LL
Substituting the values of the two DFT , the transform is given as
12N,1,0nOE)(F
n
n
Nn
2
N
n
?=??=?
+
LL
The basic methodology of the combination of the decimated signal is shown in
Figure 13 for a two point frequency spectrum combined to produce a four point
frequency spectrum [Chu 2000].
40
O
n
+
-
n
n
N
n
2
N
n
OE)(F ??=?
+
n
n
N
nn
OE)(F ?+=?
n
N
??
E
n
++
-
Figure 12 A basic FFT butterfly structure used to combine the decimated signal to obtain
the frequency spectrum.
Even indexed Two point
Frequency Spectrum
Odd indexed Two point
Frequency Spectrum
Four point Frequency Spectrum
-
+
n
N
??
+
n
N
??
-- -
Figure 13 Example of the Structural Combination of the decimated signal to produce the
frequency spectrum.
41
4.1.3 FFT Frequency Bands
Fourier transforms have been shown to be useful while studying the health of
machinery and other vibration data. The fast fourier transform is applied to the signal and
the signal data is transformed to the frequency domain. The energy of a signal in a given
frequency range (f
1
,f
2
) is defined by using frequency bands
n
2
)n21
d(F),(E
2
1
??=??
?
?
?
where )(F
n
? is the FFT of the time domain signal [ Marple 1987].
The use of FFT Frequency Bands based on Fourier signal analysis is a common
way to express and visualize the frequency content of a signal. By utilizing various
different frequency ranges the frequency band energies can be calculated and the vector
of the energies obtained is used as a feature vector to perform prognostics and damage
monitoring. The FFT frequency band energies obtained from the Fourier transform of the
various modeshapes and frequencies of vibration of the electronic assembly are the basis
for the computation of confidence values for health monitoring. The FFT frequency band
energies are the feature vector used to study the transient characteristics of the drop and
shock damage occurring in electronic assemblies. 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 strain
signal and its energy signature are shown in Figure 14 and Figure 15 respectively. The
transient strain signal obtained from the strain sensors placed on the electronic assembly
while performing drop and shock testing, as explained in section 6.1 has been used to
obtain the FFT frequency band signature.
42
Figure 14 : Transient Strain-History at Location of CSP during Drop-Event.
43
Loc 1
Loc 3
Loc 5
Loc 8
Power Spectrum Density
0
20
40
60
80
100
120
140
160
0 200 400 600 800 1000
Frequency (Hz)
PS
D (d
B/H
z
)
Power Spectrum Density
0
10
20
30
40
50
60
70
80
0 200 400 600 800 1000
Frequency (Hz)
PSD
(dB/H
z
)
Power Spectrum Density
0
50
100
150
200
250
300
350
0 200 400 600 800 1000
Frequency (Hz)
PSD
(d
B/H
z
)
Power Spectrum Density
0
20
40
60
80
100
120
140
160
180
0 200 400 600 800 1000
Frequency (Hz)
P
S
D (d
B/Hz)
(a) CSP Location 1
(b) CSP Location 3
(d) CSP Location 8(c) CSP Location 5
Fre (Hz)
PS
D (d
B/H
z
)
P
S
D (d
B/Hz)
P
S
D (dB/Hz
)
P
S
D (d
B/Hz)
Figure 15 : FFT Frequency Band Energy Feature Vector.
44
0
5
10
15
20
25
0 200 400 600 800 1000
Frequency (Hz)
Re
ce
ptan
ce
Figure 16 The Receptance Plot obtained by the Modal Analysis of the TABGA Board.
(a) Mode 1 (b) Mode 2
(c) Mode 3
Figure 17 The Mode shapes and natural frequencies of vibration of the Board.
45
4.2 Time Frequency Analysis
Many signals in real world situations have frequency content that varies over time
in the signal. Using a joint time frequency analysis on such a signal provides the exact
behavior of the frequency content and its variation over time. This provides an
opportunity to study the energy density of a signal simultaneously in time and frequency
and also helps in removing noise and interference from the signal. The joint time
frequency is classified into two categories, the Linear time frequency transforms and the
Bilinear time frequency transforms. Some of the techniques under each class have been
outlined in the chart shown in Figure 18.
Time Frequency Transform
Linear Transforms Bilinear Transforms
Short Time
Fourier Transform
Spectrogram
Continuous Wavelet
Transform
Wigner-Ville
Distribution
Cohen Class of
Distribution
Figure 18 Time Frequency Analysis Techniques.
4.3 Linear Time Frequency Transforms
The Fourier transform of a signal is performed to obtain the frequency content of
the signal and assumes that the signal is stable during the sample time. Hence the
frequency spectrum is not able represent the frequency content changing with time and
hence the frequency spectrum will not be able to uniquely represent the signal. The main
46
linear transforms used are the Short Time Fourier Transform (STFT) and the continuous
wavelet transform (CWT).
4.3.1 Short Time Fourier Transform
The STFT is a generalization of the Fourier transform. In the STFT the signal is
multiplied with a window function which emphasizes the signal at the specific time and
suppresses the signal at other times. The windowed signal is computed as
)t(h)(f(f
t
???=)?
where )?(f
t
is the modified signal, )(f ? is the original signal and )t(h ?? is the window
function centered at t. Hence the signal is suppressed at everywhere and remains
unaltered around time t.
The Fourier transform of the windowed signal at each time interval is computed
and for each time interval a different frequency spectrum is obtained. The totality of these
individual spectra is termed as the time-frequency spectrum of the signal.
?)?
?
=?
?
???
d(se
2
1
)(S
t
i
t
The spectrogram obtained from the energy density spectrum is computed as
2
tSTFT
)(S),t(P ?=?
The Spectrogram mixes the energy of the signal with the energy of the window
and hence does not give results solely for the signal but the resulting transform is
equivalent to the spectrum of the signal convolved with the spectrum of the window
[Cohen 1995]. Also the window function cannot have high resolution in time and
frequency both, hence cannot give good localization in both time and frequency. This
47
limitation is overcome by applying Quadratic time frequency transforms which allows
good localization in both the domains.
4.3.2 Continuous Wavelet Transform
The continuous Wavelet Transform (CWT) allows a variable coverage of the
time-frequency plane [Grossmann et.al. 1984]. The transform is defined as:
?
?
?
?
?
?
?
?
??
?=
?
dt
t
*)t(f
a
1
CWT
a,
where )t(? is the mother wavelet and ? nd a are the time translation and the scaling
parameter respectively. The basic algorithm is based on convolving the signal with a set
of functions which are the scaled and dilated version of the mother wavelet. The squared
magnitude of the CWT coefficients is equivalent to the power spectrum so that a typical
display of the CWT is a representation of the power spectrum as a function of time offset
? and is termed as a scalogram. The exact form of the CWT scalogram depends on the
choice of mother wavelet and therefore the extent of the relation between the squared
magnitude of the CWT and the actual signal power spectrum is dependent on the signal
structure. Since a wavelet does not precisely report information on the frequency of a
signal it may not be the best tool for analyzing transient signals containing signal
structures similar to sine waveforms. The lack of an orthonormal basis for a CWT means
that the CWT does not possess the mathematically simple method of inversion offered by
orthonormal DWT?s.
48
4.3.3 Gabor Expansion
The Gabor representation is based on expanding a signal in terms of two
dimensional time-frequency functions where the coefficients give an indication of the
relative weight of the particular expansion function [Gabor1946].
The discretization in the time-frequency plane is computed where the coordinates are
?=?
=
m
nTt
i
i
?<<?? m,n
where i and ? is the time and frequency intervals.
The proposed representation of the transient signal is given as
tjn
m,n
m,n
m,nm,n
e)mTt(h)t(h
)t(hc)t(f
?
?=
=
?
???= ,m,n
where )t(h is a one dimensional functional, mostly the Gaussian as it is most compact in
the sense of the time-frequency bandwidth. The square of the coefficients provide the
energy at the point
ii
,t ? in the time-frequency spectrum. The expansion coefficients
obtained from the Gabor expansion are not always unique to a signal and hence not a
good choice for performing the time frequency analysis for health monitoring purposes.
The biggest shortcoming of the linear time-frequency transforms is that the time-
frequency resolution is limited to the Heisenberg bound. This occurs due to imposition of
the local time window, which limits the resolution of the frequency window.
49
4.4 Quadratic Time Frequency Transforms
The quadratic time frequency transforms are computed by the multiplicative
comparison of the signal with itself which is expanded in various directions about each
point in time. The representation is in the form of a two dimensional distribution of
energy over the time-frequency spectrum. As the quadratic transforms do not use
windowing functions hence the resolution problems faced in linear transforms are
eliminated.
4.4.1 Wigner-Ville Distribution
The Wigner distribution [Wigner 1932, Ville 1948] for a given signal )t(f and its
spectrum )(F ? is given as
?
?
++
?
=
?
??
+?
?
=
d?
it?
?)e
2
1
?)F(?
2
1
(?F
2?
1
d?
i?
)e
2
?
)f(t
2
?
(tf
2?
1
?)W(t,
The Quadratic nature of the Wigner distribution is attributed to the signal entering twice
in the computation. To obtain the distribution at a given time t, the signal at a past time is
multiplied with the signal at a future time with the interval considered same, i.e. ? both
in the past and in the future. The similarity of the signals to the left of the time t and to
the right of the time t are checked and if these parts overlap then the same properties are
assumed to be present at time t. The Wigner distribution is identical in both the time and
frequency domains. As this distribution weighs the faraway times equally to the near
times hence it is highly nonlocal.
50
The characteristic function of the Wigner Distribution, the symmetric ambiguity function
is defined as
dte)
2
?
)f(t
2
?
(tf
ddt'd
i??
)e
2
?'
)f(t
2
?'
(tfe
2
1
(Addt,t(We(M
ti
iti
iti
?
??+?
??+?
?
???
??
+?
?
=
??
??
+?
?
?
=
)?,?=?)?=)?,?
The characteristic function is applied to study and construct the energy densities.
It is defined as the Fourier transform of the density
titi
edt)t(fe(M
??
==)?
?
Hence the characteristic function is the average of the term
ti
e
?
, where ?is a parameter.
The 2 dimensional characteristic function used in the Wigner and other Quadratic
transforms is just the average of the term containing both frequency and time parameters,
i.e.
??+? iti
e
??
?)?=)?,?
??+?
ddt,t(We(M
iti
And hence the density can be computed from the characteristic function by taking the
inverse Fourier transform.
??
??)?,?
?
=?
?????
dde(M
4
1
),t(W
iti
2
The basic properties of the Wigner distribution are:
1. Real Valued distribution: The Wigner distribution is always real, even when the
signal is complex. The mathematical proof of the given property can be obtained by
51
computing the conjugate of the Wigner distribution.
)?=
?
?
?
??
+=
?
??
?
??
+?=
?
?
+?=?
,t(W
-
d?
i?
)e
2
?
-(t
*
)f
2
?
f(t
2?
1
d?
i?
)e
2
?
-(t
*
)f
2
?
f(t
2?
1
d?
i?
)e
2
?
(t
*
)f
2
?
f(t
2?
1
),t(W
*
2 Symmetry: The Wigner distribution is symmetric in the frequency domain when the
signal has symmetrical spectra and symmetrical in the time domain when the signal
is symmetrical.
3 The Wigner distribution satisfies the time-frequency marginals. The energy content
in the analyzed signal and the energy content in the Wigner distribution is equal and
hence a Wigner-Ville distribution is able to represent the signal?s energy distribution
in the time-frequency domain.
???
?
??
?
??
?
??
=?)?
2
)t(fddt,t(W
The Wigner distribution for a sum of two signals, termed as the cross Wigner
distribution is given by
),t(W),t(W),t(W),t(W),t(W
)t(f)t(f)t(f
21122211
21
?+?+?+?=?
+=
Here the cross Wigner distribution is given by
?
?
2
?
?
2
?
?
?
=?
???
de)t(f)t(f
2
1
),t(W
i
2
*
112
52
As the Wigner distribution is complex in nature hence
*
2112
WW =
Due to this property of the cross Wigner distribution the sum of the two cross terms give
a real value, i.e.
)}?+)?+)?=)??
)}?=+
,t(WRe{2,t(W,t(W,t(W
,t(WRe{2WW
122211
122112
The signal can be broken into any number of parts and this causes the Wigner
distribution to contain the cross reference terms as compared to just the sum of the
individual signal Wigner distributions [Mark 1970]. This causes the correlation term to
be present in the distribution and this term is not unique and can cause errors while
studying the signal distributions. The quadratic nature of the transforms creates cross-
terms whenever multiple frequencies are superimposed. This is overcome by using the
Cohen Class of Distributions.
4.4.2 Cohen Class of Transforms
The Cohen class of transforms apply the approach of computing the time
frequency analysis using a kernel, which is a auxiliary function. The kernel can be
defined so as to represent the properties and requirements of the particular distribution.
The time-frequency representation [Cohen 1989] is given by
???
??)?,?(?
2
?
+
2
?
?
?
=?
?+?????
2
dddue)u(f)u(f
4
1
),t(C
uiiti*
where )?,?(? is the kernel function.
53
In terms of the characteristic function the distribution is represented as
)?,?(?)?,?(?=
2
?
+
2
?
?)?,?(?=?,?
??)?,?
?
=?
?
??
?
?????
2
due)u(f)u(f)(M
bygiven isfunction ticcharcteris theand
dde(M
4
1
),t(C
ui*
iti
where )?,?(A is the symmetrical ambiguity function.
A list of the various time frequency distributions, their kernels are shown in Table
4-1 shown below for the various distributions studied in this thesis. By the application of
the kernel method the time frequency distribution can be obtained such that they satisfy
the constraints for any particular distribution. The kernels are defined to satisfy certain
conditions and this cause the distributions obtained to also satisfy these conditions and as
the generation of the kernels is not very complicated this reduces the computation and
mathematical complexity involved.
4.4.3 Reduced Interference Distributions
As outlined while studying the Wigner-Ville distribution, the distribution obtained
for multicomponent signals contain correlation cross terms and hence causes the
distributions to contain data not unique to the particular signal. The time frequency
analysis methods proposed by researchers such as Choi and Williams [Choi et.al. 1989]
minimize the presence of these cross terms. The cross terms can be reduced by
developing kernels that reduce the interference or the cross reference terms, Cohen has
proposed and derived one such kernel defined as the Reduced Interference Distribution
(RID) kernels.
54
Table 4-1 Kernels applied during Time Frequency analysis [Cohen 1995].
Name Kernel: )?,?(? Distribution: )?,t(C
Cohen General
Class
)?,?(?
???
?)?,?(?
2
?
+
2
?
?
?
?+?????
2
ddue)u(f)u(f
4
1
uiiti*
Wigner-Ville 1
?
2
?
+
2
?
?
?
???
?
de)u(f)u(f
2
1
it*
Choi-Williams
Distribution
????
22
/
e
??
2
?
+
2
?
?
??
?
?????(??
2
3/2
2
)u(f)u(fe
1
4
1
*i/)tu
2
STFT
Spectogram
du)
2
u(h
e)u(h
ui*
?
+
2
?
?
?
??
2
i
d)t(h(fe
2
1
?
???)?
?
???
The derivation of the conditions required to form RID kernels has been outlined
in [Cohen 1995] and the basic mathematics of the kernel is outlined.
55
?
??
?
?????
2
2
?
+
2
?
?)?,?(?=?,?
??)?,?
?
=?
due)u(f)u(f)(M
bygiven isfunction ticcharcteris theand
dde(M
4
1
),t(C
ui*
iti
Assuming the function to be a sum of two signals as defined while studying the cross
Wigner distribution terms
ti
1
ti
121
eSeS)t(f)t(f)t(f
21
??
+=+=
Hence the distribution can be defined as
21222
CCCC(C +++=)?,?
111
Computing the characteristic function and the distribution the signal terms and the cross
reference terms are given by
)(
2
1
where
de),
2
1
(R
where
(ReSSC(ReSSC
de
2
S
Cde
2
S
C
)i
t)(i
1
*
221
t)(i
2
*
112
)i
2
2
22
)i
2
1
11
12
122
2
2112
???(??
21
????????
???(?????(??
?+?=?
????(?
?
=)?
)?=)?=
?)?(0,?
?
=?)?(0,?
?
=
?
??
1
1
Hence to minimize the cross terms the value of the term )?(R has to be minimized. The
representation of the )?(R terms shows that for the cross terms to be minimized the value
of )?,?(? away from the ? ?and axis should be small as compared to values of
? ?and respectively. The condition outlined is the property of a cross shaped low pass
filter and
56
0>>??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
?<? , and hence only one
direction of inequality of means is significant to the problem, then the one sided test is
applied.
The two sided p value is the measure of the evidence when the alternative
hypothesis is unrestricted, i.e.
21
??? . Hence the two-sided test has been performed on
the feature vectors obtained from the statistical pattern recognition techniques. The t-
value is obtained using
signal theofdeviation standard
compared being vectorsfeature signal theof smean value the,
where
t
21
12
??
???
?
???
=
Using the student-t value distribution we obtain the confidence value in our case the
reliability of the assembly. A faulty assembly gives a CV of 1 while a CV of a non faulty
assembly gives a CV of 1.
63
CHAPTER 5
CLOSED FORM ANALYTICAL MODELS
In this study, 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.
5.1 Derivation of the Lagrangian Functional
For the analytical solution for the free vibration of rectangular plates, the
Governing differential equation needs to be developed. The displacement function in the
z-direction is assumed to be of the form,
ti
y)ew(x,t)y,(x,
o
W
?
=
where
?
is the natural frequency at which the plate vibrates when the mode shape of the
plate is given by ),( yxw .
64
The displacement field from the Kirchhoff hypotheses in the pure bending case is given
by:
(5.1)
t)y,(x,
o
Wt)z,y,w(x,
,
y
o
W
zt)z,y,v(x,
,
x
o
W
zt)z,y,u(x,
LL
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?=
?
?
?=
To obtain the strains from the displacement field defined above, the assumption of small
strains and displacement has been made.
The linear strains obtained from the displacement field are:
(5.2)
0
zz
?0,
yz
?0,
xz
?
yx
o
W
2
z
xy
?
2
y
o
W
2
z
yy
?
2
x
o
W
2
z
xx
?
LL
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
===
??
?
?=
?
?
?=
?
?
?=
For the derivation of the Governing Differential equation the virtual Lagrangian
functional is developed. The Virtual Lagrangian Functional is given by
?K?V?U?L ?+=
where
Energy Kinetic Virtual?K
Energy Potential Virtual?V
Energy Strain Virtual?U
?
?
?
65
Using the Hamilton?s Principle (the dynamic version of the principle of virtual
displacements):
(5.3)0?K)dt?VU(?L
T
0
T
o
LL
?
=?+?
?
=
5.1.1 Development of the Virtual Strain energy
The virtual strain energy of a plate is
( )
()
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?=
??
?
++=
?
++=
A
dzdxdy
2
h
2
h
yx
0
?W
2
z
xy
2?
2
y
0
?W
2
z
yy
?
2
x
0
?W
2
z
xx
??U
getwe(5.2),equationfromstrainsthengSubstituti
dzdxdy
A
2
h
2
h
xy
??
xy
2?
yy
??
yy
?
xx
??
xx
?
V
dV
xy
??
xy
2?
yy
??
yy
?
xx
??
xx
??U
The moments per unit length ),,(
xyyyxx
MMM are given by
(5.5)
A
dxdy
yx
0
?W
2
xy
2M
2
y
0
?W
2
yy
M
2
x
0
?W
2
xx
M?U
ismomentstheoftermsinequationenergystraintheHence
(5.4)
2
h
2
h
dzz
xy
?
yy
?
xx
?
xy
M
yy
M
xx
M
LL
LL
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
+
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?=
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
66
Assuming linear elastic behavior of the plate material
For isotropic plate
where
()?12
E
G
66
Q
2
?1
E
22
Q
2
?1
?E
12
Q
2
?1
E
11
Q
+
==
?
=
?
=
?
=
For an orthotropic material
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
xy
2?
yy
?
xx
?
66
Q00
0
22
Q
12
Q
0
12
Q
11
Q
2
?1
E
xy
?
yy
?
xx
?
where
1
E
2
E
12
?
21
?
12
G
66
Q
21
?
12
?1
2
E
22
Q
21
?
12
?1
2
E
12
?
12
Q
21
?
12
?1
1
E
11
Q
==
?
=
?
=
?
=
Substituting the above stress-strain relation in equation (5.4) we get,
?
?
?
?
?
?
?
?
?
?
+
?
?
?=
?
?
?
?
?
?
?
?
?
?
+
?
?
?=
2
x
0
W
2
12
D
2
y
0
W
2
22
D
yy
M
2
y
0
W
2
12
D
2
x
0
W
2
11
D
xx
M
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
xy
2?
yy
?
xx
?
66
Q00
0
22
Q
12
Q
0
12
Q
11
Q
2
?1
E
xy
2?
yy
?
xx
?
2
?1
00
01?
0?1
2
?1
E
xy
?
yy
?
xx
?
67
yx
0
W
2
66
2D
xy
M
??
?
?=
where
ij
Q
12
3
h
ij
D = ; h=plate thickness
Hence the virtual strain energy is given by
(5.6)
A
dxdy
yx
0
?W
2
yx
0
W
2
66
4D
2
y
0
?W
2
2
x
0
W
2
2
x
0
?W
2
2
y
0
W
2
12
D
2
y
0
?W
2
2
y
0
W
2
22
D
2
x
0
?W
2
2
x
0
W
2
11
D
?U LL
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
??
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
+
?
?
?
?
=
5.1.2 Development of the Virtual Kinetic Energy
The Virtual Kinetic energy is given by
dxdy
A
y
0
W?
y
0
W
x
0
W?
x
0
W
2
I
0
W?
0
W
0
I
dzdxdy
A
2
h
2
h
0
W?
0
W
y
0
W?
z
y
0
W
z
x
0
W?
z
x
0
W
z??K
equationabovethein(1)indefinedfieldntdisplacemengSubstituti
(5.7))dzdxdyw?wv?vu?u
A
2
h
2
h
?(?K
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+=
??
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?=
++
??
?
=
&&&&
&&
&&
&&&&
LL&&&&&&
where ? is the mass density and I
0
, I
2
are the mass moments of inertia defined as
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
? 2
h
2
h
12
3
h
h
??dz
2
z
1
2
I
0
I
68
Hence the virtual kinetic energy is given by
(5.8)dxdy
A
y
0
W?
y
0
W
x
0
W?
x
0
W
2
I
0
W?
0
W
0
I?K LL
&&&&
&&
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+=
5.1.3 Development of the Virtual Potential Energy
Assuming a distributed load q(x, y) applied on the top surface of the plate, a
transverse edge force
n
V
?
, an normal edge moment
nn
M
?
applied on a portion of the total
boundary of the plate denoted as? and the in-plane compressive and shear forces
(
xyyyxx
NNN
?
,
?
,
?
), the virtual potential energy is given by
(5.9)
?
ds
n
o
?W
nn
M
?
o
?W
n
V
?
A
dxdy
o
y)?)q(x,
A
dxdy
x
o
?W
y
o
W
y
o
?W
x
o
W
xy
N
?
y
o
?W
y
o
W
yy
N
?
x
o
?W
x
o
W
xx
N
?
?V LL
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
+
?
?
?
?
?=
5.2 Development of Governing Differential Equation
From the virtual Strain energy, the virtual kinetic energy and the virtual potential
energy we obtain the virtual Lagrangian functional
?K?V?U?L ?+=
69
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
+
?
?
?
?
?
??
?
??
?
+
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
+
?
?
?
?
=
?
ds
n
o
?W
nn
M
?
o
?W
n
V
?
A
dxdy
o
y)?)q(x,
dxdy
A
y
0
W?
y
0
W
x
0
W?
x
0
W
2
I
0
W?
0
W
0
I
x
o
?W
y
o
W
y
o
?W
x
o
W
xy
N
?
y
o
?W
y
o
W
yy
N
?
x
o
?W
x
o
W
xx
N
?
yx
0
?W
2
yx
0
W
2
66
4D
2
y
0
?W
2
2
x
0
W
2
2
x
0
?W
2
2
y
0
W
2
12
D
2
y
0
?W
2
2
y
0
W
2
22
D
2
x
0
?W
2
2
x
0
W
2
11
D
&&&&
&&
Assumptions:
1. Assuming that the plate is undergoing free vibration, hence no external force acts
on the plate, i.e. q(x, y),
n
V
?
,
nn
M
?
,
xyyyxx
NNN
?
,
?
,
?
=0.
2. Neglecting the rotary inertia to be zero, i.e. I
2
=0
Hence
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
??
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
=
A
dxdy
0
W?
0
W
o
I
2
y
0
?W
2
2
y
0
W
2
yx
0
?W
2
yx
0
W
2
?)2(1
2
y
0
?W
2
2
x
0
W
2
2
x
0
?W
2
2
y
0
W
2
?
2
x
0
?W
2
2
x
0
W
2
D?L
&&
70
The weak form of the Lagrangian Functional for isotropic plates
() (5.10)
A
dxdy
0
W?
0
W
o
I
2
y
0
?W
2
2
y
0
W
2
yx
0
?W
2
yx
0
W
2
?)2(1
2
y
0
?W
2
2
x
0
W
2
2
x
0
?W
2
2
y
0
W
2
?
2
x
0
?W
2
2
x
0
W
2
D LL
&&
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
??
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
The weak form of the Lagrangian Functional for orthotropic plates
()(5.11)dxdy
A
o
W?
o
W
o
I
2
y
o
?W
2
2
y
o
W
2
22
D
yx
o
?W
2
yx
o
W
2
66
4D
2
y
o
?W
2
2
x
o
W
2
2
x
o
?W
2
2
y
o
W
2
12
D
2
x
o
?W
2
2
x
o
W
2
11
D
LL
&&
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
??
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
To compute the Governing Differential Equation for a Plate from the weak form, we
integrate by parts to relieve
0
W? of any differentiation.
5.2.1 Isotropic plates
Considering the Strain energy part,
dxdydt
T
0
b
0
a
0
2
y
0
?W
2
2
y
0
W
2
yx
0
?W
2
yx
0
W
2
?)2(1
2
y
0
?W
2
2
x
0
W
2
2
x
0
?W
2
2
y
0
W
2
?
2
x
0
?W
2
2
x
0
W
2
D
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
??
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
71
Integrating by parts
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
T
0
b
0
a
0
dxdydt
x
0
?W
3
x
0
W
3
D
T
o
bt
x
0
?W
2
x
0
W
2
D
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
??
?
?
?
?
?
??
?
?
+
???
?
?
??
?
?
?
?
?
?
+
???
?
?
??
?
?
?
?
?
?
dxdydt
T
0
b
0
a
0
x
0
?W
x
2
y
0
W
3
T
0
a
0
dx
x
0
?W
yx
0
W
2
v)2D(1
T
0
b
0
a
0
dxdydt
y
0
?W
y
2
x
0
W
3
T
0
t
y
0
?W
2
x
0
W
2
T
0
b
0
a
0
dxdydt
x
0
?W
x
2
y
0
W
3
T
0
t
x
0
?W
2
y
0
W
2
D?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
?
+
T
0
t
b
0
a
0
dxdyd
y
0
?W
3
y
0
W
3
D
T
0
at
y
0
?W
2
y
0
W
2
D
Assuming that all terms evaluated at t=0 and t=T are zero, and again integrating by parts.
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
??
?
?
?
?
?
?
?
?
?
?
?
??
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???
??
?
?
?
?
?
?
?
?
?
?
??
?
+
?
?
?
?
?
?
?
?
???
??
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
???
?
?
+
?
?
?
?
?
?
?
?
?
?
??
T
0
t
b
0
a
0
dxdyd
0
?W
4
y
0
W
4
D
T
0
at
0
?W
3
y
0
W
3
D
dxdydt
T
0
b
0
a
0
0
?W
2
x
2
y
0
W
4
T
0
b
0
dy
0
?W
2
yx
0
W
3
v)2D(1
T
0
b
0
a
0
dxdydt
0
?W
2
y
2
x
0
W
4
T
0
t
0
?W
y
2
x
0
W
3
T
0
b
0
a
0
dxdydt
0
?W
2
x
2
y
0
W
4
T
0
t
0
?W
x
2
y
0
W
3
Dv
T
0
b
0
a
0
dtdydx
0
?W
4
x
0
W
4
D
T
0
bt
0
?W
3
x
0
W
3
D
72
All terms evaluated at t=0 and t=T are assumed to be zero,
dxdydt
T
0
b
0
a
0
0
?W
4
y
0
W
4
D
2
x
2
y
0
W
4
v)2(1
2
x
2
y
0
W
4
2v
4
x
0
W
4
D
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
?+
??
?
+
?
?
Hence the Strain energy term after integration by parts is,
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
+
?
?
T
0
b
0
dtdydx
a
0
o
?W
4
y
o
W
4
D
2
y
2
x
o
W
4
2
4
x
o
W
4
D
Integrating by Kinetic energy term by parts, we get
[] dxdy
T
0
b
0
a
0
o
?W
o
W
o
I
T
o
ab
o
?W
o
W
o
Idxdydt
T
0
b
0
a
0
o
W?
o
W
o
I
???
?=
???
&&&&&
Hence the Kinetic Energy term after integration by parts is,
dxdy
T
0
b
0
a
0
o
?W
o
W
o
I
???
?=
&&
Hence the Hamilton?s Equation is denoted as,
() dxdydt
T
0
b
0
a
0
0
?W
0
W
o
I
4
y
0
W
4
D
2
x
2
y
0
W
4
2
4
x
0
W
4
D
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
??
?
+
?
?
&&
Applying the Fundamental of Lemma, we get the Euler-Lagrange equation or the
Governing Differential equation as
() (5.12)0
0
W
o
I
4
y
0
W
4
D
2
x
2
y
0
W
4
2
4
x
0
W
4
D LL
&&
=+
?
?
+
??
?
+
?
?
73
5.2.2 Orthotropic plates
Considering the Strain energy part,
dxdydt
T
0
b
0
a
0
2
y
0
?W
2
2
y
0
W
2
22
D
yx
0
?W
2
yx
0
W
2
66
4D
2
y
0
?W
2
2
x
0
W
2
2
x
0
?W
2
2
y
0
W
2
12
D
2
x
0
?W
2
2
x
0
W
2
11
D
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
??
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
Integrating by parts as performed in the section for Isotropic plates the Strain energy term
is computed,
dxdydt
T
0
b
0
a
0
o
?W
4
y
o
W
4
22
D
2
x
2
y
o
W
4
)
12
D
66
2(2D
4
x
o
W
4
11
D
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
++
?
?
Integrating the Kinetic energy term by parts, we get
[] dxdy
T
0
b
0
a
0
o
?W
o
W
o
I
T
o
ab
o
?W
o
W
o
Idxdyd
T
0
b
0
a
0
o
W?
o
W
o
I
???
?=
???
&&&&&
Hence the Kinetic Energy term after integration by parts is,
dxdy
T
0
b
0
a
0
o
?W
o
W
o
I
???
?=
&&
Hence the Hamilton?s Equation is denoted as,
() dxdydt
T
0
b
0
a
0
o
?W
o
W
o
I
4
y
o
W
4
22
D
2
x
2
y
o
W
4
)
12
D
66
2(2D
4
x
o
W
4
11
D
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
??
?
++
?
?
&&
Applying the Fundamental of Lemma, we get the Euler-Lagrange equation or the
Governing Differential equation as
() (5.13)0
o
W
o
I
4
y
0
W
4
D
2
x
2
y
0
W
4
)
12
D
66
2(2D
4
x
0
W
4
11
D LL
&&
=+
?
?
+
??
?
++
?
?
74
To solve the Governing Differential equation derived above, the Ritz method of solving
the free vibration problem of rectangular plates is applied [Leissa 1969, Young 1950].
The Rectangular plate is considered as a mesh of plate strips, with the plate functionals
satisfying the specific boundary conditions of the plate.
5.3 Plate Functional Derivation using Plate Strips
The method assumes that the plate strip being analyzed is very thin, i.e. a plate strip
in the x-direction has the y-axis thickness so negligible that the y-component of the
displacement field to be neglected.
5.3.1 Plate Strip Displacement Function
To assume the functional W(x, y), we analyze the free vibrations of plate strips
having the boundary conditions we desire. The Governing Differential Equation for the
plate derived for the free vibration of rectangular plates is
()0
0
W
o
I
4
y
0
W
4
D
2
x
2
y
0
W
4
2
4
x
0
W
4
D =+
?
?
+
??
?
+
?
?
&&
Considering a very thin plate strip in the x-direction such that the y-component of the
displacement function can be neglected, i.e.
tcos?W(x)t)(x,
o
W =
Hence the GDE can be written as
0
2
t
0
W
2
o
I
4
x
0
W
4
D =
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
75
Substituting the assumed solution in the governing differential equation:
0W
2
?
0
I
4
dx
W
4
d
D =?
Denoting the above equation as
0cW
2
dx
W
2
d
b
4
dx
W
4
d
a =?+
where Da = ; 0b = ;
0
I
2
?c =
Assuming
rx
AeW(x) =
0c
2
br
4
ar =?+?
Putting s
2
r =
0cbs
2
as =?+
The roots of the equation are
2
l
2
?
)4ac
2
bb(
2a
1
2
s
2
l
2
?
)4ac
2
bb(
2a
1
1
s
=++?=
=+??=
The solution can be expressed as
)14.5(
l
?x
cosh
4
c
l
?x
sinh
3
c
l
?x
cos
2
c
l
?x
sin
1
cW(x) LL+++=
where
)4ac
2
bb(
2a
1
l
?
+??= ; )4ac
2
bb(
2a
1
l
?
++?=
As b=0, hence ?=? ;
76
l
?x
cosh
4
c
l
?x
sinh
3
c
l
?x
cos
2
c
l
?x
sin
1
cW(x) +++=?
Based on the specific boundary conditions of the plate the functionals are developed
further.
5.3.2 For Simple-Simple plate strip
The boundary conditions for a free-free plate strip are given below.
Displacement at simply supported edges is zero.
0W = ; at x=0,l
Moments at the simply supported edges are zero:
0
2
x
W
2
D
xx
M =
?
?
?= ; at x=0,l
From section 5.3.1 using the displacement field derived, given by equation (5.14), the
displacement and the second level derivatives of the displacement field are calculated.
l
?x
cosh
4
c
l
?x
sinh
3
c
l
?x
cos
2
c
l
?x
sin
1
cW(x) +++=
)
l
?x
cosh
4
c
l
?x
sinh
3
c
l
?x
cos
2
c
l
?x
sin
1
c(
2
2
l
?
2
x
W
2
++??=
?
?
Applying the boundary conditions:
At x=0
2
c
4
c0
xx
M
4
c
2
c0W
=?=
?=?=
The above case is only possible if 0
4
c
2
c ==
At x=l and substituting 0
4
c
2
c ==
77
(5.16)0)sinh
3
csin?
1
c(0
xx
M
(5.15)0)sinh
3
csin?
1
(c0W
LL
LL
=+??=
=+?=
?
?
0
3
c
1
c
sinh?sin?
sinh?sin?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
??
`
For c
1
, c
3
to have distinct non-zero values the determinant should be zero
02sinh?sin =? ?
Hence the transcendental equation becomes
0sin?sinh =?
from which we obtain the eigenvalues of the plate strips using numerical techniques. We
use Matlab to obtain values of the above equation. To calculate c
1
and c
3
we solve the
equations (5.15) and (5.16)
sinh?
sin?
1
c
3
c
?
=?
Assuming c
1
=c, and substituting the calculated values of? , we get 0
3
c = for all values
of ? .
Hence strip functional is given by
l
?x
sinW(x) =
78
Table 5-1 Roots of the transcendental equation for a simply supported-simply supported
plate.
r
r
?
1 0
2 3.1416
3 6.2832
4 9.4248
5 15.7080
6 18.8496
7 21.9911
8 21.9911
9 25.1327
5.3.3 Free-Free plate strip
The boundary conditions for a free-free plate strip are given below.
Moments at the free edges are zero:
0
2
x
W
2
D
xx
M =
?
?
?= ; at x=0,a
Shear Force at free edges is zero:
0
3
x
W
3
D
x
V =
?
?
?= ; at x=0,a
From the previous section using the displacement field derived, given by equation (5.14),
79
the second and third level derivatives of the displacement field are calculated.
)
l
?x
cosh
4
c
l
?x
sinh
3
c
l
?x
cos
2
c
l
?x
sin
1
c(
2
2
l
?
2
x
W
2
++??=
?
?
)
l
?x
sinh
4
c
l
?x
cosh
3
c
l
?x
sin
2
c
l
?x
cos
1
c(
3
l
3
?
3
x
W
3
+++?=
?
?
Applying the boundary conditions:
At x=0
At x=a and substituting equations (5.17) and (5.18)
0sinh?i
2
ccosh?
1
csin?
2
ccos?
1
c(0
x
V
0cosh?o
2
csinh?
1
ccos?
2
csin?
1
c(0
xx
M
=+++??=
=++???=
0
2
c
1
c
sinh?sin?cos?cosh?
cos?cosh?sinh?sin?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
?+?
For c
1
, c
2
to have distinct non-zero values the determinant should be zero
022cosh?cos =?? ?
Hence the transcendental equation becomes
01coscosh =???
from which we obtain the eigenvalues of the plate strips using numerical techniques. We
use Matlab to obtain values of the above equation.
()
()5.18.........
3
c
1
c0
x
V
5.17......
2
c
4
c0
xx
M
=?=
=?=
80
Table 5-2 Roots of the transcendental equation of a Free-Free plate.
r
r
?
1 0
2 0
3 4.73
4 7.8532
5 10.996
6 14.137
7 17.279
8 20.42
9 23.562
10 26.704
To calculate c
1
and c
2
we solve the equations (5.17) and (5.18)
cos?cosh?
1
2
c
sinh?sin?
1
1
c
?
=?
?
=?
The value of
2
c
1
c
r
? =
The functional for a free-free plate strip used in this research is given below.
)
l
2x
(13
2
X
1
1
X
?=
=
81
?
?
?
?
?
?
+?+=
l
x
r
?
sin
l
x
r
?
sinh
r
?
l
x
r
?
cos
l
x
r
?
cosh
r
X r=3,4,5,7?.
Table 5-3 Values of ?
r
for a free-free plate.
r
r
?
3 0.9825
4 1.0008
5 0.99997
6 1
7 1
5.3.4 For a Clamped-Free Strip
The boundary conditions for a clamped-free plate strip are given below.
For Clamped end:
Displacement at clamped edge is zero.
0W(x) = ; at x=0
First Order differential of displacement at clamped edge is zero.
0
dx
dW(x)
= ; at x=0
For Free end:
Moments at the free edges are zero:
0
2
x
W
2
D
xx
M =
?
?
?= ; at x=l
82
Shear Force at free edges is zero:
0
3
x
W
3
D
x
V =
?
?
?= ; at x=l
From section 4.3.1 using the displacement field derived, given by equation (5.14), the
displacement and the first, second and third level derivatives of the displacement field are
calculated.
Applying the boundary conditions,
)(5.10=)?????????
=
=
?
?
?
?
?
?
?
?
++??=
=
?
?
?=
?=?=+?
=++?=
=
?=?=+=
9cosh
2
csinh
1
ccos
2
csin
1
c(
0
lx
)
l
?x
cosh
4
c
l
?x
sinh
3
c
l
?x
cos
2
c
l
?x
sin
1
c(
2
l
l
?
*D
lx
2
x
W
2
D
xx
M
3
c
1
c0
3
c
1
c
0sinh(0)
4
c
l
?
cosh(0)
3
c
l
?
sin(0)
2
c
l
?
cos(0)
1
c
l
?
0x
dx
dW(x)
4
c
2
c0
4
c
2
cW(0)
KK
0
2
c
1
c
sinh?sin?cos?cosh?
cos?cosh?sinh?sin?
..(5.20)0.........sinh?i
2
ccosh?
1
csin?
2
ccos?
1
c(
0
lx
)
l
?x
sinh
4
c
l
?x
cosh
3
c
l
?x
sin
2
c
l
?x
cos
1
c(
3
l
3
?
*D
lx
3
x
W
3
D
x
V
=
?
?
?
?
?
?
?
?
?
?
?
?
???
????
=??+??
=
=
?
?
?
?
?
?
?
?
+++?=
=
?
?
?=
For c
1
, c
2
to have distinct non-zero values the determinant should be zero
022cosh?cos =+??
83
Hence the transcendental equation becomes
01coscosh =+??
from which we obtain the eigenvalues of the plate strips using numerical techniques.
We use Matlab to obtain values of the above equation.
Table 5-4 Roots of the transcendental equation of a Clamped-Free plate.
r
r
?
1 1.8751
2 4.6941
3 7.8548
4 10.996
5 14.137
6 17.279
7 20.42
8 23.562
9 26.704
10 29.845
To calculate c
1
and c
2
we solve the equations (5.19) and (5.20)
cos?cosh?
1
2
c
sinh?sin?
1
1
c
+
?
=?
+
=?
The value of
2
1
c
c
r
=?
84
Table 5-5 Values of ?
r
for a Clamped-Free plate.
r
r
?
1 0.7341
2 1.0185
3 0.99922
4 1
5 1
6 1
7 1
8 1
9 1
10 1
5.3.5 For Clamped-Clamped plate strip
The boundary conditions are given below for a Clamped-Clamped plate strip.
Displacement at clamped edge is zero.
0W(x) = ; at x=0,l
First Order differential of displacement at clamped edge is zero.
0
dx
dW(x)
= ; at x=0,l
From section 5.3.1 using the displacement field derived, given by equation (5.14), the
displacement and the first level derivatives of the displacement field are calculated.
l
?x
cosh
4
c
l
?x
sinh
3
c
l
?x
cos
2
c
l
?x
sin
1
cW(x) +++=
85
)
l
?x
sinh
4
c
l
?x
cosh
3
c
l
?x
sin
2
c
l
?x
cos
1
(c
l
?
x
W
++?=
?
?
Applying the boundary conditions :
At x=0
At x=l and substituting eqns (5.21) and (5.22)
0sinh?i
2
ccosh?
1
csin?
2
ccos?
1
(c0
dx
dW(x)
0cosh?o
2
csinh?
1
ccos?
2
csin?
1
(c0W(x)
=????=
=??+?=
0
2
c
1
c
sinh?sin?cos?cosh?
cos?cosh?sinh?sin?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??+?
+??
For c
1
, c
2
to have distinct non-zero values the determinant should be zero
Hence the transcendental equation becomes
01coscosh =???
from which we obtain the eigenvalues of the plate strips using numerical techniques. We
use Matlab to obtain values of the above equation.
To calculate c
1
and c
2
we solve the equations (5.21) and (5.22)
cos?cosh?
1
2
c
sinh?sin?
1
1
c
?
=?
?
=?
The value of
2
c
1
c
r
? =
( )
()5.220.........
3
c
1
c0
dx
dW(x)
5.210......
4
c
2
c0W(x)
=+?=
=+?=
86
Table 5-6 Roots of the transcendental equation of a Clamped-Clamped plate.
r
r
?
1 4.73
2 7.8532
3 10.996
4 14.137
5 17.279
6 20.42
7 23.562
8 26.704
Table 5-7 Values of ?
r
for a Clamped-Clamped plate.
r
r
?
1 0.9825
2 1.0008
3 0.99997
4 1
5 1
5.4 Application of Ritz Method
The Rectangular plate can be considered as a mesh of plate strips in both the x and
y direction and hence the functionals for the Ritz method are same as the functionals for
plate strips depending on the boundary conditions at the edges of the plate. The Ritz
87
method has been applied to the various boundary conditions studied in this research. The
solution has been outlined for the completely free (FFFF) boundary case. All other
boundary conditions are analyzed in a similar manner as shown below.
5.4.1 Completely Free (FFFF) Plate
In this case, i.e. the FFFF plate, we have to consider 2 plate strips with Free-Free
(FF) boundary conditions, one in the x-direction and the other in y-direction.
(y)
n
(x)Y
m
X
mn
Ay)(x,
0
W
iwt
y)e(x,
o
Wt)y,W(x,
=
=
where (x)
m
X is the functional for FF strip in the horizontal direction and (y)
n
Yis the
functional for FF strip in the vertical direction.
)
a
2x
(13
2
X
1
1
X
?=
=
?
?
?
?
?
?
+?+=
a
x
m
?
sin
a
x
m
?
sinh
m
?
a
x
m
?
cos
a
x
m
?
cosh
m
X m=3,4,5,7?.
Similarly
)
b
2y
(13
2
Y
1
1
Y
?=
=
?
?
?
?
?
?
+?+=
b
y
n
?
sin
b
y
n
?
sinh
n
?
b
y
n
?
cos
b
y
n
?
cosh
n
Y n=3,4,5,7?.
Applying the Ritz method using the above assumed solution to the weak form of the
GDE i.e.
88
dxdy
a
0
b
0
W?W
o
I
2
y
?W
2
2
y
W
2
yx
?W
2
yx
W
2
?)2(1
2
y
?W
2
2
x
W
2
2
x
?W
2
2
y
W
2
?
2
x
?W
2
2
x
W
2
D
??
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
??
?
??
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
=
&&
?
=
?
=
?
?
=
?
?
++
?
?
+
?
?
+
?
?
=
m
1i
n
1j
ij
?A
ij
A
?
mn
?A
mn
A
?
.........
13
?A
13
A
?
12
?A
12
A
?
11
?A
11
A
?
??
The Ritz method states that for {A} to be linearly independent
2....n1,j2.....m;1,ifor0
ij
A
?
===
?
?
Assuming
?
=
?
=
=
?
=
?
=
=
p
1p
q
1q
n
Y
m
X
mn
?A
0
?W
p
1i
k
Y
q
1j
i
X
ik
A
0
W
dxdy
a
0
b
0
n
Y
m
X
k
Y
i
X
0
I
2
?
2
y
n
Y
2
m
X
2
y
k
Y
2
i
X
y
n
Y
x
m
X
y
k
Y
x
i
X
v)2(1
2
y
n
Y
2
m
X
k
Y
2
x
i
X
2
n
Y
2
x
m
X
2
2
y
k
Y
2
i
X
v
n
Y
2
x
m
X
2
k
Y
2
x
i
X
2
D
ik
A
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
+
?
?
?
?
=
?
?
89
Multiplying the whole equation by a
2
to help simplify the mathematics we get
()
()
()()[]
dxdy
a
0
b
0
n
Y
k
Y
m
X
i
X
2
a
0
I
2
?
2
y
n
Y
2
2
y
k
Y
2
m
X
i
X
2
a
y
n
Y
y
k
Y
b
x
m
X
x
i
X
a
b
a
v)2(1
2
y
n
Y
2
k
bY
m
X
2
x
i
X
2
a
n
Y
2
y
k
Y
2
b
2
x
m
X
2
i
aX
v
b
a
n
Y
k
Y
2
x
m
X
2
2
x
i
X
2
2
a
D
ik
A
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
The integrals required for the above analysis are:
miwhen0
miwhen
3
a
4
i
?a
0
dx
2
x
m
X
2
2
x
i
X
2
?=
==
?
?
?
?
?
This relation is obtained as the FF strip functionals are orthogonal and their second
derivatives are also orthogonal.
miwhen0
a
0
miwhenadx
m
X
i
X
?=
?
==
nkwhen0
nkwhenbdy
b
0
n
Y
k
Y
?=
==
?
90
nkwhen0
n.kwhen
3
b
4
n
b
0
dy
2
y
n
Y
2
2
y
k
Y
2
?=
=
?
=
?
?
?
?
?
To simplify the mathematical representations the integrals are represented by the
following terms [Young 1950],
dx
a
0
2
dx
m
X
2
d
i
Xa
im
E
?
=
?
=
a
0
dx
2
dx
i
X
2
d
m
Xa
mi
E
dy
b
0
2
dy
k
Y
2
d
n
Yb
nk
F
?
=
?
=
a
0
dx
dx
m
dX
dx
i
dX
a
im
H
?
=
b
0
dy
dy
n
dY
dy
k
dY
b
kn
K
Hence for i=m, the equation comes out to be
0dy
a
0
b
0
k
Y
k
Ydx
i
X
i
X
2
a
0
I
2
?
b
0
dy
2
y
k
Y
2
2
y
k
Y
2
a
0
dx
i
X
i
X
2
Da
a
0
b
0
dy
y
k
Y
y
k
Y
bdx
x
i
X
x
i
X
a
b
a
v)D2(1
a
0
b
0
dy
k
Y
2
y
k
Y
2
bdx
2
x
i
X
2
i
aXv
b
a
2D
a
0
b
0
dy
k
Y
k
Ydx
2
x
i
X
2
2
x
i
X
2
2
Da
=
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
??
?
?
?
?
+
?
?
?
?
?
?
?
?
??
?
?
?
?
()( )[] ()( )[]
() () 0
2
b
a
2
a
0
I
2
?
4
b
k?
2
b
a
2
Da
kk
K
ii
H
b
a
v)D2(1
kk
F
ii
Ev
b
a
2D
2
b
3
a
4
i
?
2
Da
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
dy
b
0
2
dy
n
Y
2
d
k
Yb
kn
F
?
=
91
()
()
0
2D
b
3
a
0
I
2
?
kk
K
ii
H
b
a
v)2(1
kk
F
ii
vE
b
a
2
3
2b
4
k?
3
a
2a
4
i
b?
0
2
b
3
a
0
I
2
?
kk
K
ii
H
b
a
v)2(1
kk
F
ii
vE
b
a
2
3
2b
4
k?
3
a
2a
4
i
b?
D
=?
?
?
?
?
?
?
?
?
?+++?
=?
?
?
?
?
?
?
?
?
?+++?
For mi ? the equation comes out to be
0dy
a
0
b
0
n
Y
k
Ydx
m
X
i
X
2
a
0
I
2
?
b
0
dy
2
y
n
Y
2
2
y
k
Y
2
a
0
dx
m
X
i
X
2
Da
a
0
b
0
dy
y
n
Y
y
k
Y
bdx
x
m
X
x
i
X
a
b
a
v)D2(1
dy
a
0
b
0
2
y
n
Y
2
k
bYdx
m
X
2
x
i
X
2
ady
b
0
n
Y
2
y
k
Y
2
b
a
0
dx
2
x
m
X
2
i
aXv
b
a
D
a
0
b
0
dy
n
Y
k
Ydx
2
x
m
X
2
2
x
i
X
2
2
Da
=
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
??
?
?
?
?
+
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
??
?
?
?
?
()()[] ()()( )( )[]( )( )[ ]
()()[] ()()[]
0
kn
K
im
H
b
a
v)2(1)
kn
F
mi
E
nk
F
im
v(E
b
a
0
kn
K
im
H
b
a
v)2(1)
kn
F
mi
E
nk
F
im
v(E
b
a
D
000
2
a
0
I
2
?00
2
Da
kn
K
im
H
b
a
v)D2(1
kn
F
mi
E
nk
F
im
Ev
b
a
D00
2
Da
=
?
?
?
?
?
?
?++?
=
?
?
?
?
?
?
?++?
=?+
?+++?
92
This can be represented as an Eigenvalue problem, i.e.
0
mn
??
(ik)
mn
C =?
where
()
mnikwhenKH
b
a
)v1(2)FEFE(v
b
a
mnikwhenKH
b
a
)v1(2FvE
b
a
2
b2
ka
a2
b
C
knimknminkim
kkiikkii
3
434
i)ik(
mn
?
?
?
?
?
?
?
?++
=
?
?
?
?
?
?
?
?
?++
?
+
?
=
D2
baI
3
0
2
?
=?
Hence the equation reduces out to
where:
2D
b
3
a
0
I
2
?
? =
where h
0
?
0
I =
1
mn
? = for mn=ik
= 0 for ikmn ?
The Eigenvalues for equation (5.23) are obtained using iterative techniques, in this case
by using Matlab.
The coefficients A
mn
can be calculated separately for the symmetrical and asymmetrical
deflections by an iterative method devised by Ritz.
).....(5.23..........
p
1m
0
q
1n
mn
A
mn
??
(ik)
mn
C?
=
=?
=
?
?
?
?
?
?
?
93
The coefficients for symmetrical group are calculated as follows:
As
0............)(
0............)(
21
13
2115
13
1513
13
1311
13
11
21
11
2115
11
1513
11
1311
11
11
=++?+
=+++?
ACACACAC
ACACACAC
?
?
These are the two of the nine equations that can be written for the symmetric modes of
the SSFF plate. To calculate these coefficients for each mode, we assume one of the
constants to be 1, which here are the constants being multiplied to the diagonals of the C
matrix.
For example we assume
11
A
=1 and the rest of the constants as 0.
.......
21
11
2115
11
1513
11
13
11
11
++++=? ACACACC?
Next we calculate each of the other constants using the above
?
value:
?
?
?+++?=
?+++?=
)/()..........(
)/()..........(
11
1521
11
2113
11
13
11
1115
11
1321
11
2115
11
15
11
1113
?
?
CACACCA
CACACCA
The values obtained from this first iteration are substituted back into the above described
process until the successive values obtained for
?
and A?s are close to the desired
accuracy.
The same process is repeated for the antisymmetrical modes.
94
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Length (inches)
Mode 2
W
idt
h (
inc
he
s
)
0
2
4
6
8
10 0
2
4
6
8
10
-3
-2
-1
0
1
2
3
Width (inches)
Mode 2
Length (inches)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Length (inches)
Mode 1
W
idt
h (
inc
h
e
s
)
0
2
4
6
8
10 0
2
4
6
8
10
-3
-2
-1
0
1
2
3
Width (inches)
Mode 1
Length (inches)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Length (inches)
Mode 4
W
idt
h (
inc
h
e
s
)
0
2
4
6
8
10 0
2
4
6
8
10-3
-2
-1
0
1
2
3
Width (inches)
Mode 4
Length (inches)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Length (inches)
Mode 3
W
idt
h (
inc
h
e
s
)
0
2
4
6
8
10 0
2
4
6
8
10
-3
-2
-1
0
1
2
3
Width (inches)
Mode 3
Length (inches)
Mode 1 Mode 2
Mode 3 Mode 4
Figure 22 Modeshape Correlation of a Completely Free plate with [Leissa 1969].
95
5.5 Point Mass Components on the PCA
As derived in the sections above, the equation of motion of a uniform rectangular
free plate is given by
)24.5()t,y,x(p
t
)t,y,x(w
)t,y,x(wD
2
2
4
E
KKK=
?
?
?+?
where
4
4
22
4
4
4
4
yyx
2
x ?
?
+
??
?
+
?
?
=? is the biharmonic operator.
)]v1(12[
Eh
D
2
3
E
?
= is the Flexural Rigidity.
p (x, y, t) is the transverse external loading.
E is the Young?s Modulus, h is the thickness, v is the Poisson ratio, ? is the mass per unit
area of the plate, w(x, y, t) is the transverse deflection at position (x, y) and time t.
Applying Mode superposition theory, the value of w(x, y, t) due to forced
vibration may be obtained from [Wu et.al. 1997a, b]
)25.5()t(q)y,x(W)t,y,x(w
'n
1i
ii
KKK
?
=
=
where )y,x(W
i
is the i
th
normal mode shape of the free plate, q
i
(t) is the i
th
generalized
co-ordinate, and n? is the mode number. The mode shapes are arranged in order of the
modal frequency. The normalized mode shapes are given by
)y,x(WC)y,x(W
iii
=
96
where C
i
is determined by
1dAWW
i
A
i
=?
?
Substituting equation (5.25) into equation (5.24), and multiplying both sides by )y,x(W
j
.
Integrating the obtained expression over the area A of the plate and applying the
orthogonality of the mode shapes we obtain
dA
A
j
Wt)y,p(x,(t)
j
PdA
j
W
A
4
E
D
j
W
jj
KdA
j
W
A
?
j
W
jj
M
where
(5.26)n'1,j(t),
j
P(t)
j
q
jj
K(t)
j
q
jj
M
?
=
?
?=
?
=
==+ KKK&&
which represent the generalized mass, generalized stiffness and generalized force,
respectively.
If
j
W is a normal mode shape (with respect to ?), then 1M
jj
= and equation (5.26)
reduces to
n'1,j(t).
j
P(t)
j
q
2
j
?(t)
j
q K&& ==+
where
jj
K
jj
M
jj
K
j
? == is the j
th
natural frequency of the free plate.
5.5.1 Eigenvalue Equation of a Constrained Plate
If the inertia forces of the concentrated masses are considered the external exciting
forces, then the forced vibration equation for a free plate may be used to determine the
natural frequencies and mode shapes of the constrained plate.
97
The external forces on the plate during free vibrations are given by
?
= ?
?
?=
?
?
?=
n'
1i
)
a
y,
a
(x
i
W
2
t
(t)
i
q
2
m
2
t
t),
a
y,
a
w(x
2
mt),
a
y,
a
(xP
where m is the point mass present at point (x
a
, y
b
).
Hence the equation of motion obtained for a constrained plate is
n'1,j(t)
i
q
n'
1i
)
a
y,
a
(x
i
W)
a
y,
a
(x
j
Wm(t)
j
q
2
j
?(t)
j
q K&&&& =?
=
?=+
For the constrained plate to perform harmonic free vibration, the generalized co-ordinate
takes the form
t?i
e
j
q(t)
j
q =
where
j
q is the amplitude of q
j
(t) and ? is the natural frequency of vibration of the
constrained plate.
Hence
n'1,j
i
q)
a
y,
a
(x
i
W
n'
1i
)
a
y,
a
(x
j
Wm
2
?
j
q
2
?
j
q
2
j
? K=?
=
+=
Let
{} { } { } { }
[]
?? ??
{}{} )]
a
y,
a
(xWm[
n'xn'
[A]
T
WW
n'xn'
]W[
n'xn'
11,1,
n'xn'
[I]
n'xn'
2
n'
?
2
2
?,
2
1
?
n'xn'
2
?
x1n'n'
q,
2
q,
1
q
x1n'
q
x1n'n'
W,
2
W,
1
W
x1n'
W
==
===
==
KKKK
KKKK
Hence the equation can be written as
{} {}q[A])([I]
2
?q]
2
[? +=
98
Putting [B][A][I] =+ , we get the eigenvalue problem
{} {}q[B]
2
?q]
2
[? =
We can solve the above eigenvalue problem for
j
? , the j
th
natural frequency of the
constrained plate and the corresponding eigenvector{} )n'1,....,(j
(j)
q = .
Calculating the Eigen vectors:
For first eigenvector ?
1
[]
0
30
q
.
.
2
q
1
q
3030
A'
1
?
3030
B'...
3030
A'
1
?
3030
B'
.....
.....
.....
130
A'
1
?
130
B'..
12
A'
1
?
12
B'
11
A'
1
?
11
B'
0{q}][A'
1
?][B'
]{q}[A'
1
?]{q}[B'
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
???
?
=??
=
Hence we get 30 equations in 30 unknowns which are be solved to get the eigenvectors
corresponding the the eigenvalue ?
1
.
The Mode shapes are given by
{}
(j)
}q{
T
y)(x,W
(j)
i
q
n'
1i
y)(x,
i
Wy)(x,
j
w
~
=?
=
=
The closed form model for the vibration of electronic assemblies with components
represented as point masses is used to study the affect of the location of these electronic
components on the dynamic characteristics of the assemblies during drop and shock.
Case studies have been performed to study the affect of the fall off of components from
the assembly on the transient response of the PCA. An analytical model for the vibration
99
characteristics of a 0.5 mm pitch, 132 I/O, 8 mm flex-substrate CSP has been created.
The model structure of the assembly has been shown in Figure 23.
Figure 23 Point Mass representation of the Electronic Assembly.
100
CHAPTER 6
APPLICATION AND VALIDATION OF PREDICTIVE MODEL
In this study, 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. Closed-form models that have been developed for the eigen-
frequencies and mode-shapes of electronic assemblies with various boundary conditions
and component placement configurations. 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.
6.1 Development of Training Signal and High-Speed Measurement Transient Dynamic
Response
Three test boards have been used to study the reliability of fine-pitch ball-grid
arrays in transient-shock. The packages on one of the test board are 27 mm ball-grid
array, 1 mm pitch, 676 I/O; 10 mm Tape Array, 0.8 mm pitch, 144 I/O; 16 mm Flex
BGA, 0.8 mm pitch, 280 I/O; 7 mm CABGA, 0.5 mm pitch, 84 I/O; 15 mm ball-grid
array, 1 mm pitch, 196 I/O; 6mm Tape Array, 0.5 mm pitch, 64 I/O. (Table 6-1). Each
101
10 mm,
144 I/O Tape Array
27mm,
676 I/O PBGA
16 mm,
280 I/O Flex BGA
7mm,
84 I/O CABGA
15 mm,
196 I/O PBGA
6 mm,
64 I/O Tape Array
Figure 24 Interconnect array configuration for Test Vehicles.
102
Table 6-1: Test Vehicles.
10 mm
TABGA
27 mm
PBGA
7 mm
CABGA
16 mm
Flex
BGA
15 mm
PBGA
6 mm
TABGA
I/O 144 676 84 280 196 64
Pitch (mm) 0.8 1 0.5 0.8 1 0.5
Die Size (mm) 7 6.35 5.4 10 6.35 4
Substrate
Thick (mm)
0.36 0.36 0.36 0.36 0.36 0.36
Pad Dia. (mm) 0.30 0.38 0.28 0.30 0.38 0.28
Substrate Pad NSMD SMD NSMD NSMD SMD NSMD
Ball Dia. (mm) 0.48 0.63 0.48 0.48 0.5 0.32
103
component has multiple components. All the components are mounted on one side of the
board. The test board is made of FR-4. The test board is based on standard PCB
technology with no build-up or HDI layers. The test Board is 2.95" by 7.24" by 0.042"
thick
Two of the test boards have been used to study the reliability of chip-scale
packages and ball-grid arrays. Test board A has 10 mm ball-grid array, 0.8 mm pitch, 100
I/O. It has 10 components on one side of the board (Figure 25). Test board B includes
8mm flex-substrate chip scale packages, 0.5 mm pitch, 132 I/O (Table 6-2). The number
of components varies from 6 to 10 on some of the boards. All the components are on one
side of the board. For the 8 mm CSP, conventional eutectic solder, 63Sn/37Pb and lead-
free solder balls 95.5Sn4.0Ag0.5Cu have been studied. Test boards A and B are made of
FR-4. These test boards were based on standard PCB technology with no build-up or HDI
layers. Test Board A and B was 2.95" by 7.24" by 0.042" thick.
10 mm, 100 I/O BGA 8mm 132 I/O BGA
Figure 25: Interconnect array configuration for 95.5Sn4.0Ag0.5Cu and 63Sn37Pb Test
Vehicles.
104
Table 6-2: Test Vehicles.
10mm
63Sn37Pb
8mm
62Sn36Pb2Ag
8mm
95.5Sn4.0Ag 0.5Cu
Ball Count 100 132 132
Ball Pitch 0.8 mm 0.5 mm 0.5 mm
Die Size 5 x 5 3.98 x 3.98 3.98 x 3.98
Substrate Thickness 0.5 mm 0.1 mm 0.1 mm
Substrate Pad Dia. 0.3 mm 0.28 mm 0.28 mm
Substrate Pad Type SMD Thru-Flex Thru-Flex
Ball Dia. 0.46 mm 0.3 mm 0.3 mm
105
The test boards were subjected to a controlled drop. Repeatability of drop
orientation is critical to measuring a repeatable response and to develop a training signal
for statistical pattern recognition. Small variations in the drop orientation can produce
vastly varying transient-dynamic board responses. Significant effort was invested in
developing a repeatable drop set-up. The drop height was varied from 3 feet to 6 feet.
Component locations on the test boards were instrumented with strain sensors. Strain and
continuity data was acquired during the drop event using a high-speed data acquisition
system at 2.5 to 5 million samples per second. The drop-event was simultaneously
monitored with ultra high-speed video camera operating at 50,000 frames per second.
Targets were mounted on the edge of the board to allow high-speed measurement of
relative displacement during drop.
The test boards were dropped in their vertical orientation with a weight attached
to its top edge (Figure 26). The board orientation during drop has been maintained to be
close to zero degrees with the vertical. In addition, the boards were dropped in the
horizontal orientation per the JESD22-B111. Strain, displacement, orientation angle,
velocity, acceleration, and continuity data has been acquired simultaneously.
An image tracking software was used to quantitatively measure displacements during the
drop event. Figure 26 shows a typical angle, and relative displacement plot measured
during the drop event. The position of the vertical line in the plot represents the present
106
Figure 26 Measurement of Velocity, Acceleration, and Relative Displacement During
Impact.
Figure 27 Relative Displacement and Strain Measurement in Horizontal Orientation.
107
Figure 28 Transient Strain-History at Location of CSP during Drop-Event.
108
location of the board (i.e. just prior to impact in this case) in the plot with ?pos (m)? as
the ordinate axis. The plot trace subsequent to the white scan is the relative displacement
of the board targets w.r.t. to the specified reference. Figure 27 shows the board
instrumentation for strain and relative displacement during horizontal JEDEC drop. In
addition to relative displacement, velocity, and acceleration of the board prior to impact
was measured. This additional step was necessary since, the boards were subjected to a
controlled drop, in order to reduce variability in drop orientation. The measured velocity
prior to impact was used to correlate the controlled drop height to free-drop height (
2ghv = ). Thus velocity prior to impact for a 6ft drop (?1.83 meter) will be 5.99 m/s.
6.2 Training of the Predictive Model
Repeatability of drop orientation is critical to measuring a repeatable response and
to develop a training signal for statistical pattern recognition. Small variations in the drop
orientation can produce vastly varying transient-dynamic board responses. To train the
predictive model the test board were dropped and the strain signals obtained when no
failure occurred in the board were compared using the four statistical pattern recognition
technique. The strain signals obtained during these repeatable drops as shown in Figure
29 have been used to train the model to perform prognostics and damage monitoring.
The feature vectors and the Confidence Values obtained by applying the Wavelet Packet
transform, the Mahalanobis Distance measure, the FFT Frequency band energy
computation and the Time frequency analysis Moment calculation have been shown in
Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35, and Figure 36, Figure
37, Figure 38, respectively.
109
Repeatable Test Drop Strains
-2000
-1500
-1000
-500
0
500
1000
1500
-0.001 0.004 0.009 0.014 0.019 0.024 0.029 0.034
Time (sec)
Strain (microstrains)
drop 1
drop 2
drop 3
drop 4
drop 5
drop 6
Figure 29 Strain data for Repeatable Drops of an electronic Assembly.
110
Wavelet Packet Energy Feature Vectors
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100 120 140
Packet Number
Packet E
n
ergy
drop 1
drop 2
drop 3
drop 4
drop 5
drop 6
Figure 30 Repeatable Feature Signatures obtained using Wavelet Packet Energy Vectors.
111
Wavelet Packet Analysis
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1234567
Drop Number
Confidence Value
Figure 31 Confidence Values obtained by applying Wavelet Packet Energy Approach to
Repeatable Drops (No Failure).
112
Mahalanobis Distance Feature Vector
0
2
4
6
8
10
12
14
16
0 5000 10000 15000 20000 25000 30000 35000
Sample Number
Maha
lanobis Dista
n
ce
drop 1
drop 2
drop 3
drop 4
drop 5
drop 6
Figure 32 Repeatable Feature Signatures obtained using Mahalanobis Distance Vectors.
113
Mahalanobis Distance Analysis
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1234567
Drop Number
Confidence Value
Figure 33 Confidence Values obtained by applying Mahalanobis Distance computation to
Repeatable Drops (No Failure).
114
FFT Frequency Band Energy Feature Vector
0.E+00
1.E+06
2.E+06
3.E+06
4.E+06
5.E+06
6.E+06
7.E+06
8.E+06
0 100 200 300 400 500
Frequency
En
erg
y
drop 1
drop 2
drop 3
drop 4
drop 5
drop 6
Figure 34 Repeatable Feature Signatures obtained using FFT Frequency Bands Energy
Vectors.
115
FFT Frequency Band Analysis
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1234567
Drop Number
Confidence Value
Figure 35 Confidence Values obtained by applying FFT Frequency Band Energy
computation to Repeatable Drops (No Failure).
116
Time Frequency Analysis
-1.000E+08
0.000E+00
1.000E+08
2.000E+08
3.000E+08
4.000E+08
5.000E+08
6.000E+08
7.000E+08
8.000E+08
0 200 400 600 800 1000 1200
Moment Number
Time Moment
drop 1
drop 2
drop 3
drop 4
drop 5
drop 6
Figure 36 Repeatable Feature Signatures obtained using Time Moment Vectors.
117
Time Frquency Analysis
-5.E+02
2.E+11
4.E+11
6.E+11
8.E+11
1.E+12
1.E+12
1.E+12
2.E+12
2.E+12
2.E+12
0 200 400 600 800 1000
Moment Number
Freque
ncy Moment
drop 1
drop 2
drop 3
drop 4
drop 5
drop 6
Figure 37 Repeatable Feature Signatures obtained using Frequency Moment Vectors.
118
Time Frequency Moments
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
12
Drop Number
Confidence Value
Frequency Moments
Time Moments
Figure 38 Confidence Values obtained by applying Time Frequency Analysis to
Repeatable Drops (No Failure).
119
6.3 Closed Form Model Results
Closed form models are used to study the damage progression in electronic
devices at product-level. Damage progression has been monitored through correlating
changes in structural parameters with the dynamic response of the assembly. Effective
parameter changes implying damage in an electronic assembly include changes in
flexural rigidity, changes in effective properties, and the change in boundary conditions.
Partial release or complete failure of constraints may produce change in the aspect ratio
or eventual shift of boundary conditions of the assembly. Further, failure of the
interconnects or fall-off of the components may produce changes in the flexural rigidity
and smeared properties of the PCA. Case study has been analyzed using closed-form
models, in which the PCB is press fitted in a product casing, which might open due to
shock and drop causing the boundary conditions for vibration to change from clamped to
free conditions. Clamped condition can be realized in a product with a printed circuit
assembly in which the printed circuit assembly is held between two snap-fit housings. In
the study, various combinations of change in boundary conditions and effective
parameters have been studied in the form of case-studies. In the case-study, the boundary
conditions gradually change to from the original assembly condition to a faulty condition
coupled with effective parametric changes.
6.3.1 CFFF to FFFF Boundary Condition change with change in aspect ratio
Partial release or complete failure of constraints may produce change in the aspect
ratio or eventual shift of boundary conditions of the assembly. Further, failure of the
interconnects or fall-off of the components may produce changes in the flexural rigidity
120
and smeared properties of the PCA. Case study has been analyzed using closed-form
models, in which the PCB is press fitted in a product casing, which might open due to
shock and drop causing the boundary conditions for vibration to change from clamped to
free conditions. Clamped condition can be realized in a product with a printed circuit
assembly in which the printed circuit assembly is held between two snap-fit housings. In
the example case-study, the boundary conditions gradually change to FFFF, from CFFF
in the original assembly. The confidence value degradation for change in boundary
conditions is quite significant as shown in Figure 39 and Figure 40.
Table 6-1 The Case Study Parameters.
Parameter
Initial
Conditions
After Drop
and Shock
Boundary
Condition
CFFF FFFF
Aspect Ratio
(a/b)
7/3
Compared for
up to 10%
change
Flexural
Rigidity
12.386 N-m
2
No Change
Density
1800 kg/m
3
No Change
Poisson?s ratio 0.3 No Change
121
Mode 1
0
0.2
0.4
0.6
0.8
1
1.2
02468
Percentage Change in Aspect Ratio (%)
Confiden
ce Value
Wavelet Packet Energy
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moments
Frequency Moments
Figure 39 Confidence Value Degradation with Change in Aspect Ratio for Mode 1.
122
Mode 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
02468
Percentage Change in Aspect Ratio (%)
Confidence Value
Wavelet Packet Energy
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moments
Frequency Moments
Figure 40 Confidence Value Degradation with Change in Aspect Ratio for Mode 2.
123
6.3.2 Point Mass Fall off from Assembly corresponding to Package Falloff
The location of point masses on the plate has been shown to cause a significant
change in the vibration frequencies and mode shapes of an assembly. To fully understand
the dynamic characteristics of the affect of the presence of point mass components on the
board, the pristine configurations of the board have been perturbed to simulate failure
corresponding to package fall off and the damage progression has been monitored using
SPR techniques. The affect of component fall off, i.e. point mass removal with respect to
the location of the point mass has been studied. The assembly as described in the section
5.5 on point mass modeling is shown in Figure 41 and the various point masses on the
assembly have been labeled using numbers. The point masses have been removed one at
a time and the vibration characteristics of the perturbed assembly has been simulated to
study the sensitivity of the assembly to the location of the failure. The damage
monitoring based on the confidence value graph obtained in this study is shown in Figure
42. The sensitivity of failure detection irrespective to the sensor location has been studied
and shown in Figure 43, the package fall off occurs at location 1 and the displacement is
monitored at each package location i.e. the possible sensor locations.
The point mass assembly has also been used to study the affect of the removal of
a number of point masses from the assembly to study the affect on the modeshapes. As
shown in Figure 44 the removal of two point masses from the assembly causes a
significant change in Mode shapes of the electronic assembly.
124
Loc 1
Loc 2
Loc 3
Loc 4
Loc 5
Loc 10
Loc 9
Loc 8
Loc 7
Loc 6
Figure 41 Point Mass Closed Form Model and Numbering of Location of Packages.
125
Degradation in Mode Shape 1 with respect to Package
Fall off Location
0
0.2
0.4
0.6
0.8
1
0246810
Package Fall off Location
Confidence Value
Wavelet Packet Analysis
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moments
Frequency Moments
Figure 42 Degradation in Confidence Value with respect to Location of Package Fall off.
126
Sensitivity of Package Fall off
Irrespective of Sensor Placement
0
0.2
0.4
0.6
0.8
1
1.2
0246810
Package Location at which Signal
Measured
Confidence Value
Wavelet Packet Analysis
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moments
Frequency Moments
Figure 43 Degradation in Confidence Value with Package Fall off from Location 1.
127
0
0.02
0.04
0.06
0
0.05
0.1
0.15
-15
-10
-5
0
5
10
15
Width
Mode 3
Length
0
0.02
0.04
0.06
0
0.05
0.1
0.15
-15
-10
-5
0
5
10
15
Width
Mode 3
Length
2 Packages
Missing
Package Missing
Packages Missing
Figure 44 Effect of Package Fall off on Modeshape of Assembly.
128
6.4 Model Based Correlation of Damage
The closed form modeling approach described above shows us the affect of the
change in effective parameters on the reliability of the confidence values. To provide a
physical relevance to the above approach an explicit finite element model for the free
drop and horizontal drop of a 0.5 mm pitch, 132 I/O, 8 mm flex-substrate CSP has been
created. A reduced integration shell elements (S4R) is used for the PCB, and the various
component layers such as the substrate, die attach, silicon die, mold compound have been
modeled with C3D8R elements. The interconnects are modeled using two-node beam
elements (B31) in place of solder balls as shown in Figure 46.and Figure 46 Smeared
properties have been derived for the CSP considering all the individual components
mentioned above. The concrete floor has been modeled using rigid R3D4 elements. A
weight has been attached on the top edge of the board.
The explicit models created for the study for the Vertical free drop and the Horizontal
drop are shown in Figure 47and Figure 48 respectively. Some of the common faults that
occur in an electronic assembly due to a drop and shock event have been modeled, and by
applying the Statistical Pattern recognition techniques the degradation in confidence
value is physically correlated to the occurrence of damage in an assembly. The faults
simulated in this study are Solder ball cracking, solder ball failure, chip cracking, chip
delamination and Package Fall off. The faults have been simulated for both the vertical
and horizontal drops.
129
Figure 45 Package with Solder Beam Array.
Figure 46 Solder Bam Array modeled to represent Solder Balls.
130
Detailed
Package
Packages
based on
calculated
Smeared
Properties
Figure 47 Vertical Drop Model developed for the Study.
Figure 48 Horizontal Drop Model developed for the Study.
131
6.4.1 Solder Ball Cracking and Failure
The above described model for the free drop was analyzed several times, with all
solder beams intact and with the various corner solder beams damaged (cracked) and
failed. The statistical pattern recognition methods described in this study have been
applied to the time history output of the strain signal obtained at the PCB surface below
the centre of the package, where the sensor would have been mounted in an experimental
setup. The solder beam array in the failure simulations for solder ball failure is shown in
Figure 49 and the various solder beams missing have been marked and shown for the
models developed. The solder beam array for the damaged beam model is also shown in
Figure 50 and the damaged beam can be seen at the corners of the solder array.
The failure of corner solder balls has been simulated by removing the corner
interconnects as the corner solder balls have the most stress concentration and hence are
mostly the first to fail in a solder array. The cracking of the solder ball has been simulated
by reducing the cross sectional area of the corner solder beams and the model has been
simulated for up to all four corner ball damaged, for both vertical and horizontal drop.
The confidence values computed for solder ball failure are shown in Figure 51, for a
vertical drop and in Figure 52, for a horizontal drop orientation. The damage monitoring
of solder ball damage is shown in Figure 53, for a vertical drop and in Figure 54, for a
horizontal drop orientation. The confidence value shows a drop in confidence with the
degradation of reliability, thus showing the applicability of the above stated damage
monitoring methods to the reliability studies of Electronic assemblies.
132
(a) One Interconnect Missing
(b) Two Interconnects Missing
(c) Three Interconnects Missing (d) Four Interconnects Missing
Figure 49 Model Configurations for Correlation of Interconnect Failure to Confidence
Value Degradation.
133
(a) One Interconnect Damaged (b) Two Interconnects Damaged
(c) Three Interconnects Damaged (d) Four Interconnects Damaged
Figure 50 Model Configurations for Correlation of Interconnect Damage to Confidence
Value Degradation.
134
Damage Monitoring for a Vertical Drop
0.0
0.2
0.4
0.6
0.8
1.0
01234
Number of Corner Interconnect Failure
Confidence Value
Wavelet Packet Energy
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moments
Frequency Moments
Figure 51 Confidence Value degradation in Transient PCB Strain with Solder Ball
Failure for Vertical Drop.
135
Damage Monitoring for a Horizontal Drop
0
0.2
0.4
0.6
0.8
1
01234
Number of Corner Interconnect Failure
Confidence Value
Wavelet Packet Analysis
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moments
Frequency Moments
Figure 52 Confidence Value degradation in Transient PCB Strain with Solder Ball
Failure for Horizontal Drop.
136
Damage Monitoring for a Vertical Drop
0
0.2
0.4
0.6
0.8
1
1234
Number of Corner Interconnects
Damaged
Confidence Value
Wavelet Packet Energy
Mahalanobis Distance
Time Moments
Frequency Moments
Figure 53 Confidence Value degradation in Transient PCB Strain with Solder Ball
Damage for Vertical Drop.
137
Damage Monitoring for a Horizontal Drop
0
0.2
0.4
0.6
0.8
1
01234
Number of Corner Interconnects
Damaged
Confidence Value
Wavelet Packet Analysis
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moments
Frequency Moments
Figure 54 Confidence Value degradation in Transient PCB Strain with Solder Ball
Damage for Horizontal Drop.
138
6.4.2 Chip Fracture
The model has also been modified to represent the cracking of the silicon chip due
to drop and shock of the electronic assembly. The cracked surface on the chip is
represented by detaching the chip and creating a contact surface representing the chip
fracture between two parts of the detached chip. The chip fracture and the contact surface
representing the chip crack has been shown in Figure 55.
Statistical pattern recognition is applied to the time history output of the strain
signal obtained at the PCB surface below the centre of the package, where the sensor
would have been mounted in an experimental setup. The confidence value obtained for
the vertical drop and the horizontal drop orientation has been plotted in Figure 56 and
Figure 57 respectively, and shows the expected degradation in confidence value with the
fracture of the chip in the assembly.
6.4.3 Chip Delamination
Shock and drop of an assembly can also cause delamination between the chip and
the substrate, thus causing failure in the assembly. The delamination is modeled by
detaching part of the substrate from the chip. A contact surface is then created to
represent the delaminated surface between the substrate and the chip attach. The
delaminated chip modeled by the contact surface between the chip and the substrate has
been shown in Figure 58.
The strain is obtained at the PCB surface below the centre of the package, and
statistical pattern recognition is applied to the obtained signals. The confidence value
139
drop obtained for the horizontal drop orientation and the vertical drop orientation is
shown in Figure 59 and Figure 60 respectively.
Contact surface at
crack location
Figure 55 Model configuration for Chip Fracture (cracking).
140
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelet
Packet
Mahalanobis
Distance
FFT Energy Time
Moments
Frequency
Moments
Damage Monitoring for Vertical Drop
Uncracked Chip
Cracked Chip
Confidence Value
Figure 56 Confidence Value degradation in Transient PCB Strain with Chip Failure for
Vertical drop orientation.
141
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelet
Packet
Mahalanobis
Distance
FFT Energy Time
Moments
Frequency
Moments
Damage Monitoring for Horizontal Drop
Uncracked Chip
Cracked Chip
Confidence Value
Figure 57 Confidence Value degradation in Transient PCB Strain with Chip Failure for
Horizontal drop orientation.
142
Contact Surface
representing
Delamination
Figure 58 Model configuration for Chip Delamination.
143
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelet
Packet
Mahalanobis
Distance
FFT Energy Time
Moments
Frequency
Moments
Damage Monitoring for Vertical Drop
No Delamination
Delamination
Confidence Value
Figure 59 Confidence Value degradation in Transient PCB Strain with Chip
Delamination for Vertical drop orientation.
144
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelet
Packet
Mahalanobis
Distance
FFT Energy Time
Moments
Frequency
Moments
Damage Monitoring for Horizontal Drop
No Delamination
Delamination
Confidence Value
Figure 60 Confidence Value degradation in Transient PCB Strain with Chip
Delamination for Horizontal drop orientation.
145
6.4.4 Package Fall Off
The loss of package from the board is also a scenario encountered in the drop and
shock testing of electronic assemblies, and thus has been considered. The finite element
explicit model has been developed for the case of a package fall off by removing one
package from the assembly, as simulated using closed form models in section 6.3.2. The
model developed for failure, the removal of the package from the assembly to simulate
failure has been shown in Figure 61. The failure model has been developed for both the
vertical drop and the horizontal drop.
The strain is obtained at the PCB surface below the centre of the package falling
off, where the sensor would have been mounted in an experimental setup, and statistical
pattern recognition is applied to the obtained signals. The confidence value drop obtained
in the vertical drop failure simulation is shown in Figure 62 . As would be expected, as
soon as the package falls off the assembly the confidence value drops to zero, thus
showing the applicability of the statistical pattern recognition techniques for damage
monitoring in electronic assemblies. The health degradation due to package fall off in the
horizontal model can be seen for the horizontal drop in Figure 63.
6.5 Experimental Validation
The damage monitoring of electronic assemblies by statistical pattern recognition has
been applied to experimental results obtained by the JESD22-B11 drop and shock testing
of PCB011. The boards were thermal cycled for 750 cycles and then subjected to JEDEC
drop testing. The strain signal obtained from sensors placed on the board at appropriate
locations are employed to perform statistical pattern recognition.
146
(a)
(b)
Figure 61 Model configuration for loss of package from assembly (a) Vertical Drop (b)
Horizontal Drop.
147
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelet
Packet
Mahalanobis
Distance
FFT Energy Time
Moments
Frequency
Moments
Damage Monitoring for Vertical Drop
No Failure
Package Fall off
Co
nf
id
en
ce Value
Figure 62 Confidence Value degradation in Transient PCB Strain with Package Loss for
Vertical drop orientation.
148
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelet
Packet
Mahalanobis
Distance
FFT Energy Time
Moments
Frequency
Moments
Damage Monitoring for Horizontal Drop
No Failure
Package Fall off
Co
nf
id
en
ce Value
Figure 63 Confidence Value degradation in Transient PCB Strain with Package Loss for
Horizontal drop orientation.
149
The feature vectors obtained using the Wavelet packet analysis, the Mahalanobis
distance computation, the FFT frequency band analysis and the time moments and
frequency moment distribution are shown in Figure 64, Figure 65, Figure 66, Figure 67,
and Figure 68 respectively. As shown the feature vectors used in failure pattern
classification are seen to show significant change in case of failure of a package
irrespective to the placement of the sensor. Statistical pattern recognition has been
applied to the strain signals taken at the various drops of the board. A linear regression fit
of the confidence values obtained show a progressive degradation of reliability of the
assembly with the number of drops. A significant drop of reliability occurs with failure of
individual packages on the assembly. The confidence value plots, as shown in Figure 69
and Figure 70, computed by statistical pattern recognition of strain signals obtained from
the strain sensors show a good correlation of the reliability degradation with the
experimental values. The above correlation is performed to exhibit the applicability of
statistical pattern recognition techniques to monitor damage progression in experimental
setups. The degradation in confidence values demonstrates the capability of the approach
in sensing damage progression and failure at the same location and other locations in the
board assembly. Package-3 transient strain shows degradation in confidence value
because of failure at package location-1. Package-1 transient strain shows degradation in
confidence value due to failure at package location-1 and package-location-3.
150
Feature Vector At Drop 1 and at Failure Drop
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2040608010
Packet Number
Ener
gy Vector
drop 1
drop 63
Figure 64 Wavelet Packet Energy Feature Vector used in Failure Classification.
151
Distance Feature Vectors at Drop1 and
Failure Drop
0
10
20
30
40
50
60
-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
time (sec)
Ma
hala
nobi
s Distance drop 1
drop 63
Figure 65 Mahalanobis Distance Feature Vector used for Failure Classification.
152
FFT Frequency Band Energy Feature Vector at
Drop 1 and Failure Drop
0
1E+11
2E+11
3E+11
4E+11
5E+11
6E+11
7E+11
8E+11
9E+11
0 200 400 600 800
Frequency (Hz)
Band Energy
drop 1
drop 63
Figure 66 FFT Frequency Band Energy Feature Vector used for Failure Classification.
153
Time Moment Feature Vector
0
1000
2000
3000
4000
5000
6000
0 0.01 0.02 0.03 0.04 0.05
Time (sec)
Instantane
ous Frequency (Hz)
drop 1
drop 63
Figure 67 Time Moment Feature Vector used for Failure Classification.
154
Frequency Moment Feature Vector
-5
-3
-1
1
3
5
7
9
11
13
15
0 100 200 300 400
Frequency (Hz)
Instanaeous Time (sec)
drop1
drop 63
Figure 68 Frequency Moment Feature Vector used for Failure Classification.
155
Damage Degradation
(Sensor under Package-1)
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200
Drop Number
Con
fidence Value
Wavelet Packet Energy
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moment
Frequency Moment
Package-1
Fails
Package-2
Fails
Package-3
Fails
Package-4
Fails
Con
fidence Value
Figure 69 Confidence value degradation showing progressive damage with the Drops.
156
Damage Degradation
(Sensor Under Package-3)
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200
Drop Number
Confidence Value
Wavelet Packet Energy
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moment
Frequency Moment
Package-1
Fails
Package-2
Fails Package-3
Fails
Package-4
Fails
Confidence Value
Figure 70 Confidence value degradation showing progressive damage with the Drops.
157
6.6 Solder Joint Built in Reliability Test (SJ-BIRT)
Solder joints in electronic assemblies subjected to shock and drop are prone to
failure due to high strain rate transient bending produced by such loads. The damage in
the solder joint occurs due to overstress or cumulative fatigue and occurs in the form of
plastic work and cracks in the solder balls. The Built in self test (BIST) approach is
currently being applied to the testing of Digital chips and systems [Hashempour et.al.
2004], Internal and External Memories like DDR, QDR, Double DDR SRAM, FCRAM,
and RLDRAM [Kim et.al 2004], ASICs [Sato et.al. 2001], and Mixed-Signal BIST
applied on devices like ADCs, DACs, filters, amplifiers, power regulators, mixers [Sunter
2002, Huang et.al. 2000]. Built-In self tests have also been applied to the functional
testing of FPGAs, testing the core logic block, the memory and various interconnect
faults [Dutt 2006, Abramovici et.al. 2004, Liu et.al. 2003, Sun et.al. 2000, Stroud et.al.
1998]. The BIST approach has also been applied to testing of MEMS Accelerometers
[Deb et.al. 2006] and Industrial circuits [Kiefer et.al. 2000] but the current version of
BIST approach is focused on reactive failure detection and provides limited insight in to
solder joint reliability and residual life.
The stress magnitudes occurring in the solder is difficult to predict and quantize
and hence the prognostic approach based solely on accelerated life testing can give
inaccurate life predictions [Lall 2004]. With the current testing methodologies utilizing
the functional performance as a measure of health, the in-situ Solder joint Built in
Reliability test (SJ-BIRT) provides prognostics even when the damaged solder joint does
not cause immediate operational failure. Built in Self Tests (BIST) are applied at
component levels to perform reliability tests to isolate manufacturing failures that might
158
occur, but with the use of SJ-BIRT the failure modes caused during the PCB-FPGA
assembly can also be identified. The failure in a solder-joint manifests itself in the form
of an increase in the resistance of the joint from milliohms to tens and hundreds of ohms.
The failure might be an open circuit or might be due to the solder cracking, which results
in different values of the resistance of the joint. A crack progressing with time becomes
longer until it causes the solder joint to open and the device to fail completely, thus by
using an in-situ BIRT we can predict the degradation of the solder joint reliability.
The SJ BIRT is performed by attaching a small capacitor to the I/O port and by
programming the FPGA to write a voltage to the capacitor through a solder joint being
tested and reading the voltage from the another solder joint. A prototype circuit has been
modeled and analyzed in PSPICE to study the behavior of a damaged solder interconnect
compared to an undamaged interconnect as shown in Figure 71.
The undamaged solder interconnect is modeled as a resistance having a value to 20
milliohms while a failed interconnect is modeled as a 20 ohms. The voltage across the
1?F capacitor is measured and observed to identify failures in the device. The voltage
characteristics of the capacitor are shown below in Figure 72 for both damaged and
undamaged interconnects. The basic algorithm behind the SJ BIRT technique is the
writing of a logical high i.e. ?1? to charge the capacitor and then reading the voltage
across the charged capacitor. The capacitor gets fully charged when the solder
interconnect is undamaged and a logical high ?1? is read by the BIRT. When due to the
high resistance produced by the damaged interconnect, the RC time constant becomes
high and the capacitor does not get fully charged, a logical low i.e. ?0? is read by the SJ
BIRT.
159
The voltage signals obtained from the SJ BIRT circuit has been analyzed for
various values of the resistance value and statistical pattern recognition has been applied
to these voltage signals for prognostics and health monitoring of the device. The
application of statistical pattern recognition provides a quantified damage index for the
reliability and performance of the FPGA as even though there might be damage in the
device, the SJ BIRT might get a logical high of ?1? if the damage is not complete failure
and the resistance value hasn?t increased enough to cause the capacitor not to charge to
the threshold set for the testing.
The confidence value graphs shown in Figure 73 show that the above outlined method
can be applied successfully to the prognostics of FPGA based on a SJBIST approach.
Figure 71 Solder Joint Built in Reliability test Circuit Design.
160
Voltage Characteristics
0
1
2
3
4
5
6
100 110 120 130 140 150
Time (nanosec)
V
o
lat
g
e acr
o
ss C
a
pacitor
(V
olts
)
0.02 ohms (Undamaged)
0.2 ohms
2 ohms
20 ohms
100 ohms
Figure 72 Voltage Characteristics obtained due to variation in Solder interconnect
resistance due to heath degradation.
161
SJ-BIRT Confidence Value
0
0.2
0.4
0.6
0.8
1
1.2
0.01 0.1 1 10 100
Resistance Value (Logarithmic Scale)
Confidence Value
Wavelet Packet Energy
Mahalanobis Distance
FFT Frequency Band
Energy
Time Moment
Frequency Moment
Figure 73 Damage Detection using a Solder Joint Built in Reliability Test (SJ-BIRT).
162
CHAPTER 7
SUMMARY & CONCLUSIONS
In this work an approach for health monitoring in electronic products, based on
closed-form energy-method based models, explicit finite elements, and statistical pattern
recognition has been developed. Statistical pattern recognition techniques previously
employed in several engineering and scientific disciplines such as biology, psychology,
medicine, marketing, artificial intelligence, computer vision and remote sensing, have
been used for monitoring damage progression in electronic assemblies. Damage proxies
have been developed based on the Wavelet Packet Energy, the Mahalanobis Distance
approach, the FFT Frequency Band energy and the Time Frequency Analysis methods
and applied to the experimental and simulation data.
The above approach addresses detection and monitoring of shock-even damage,
without requiring continuous high-speed interconnect resistance monitoring during drop
and shock events. Detected changes in damage have been demonstrated with changes in
the energy feature vector and the change in the confidence value of the signal. The
approach has been demonstrated for several test cases including
�? Failure and damage of one-or-more interconnects in an area array package
�? Chip Delamination
�? Chip cracking
163
�? Package Fall off
�? failure of packages at the location of transient strain measurement or at another
location
�? change in the structural properties and the transient dynamic response of the
structure.
The various damage proxies have been developed by simulating the studied faults in
the explicit finite element models and the closed form analytical models developed for
the study. The confidence value is shown to quantify the reliability degradation and
provides an effective measure for the occurrence of a fault in the assembly. As shown in
Figure 74 the drop in confidence value is directly related to the amount of damage that
occurs in the assembly.
A Solder joint Built in Reliability Test (SJ-BIRT) has been studied and statistical
pattern recognition has been applied to the voltage signals obtained from the test. The
Built in self test (BIST) approach currently being applied to the testing of Digital chips
and systems, Internal and External Memories like DDR, QDR, Double DDR SRAM,
FCRAM, and RLDRAM, ASICs, and Mixed-Signal BIST applied on devices like ADCs,
DACs, filters, amplifiers, power regulators, mixers, functional testing of FPGAs, testing
the core logic block, the memory and various interconnect faults, testing of MEMS
Accelerometers and Industrial circuits has been applied to study the Solder joint
Reliability in a FPGA device.
164
Damage Degradation
0
0.2
0.4
0.6
0.8
1
1.2
0246
Confidence Value
Repeatable
Partial Corner I/O Damage
Corner I/O Failure
Chip Failure
Chip Delamination
Packg Fall off
Figure 74 The Degradation in Confidence Value relative to Damage occurrence in
Assembly.
165
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