damage diagnosis of algorithms for wireless structural

Department of Civil and Environmental Engineering

Stanford University

Report No. 165 November 2007

Damage Diagnosis of Algorithms for Wireless Structural Health Monitoring

By

Krishnan Nair Kesavanand Anne S. Kiremidjian

The John A. Blume Earthquake Engineering Center was established to promote research and education in earthquake engineering. Through its activities our understanding of earthquakes and their effects on mankind’s facilities and structures is improving. The Center conducts research, provides instruction, publishes reports and articles, conducts seminar and conferences, and provides financial support for students. The Center is named for Dr. John A. Blume, a well-known consulting engineer and Stanford alumnus.

Address:

The John A. Blume Earthquake Engineering Center Department of Civil and Environmental Engineering Stanford University Stanford CA 94305-4020

(650) 723-4150 (650) 725-9755 (fax) [email protected] http://blume.stanford.edu

©2007 The John A. Blume Earthquake Engineering Center

ii

Abstract

Recent research efforts in wireless structural health monitoring have resulted in an

explosion in the development of new sensors. Little attention, however, has been focused

on the efficient and effective use of the data collected by these sensors. While these

wireless sensor networks enable dense instrumentation, the amount of data that needs to

be transmitted can prove to be prohibitive. The main difficulty arises from the low data

rates associated with low power ad-hoc wireless sensor networks. Thus, data transmission

over the wireless network is demanding, time consuming and can significantly reduce

power source life. Typically these data are required because current damage detection

algorithms perform global system level analysis rather than local sensor level analysis. In

this dissertation, three local sensor based damage diagnosis algorithms using statistical

signal processing and pattern classification techniques have been developed. The main

features of these algorithms are that they are simple, robust and computationally efficient.

The first algorithm uses time series to model the vibration signal and defines a damage

sensitive feature DSF using the first three autoregressive (AR) coefficients. A t-test on

the DSF’s is used to discriminate between an undamaged state and a damaged state. This

algorithm is valid for linear and stationary signals.

The second algorithm utilizes the first three AR coefficients as the feature vector.

Damage detection is performed using the Gaussian Mixture Models (GMM’s) and the

gap statistic. This algorithm, like the first algorithm described above, is valid for linear,

iii

stationary signals. This algorithm is shown to be more effective in detecting minor

damage patterns in comparison to the first algorithm. A damage measure has been

developed using the Mahalanobis distance between the means of the damaged and

undamaged datasets.

The third algorithm uses the wavelet energies at the fifth, sixth and seventh dyadic scales

as feature vectors. This algorithm allows the use of non-stationary signals. This algorithm

requires a creation of a database of baseline signals. The first part of this algorithm

requires finding that signal in the database closest to the new signal. The second part of

this algorithm is to obtain the feature vectors. Both of these steps are performed using

principal components analysis. Damage detection is performed using the k-means

algorithm in conjunction with the gap statistic. A damage measure has been developed

using the Euclidean distance between the means of the damaged and undamaged feature

vector.

The performance of the developed algorithms is validated using the datasets of the ASCE

Benchmark Structure. It is observed that the damage patterns as defined in the ASCE

Benchmark Structure are consistently identified using these algorithms. The damage

measures are also shown to correlate well with the extent of damage.

iv

Acknowledgments

This research was supported in part by the John A. Blume Research Fellowship, the

National Science Foundation Grant No. CMS-0121841, the National Science Foundation

- George E. Brown, Jr. Network for Earthquake Engineering Simulation Research Grant

No. 15BBK16379 and by the John A. Blume Earthquake Engineering Research Center.

v

Table of Contents

Abstract ii

Acknowledgments iv

List of Tables viii

List of Figures ix

1 Introduction 1

1.1 Motivation .........................................................................................................1

1.2 Objectives..........................................................................................................6

1.3 Organization of the Thesis ................................................................................6

2 A Time Series Based Damage Detection Algorithm with Hypothesis Testing 8

2.1 Description of the Damage Detection Algorithm .............................................9

2.1.1 Modeling of the Vibration Signals ........................................................10

2.1.2 Development of Damage Sensitive Feature (DSF) ...............................17

2.1.3 Correlation of the AR Coefficients to the Structural System ................18

2.2 Damage Detection Algorithm Synthesis .........................................................24

2.3 Application Results .........................................................................................25

2.3.1 Damage Detection .................................................................................28

2.4 Summary .........................................................................................................33

3 A Time Series Based Structural Damage Detection Algorithm Using

Gaussian Mixture Modeling 37

3.1 Overview of the Damage Diagnosis Algorithm ..............................................38

vi

3.2 Modeling of Vibration Signals ........................................................................40

3.3 Gaussian Mixture Modeling ...........................................................................41

3.4 Damage Diagnosis using Gaussian Mixture Models ......................................45

3.4.1 Damage Identification using the Gap Statistic ......................................45

3.5 Damage Extent using the Mahalanobis Metric ...............................................47

3.6 Application ......................................................................................................48


3.6.2 Damage Extent ......................................................................................56

3.6.3 Effect of Noise on the Damage Diagnosis.............................................58

3.7 Summary .........................................................................................................60

3.8 Appendix: The EM Algorithm ........................................................................61

4 Damage Feature Extraction from Wavelet Transform of Vibration Signals 65

4.1 Properties of the Continuous Wavelet Transform...........................................66

4.1.1 Haar Wavelet .........................................................................................69

4.1.2 Morlet Wavelet ......................................................................................70

4.2 Derivation of the Damage Sensitive Feature using Wavelet Coefficients

of Acceleration Signals for a SDOF System ...................................................71

4.2.1 Wavelet Transform of Acceleration Signals .........................................74

4.2.1.1 Haar Wavelet Coefficients of Acceleration Signals ................74

4.2.1.2 Morlet Wavelet Coefficients of Acceleration Signals .............75

4.2.2 Damage Sensitive Feature .....................................................................77

4.2.2.1 Haar Basis ................................................................................77

4.2.2.2 Morlet Basis ............................................................................79

4.3 Derivation of the Damage Sensitive Feature using Wavelet Coefficients

of Acceleration Signals for a MDOF System .................................................81

4.3.1 Wavelet Coefficients of Acceleration Signals .......................................82

4.4 Application ......................................................................................................85


4.4.1.1 Sensor 2 ...................................................................................86

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4.4.1.2 Sensor 3 ...................................................................................86

4.4.1.3 Sensor 9 ...................................................................................86

4.4.1.4 Effect of Noise .........................................................................96

4.5 Summary .........................................................................................................98

4.6 Appendix: Derivation of the Integral IH ..........................................................99

5 A Wavelet Based Damage Detection Algorithm 103

5.1 Overview of Algorithm .................................................................................104

5.2 Application of Principal Components Analysis in Optimal Selection of

Baseline Signal and Feature Extraction ........................................................107

5.2.1 Principal Components Analysis ..........................................................107

5.2.2 Selection of the Closest Baseline Signal .............................................108

5.2.3 Feature Extraction ...............................................................................112

5.3 Damage Diagnosis ........................................................................................115

5.3.1 Damage Detection using the k-means Algorithm and the Gap

Statistic ................................................................................................115

5.3.2 Damage Extent Measure ......................................................................120

5.4 Summary .......................................................................................................121

6 Summary, Conclusions and Future Work 123

6.1 Summary .......................................................................................................124

6.2 Conclusions ...................................................................................................126

6.3 Future Work and Research Needs .................................................................127

6.3.1 Damage Diagnosis ...............................................................................127

6.3.2 Damage Prognosis ...............................................................................128

viii

List of Tables

Number Page

Table 2.1: Sensitivity of AR coefficients to the number of data points .............................17

Table 2.2: Results of damage decision for damage pattern 1 ............................................30



Table 2.5: Results of damage decision for damage pattern 4 and 5 ..................................31


Table 3.1: Results from the EM Algorithm for various damage patterns (DP) .................55

Table 3.2: Variation of DM for various sensors and different damage patterns ................57

Table 3.3: Variation of the means of the undamaged and the damaged data obtained

from sensor 2 with different noise to signal ratios (NSR) ...............................58

Table 3.4: Variation of the damage metric DM with noise to signal ratio (NSR) .............59

Table 4.1: Variation of DM for the Morlet wavelet based damage sensitive feature for

various sensors and different damage patterns ................................................96

Table 4.2: Variation of DM for sensor 2 with different noise to signal ratios (NSR) for

damage patterns DP 1-6 ...................................................................................97

Table 5.1: Variation of DM for the DB4 Wavelet based damage sensitive feature for

all sensors and different damage patterns ......................................................120

Table 6.1: Summary of damage detection algorithms developed in this dissertation .....125

ix

List of Figures

Number Page

Figure 2.1: Plot of a typical raw acceleration time history from an undamaged case

serving as the reference signal for subsequent damage detection (Sensor 2) ..10

Figure 2.2: Autocorrelation Function of the Normalized Signal .......................................12

Figure 2.3: Determination of Optimal AR model order (a) Variation of AIC with AR

model order for MA orders varying from 0 to 3 and (b) Variation of Cross

Validation Error with AR model order ............................................................14

Figure 2.4: Verification of the i.i.d characteristics and normality of residuals: (a)

Variation of residuals with time. (b) Normal probability plot of the

residuals. (c) Variation of the autocorrelation function of the residuals

with lag.............................................................................................................16

Figure 2.5: Variation of DSF with record number for different damage patterns for

Sensor 2: (a) Damage Pattern 1, (b) Damage Pattern 2, (c) Damage Pattern

3, (d) Damage Pattern 4, (e) Damage Pattern 5 and (f) Damage Pattern 6. .....20

Figure 2.6: ASCE Benchmark Structure (Johnson et al., 2004) ........................................26

Figure 2.7: Placement of sensors and direction of acceleration in the ASCE

Benchmark Structure (http:// wusceel.cive.wustl.edu/ asce.shm/

benchmarks.htm) ..............................................................................................27

Figure 2.8: Dispersion of Values of DSF’s for Damage Pattern 6 sensors along (a)

Face 1 and (b) Face 2 .......................................................................................34

x

Figure 2.9: Dispersion of values of DSF’s for Damage pattern 2 sensors along (a)

Face 1 and (b) Face 2 .......................................................................................35

Figure 3.1: Migration of feature vectors (defined by the first three AR coefficients)

from an undamaged state to damage patterns 1 and 2 as defined by the

ASCE Benchmark Structure ............................................................................39

Figure 3.2: Variation of log-likelihood with number of mixtures in the dataset ...............44

Figure 3.3: Illustration of within cluster distance ..............................................................46

Figure 3.4: Migration of the feature vectors with damage for minor patterns (a)

Damage pattern 6 and (b) Damage Pattern 3 ...................................................49

Figure 3.5: Migration of the feature vectors with damage for moderate patterns (a)


Figure 3.6: Migration of the feature vectors with damage for major patterns (a)


Figure 3.7: Illustration of the gap statistic for a damaged case (a) Distribution of AR

coefficients (b) Variation of the observed and expected value of log(Wk)

with number of mixtures (c) Variation of the gap statistic with number of

mixtures............................................................................................................53

Figure 3.8: Illustration of the gap statistic for an undamaged case (a) Distribution of

AR coefficients (b) Variation of the observed and expected value of

log(Wk) with number of mixtures (b) Variation of the gap statistic with

number of mixtures ..........................................................................................54

Figure 3.9: Variation of the damage metric DM with damage pattern for sensor 2 ..........56

Figure 4.1: Haar Wavelet (a) Haar Basis Function and (b) its Fourier Transform ............72

Figure 4.2: Morlet Wavelet (a) Morlet Basis Function and (b) its Fourier Transform ......73

Figure 4.3: Migration of the Morlet wavelet based damage sensitive feature E7 for

sensor 2 with damage for minor patterns (a) Damage pattern 6 and (b)

Damage Pattern 3 .............................................................................................88

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Figure 4.4: Migration of Morlet wavelet based damage sensitive feature E7 for sensor

2 with damage for major patterns (a) Damage pattern 1 and (b) Damage

Pattern 2 ...........................................................................................................89


3 with damage for (a) Damage pattern 4 and (b) Damage Pattern 5

(Undamaged ; Damaged +) ............................................................................90


9 with damage for (a) Damage pattern 3 and (b) Damage Pattern 4


Figure 4.7: Migration of the Haar wavelet based damage sensitive feature E6 for

sensor 2 with damage for minor patterns (a) Damage pattern 6 and (b)

Damage Pattern 3 .............................................................................................92


sensor 2 with damage for major patterns (a) Damage pattern 1 and (b)

Damage Pattern 2 .............................................................................................93


sensor 3 with damage for (a) Damage pattern 4 and (b) Damage Pattern 5



sensor 9 with damage for (a) Damage pattern 3 and (b) Damage Pattern 4


Figure 4.11: Illustration of the Proof of the Contour Integration Formula ........................99

Figure 5.1: Illustration of a similar and dissimilar cloud by comparing E1,baseline and

E1,new ...............................................................................................................109

Figure 5.2: Histogram of for sensor 2 for (a) similar loading condition with DP2 and

(b) dissimilar loading conditions for undamaged cases .................................111

Figure 5.3: Variation of 0 for similar loading conditions comparing undamaged case

and damage pattern 2 .....................................................................................112

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Figure 5.4: Illustration of damaged and undamaged cloud using principal components

analysis ...........................................................................................................113

Figure 5.5: Variation of the damage sensitive feature vectors for damage patterns

(DP) 0, 1 and 6 as defined in the ASCE Benchmark Experiment .................114

Figure 5.6: Migration of the feature vectors κ with damage for minor patterns (a)

Damage pattern 6 and (b) a zoom in of the undamaged cloud (Undamaged

; Damaged +) ...............................................................................................117

Figure 5.7: Migration of the feature vectors with damage for damage patterns (a)

Damage pattern 3 and (b) Damage Pattern 4 (Undamaged ; Damaged +) ..118

Figure 5.8: Migration of the feature vectors with damage for major patterns (a)

Damage pattern 1 and (b) Damage Pattern 2 (Undamaged ; Damaged +) ..119

CHAPTER 1. Introduction

1

Chapter 1

Introduction

1.1 Motivation

In the civil engineering community, it is accepted that there exists a clear need to

efficiently monitor the health of civil engineering structures over their operational lives.

Aging, corrosion, scour and fatigue reduce the life span of the structure, by gradually

deteriorating structural integrity. Moreover, extreme events such as earthquakes,

hurricanes and blasts can severely damage civil infrastructure. Structural health

monitoring (SHM) has received considerable attention from the civil engineering

community and research activities can be found in the conference proceedings edited by

Chang (1999, 2001, 2003 and 2005).

A structural health monitoring system involves the development and implementation of

damage diagnosis and prognosis methodologies for a civil / mechanical infrastructure

(Rytter, 1993). In the context of SHM, damage is defined as changes in the parameters of

the system. These parameters include material properties such as stiffness, damping and

mass; and geometric properties such as boundary conditions such as bolt connectivity


2

(Doebling et al., 1996). Damage diagnosis involves three key aspects: damage detection,

damage localization and damage extent. Damage prognosis involves the calculation of

the structure's residual capacity and residual life forecasting. To this end, data have to be

collected from an array of sensors deployed in the structure. Data collected can include

accelerations, strains, temperature and humidity. These would be the inputs to the SHM

system, where an inverse problem has to be solved. The outputs obtained from the

solution of the inverse problem are the system parameters, viz., the mass, damping and

stiffness matrices.

One common approach to the above problem is vibration based damage detection

algorithms (Doebling et al., 1994). Vibration based methods are divided into model based

and non-model based methods (Doebling et al., 1994). Model based methods give a

quantitative assessment of damage. However, these are computationally expensive and

require a finite element model, which has to be suitably updated at each stage of damage.

Non-model based methods are less computationally intensive, but require extensive

calibration to provide a quantitative damage assessment.

Most currently available damage detection methods are global in nature, i.e., the dynamic

properties (natural frequencies and mode shapes) are obtained for the entire structure

from the input-output data using a global structural analysis. However, global damage

measures are not sensitive to minor damage and local damage. Also these damage

detection methods require that all the data be transmitted to the host data acquisition

system, be it a server or a desktop. Then the data are analyzed using finite element

modeling and system identification techniques to track changes in the global dynamic

properties of the structure (Doebling et al., 1994). More importantly, these methods are

computationally expensive and thus do not lend themselves to be embedded at the

sensing nodes.

Recent research has demonstrated that wireless sensing networks can be successfully

used for structural health monitoring (Straser and Kiremidjian, 1998; Lynch et al., 2003).


3

Low cost microeletromechanical (MEMS) sensors and wireless solutions have been

fabricated for structural measurement and this allows for a dense network of sensors to be

deployed in structures. As a result, data collected by these sensors will be voluminous

and data transmission over the wireless network will be demanding since sensor radios

are designed with small transmission rates in mind. In order to reduce the amount of data

transmission yet provide the information on the state of the structure, it is highly

desirable that data processing and information extraction be performed at the sensing

nodes. Thus, results of the data analysis are only transmitted resulting in significant data

compression. For this purpose, digital signal processing techniques coupled with

statistical pattern classification methods can be used. The main advantage of these

methods is that they rely on the comparison of a base signal and its characteristics

obtained at a sensor location, to a signal arising from a fault or a change in the system, at

the same sensor location. Thus, these methods can be embedded at the sensing nodes.

In this dissertation, the focus is on the development of pattern classification based

damage detection algorithms for SHM that can be embedded at the sensor level. The

main premise of these algorithms is as follows: structural damage will alter the dynamic

properties of a structure, which will in turn, change the dynamic response of the structure.

Thus, structural damage can be detected based on the time domain or spectral analysis of

the vibration signals measured from the pre-damaged and post damaged states of the

structure.

Statistical pattern classification methods have been developed over past couple of

decades for applications in engineering, biology and finance. In the past decade,

developments in the engineering field have been fueled by the need for image

reconstruction for medical and computer visualization applications, automated speech

recognition, finger print identification, and much more (Duda et al., 2001). Classification

schemes are broadly classified into supervised learning and unsupervised learning

schemes (Hastie et al., 2001). In supervised learning schemes, the algorithm is trained on

a dataset whose outcome variables are observed and predictions are made with respect to


4

the training dataset. On the other hand, unsupervised learning schemes are algorithms

where no outcome variables are observed and thus, the main aim is to classify or cluster

the data.

In the context of structural health monitoring, a pattern classification framework was first

proposed by Sohn and Farrar, 2001. Such methods rely on the signatures obtained from

the recorded vibration, strain or other data to extract features that change with the onset

of damage. A pattern classification algorithm in the context of SHM involves the

following steps: (i) the acquisition of structural response measurements and data

preprocessing, (ii) the extraction of features that are sensitive to damage, and (iii) the

development of statistical models for feature discrimination.

Although pattern classification techniques have been applied to identifying faults in

machinery or discrimination of vibrations arising from different rotating components

(Farrar and Duffey, 1999), there are many challenges in extending this paradigm to civil

engineering structures. Civil structures generally have a complicated geometry. Also, a

number of different materials such as steel, reinforced concrete and composites are used.

Civil structures are interconnected systems of substructures, where damage to one

substructure would lead to force redistribution, a phenomenon not generally observed in

mechanical systems. Also, boundary conditions such soil conditions can have a major

impact on the structural response. Environmental effects such as temperature, humidity

can affect the damage diagnosis process (Sohn et al., 1999). Similarly, loading conditions

can affect the damage diagnosis process. Strong motion and ambient vibration datasets

are available for some structures that can be used to investigate the effectiveness of the

pattern classification based methods. Strong motion accelerations are non-stationary, but

excite higher modes of vibration in the structure. Ambient vibration datasets are more

practical since these are obtained under normal operating conditions. Thus, one strategy

is to compare the ambient vibration datasets before and after an extreme event. However,

a disadvantage with ambient vibration datasets is that these do not excite higher modes.

Ideally, it is desirable to develop algorithms that can be used for both types of vibration


5

signals. However, if this is not possible, algorithms should be developed for each type of

vibration dataset. In this thesis, the main focus is on dealing with ambient vibration

signals before and after damage. The basic steps involved in a pattern classification based

damage detection algorithm are outlined below:

Populate a database with signals obtained from the undamaged structure under

different operational conditions

Process the measurement signal by using standard denoising techniques and suitably

normalize (or standardize) the signal

Extract damage sensitive features from the preprocessed measurement signal that can

be correlated to physical system characteristics

For new signals that are extracted from the same sensor location at a later time,

perform the above three steps and

o Use a statistical pattern classification algorithm to discriminate between a

damaged and undamaged state

o Obtain the most probable location of damage

o Calculate the extent of damage by using an appropriate measure of the

difference between the extracted feature vectors

Report the damage decision, the location of damage and the extent of damage


6

1.2 Objectives

This study deals with the development of pattern classification based damage detection

algorithms that are proposed primarily for ambient vibration signals. For that purpose,

three algorithms are proposed. Thus, the objectives of this study are as follows:

Develop damage sensitive feature vectors defining various characteristics of the

measurement signal using statistical signal processing methods. Here, damage

sensitive features are extracted from the modeling of the measurement signals, which

correlates to a physical feature of the structure.

Develop statistical pattern classification schemes to detect damage by identifying

changes in signals that result from changes in the parameters of the system.

Quantify the amount of damage using differences in measured changes.

The main emphasis is in developing algorithms that are computationally efficient and

provide robust damage detection. Finally these algorithms are tested on the ASCE

Benchmark Structure Phase I datasets (Johnson et al., 2004).

1.3 Organization of the Thesis

The dissertation is organized as follows:

Chapter 2 discusses the development of a time series based damage detection

algorithm using autoregressive (AR) and autoregressive moving average (ARMA)

time series models. A damage sensitive feature is developed and a t-test is used as the

classification algorithm.


7

Chapter 3 presents a similar algorithm as discussed in the previous chapter. Here, AR

/ ARMA models are used for extracting feature vectors. The feature vectors are fitted

using the Gaussian mixture model (GMM) and the gap statistic is used as the

classification algorithm. A damage extent measure based on the Mahalanobis metric

is developed and tested.

Chapter 4 introduces the use of wavelet decomposition to take into account the non-

stationarities of the measurement signal. The feature vectors, based on the wavelet

energies at higher dyadic scales are formulated. Closed form expressions for the

feature vectors connecting them to the physical parameters of the structure are also

established for the Haar and Morlet wavelets.

Chapter 5 introduces a new normalization scheme for distinguishing between

different loading conditions using wavelet decomposition of the vibration signal and

principal components analysis. The features extracted are also based on the wavelet

decomposition of the vibration signals at higher dyadic scales. Following this, the k-

means algorithm is used in conjunction with the gap statistic for damage

identification. A damage extent measure based on the Euclidean distance is also

presented.

Chapter 6 provides a summary of the dissertation and also the future extensions of the

research undertaken.

Chapters 2-5 end with the results obtained from applying the developed algorithm on the

datasets of the ASCE Benchmark Structure. The effect of noise on the damage decision

and extent is also reported.

CHAPTER 2. A Time Series Based Damage Detection Algorithm Using Hypothesis Testing

8

Chapter 2

A Time Series Based Damage Detection Algorithm with Hypothesis Testing

This chapter describes a damage detection algorithm based on time series modeling of the

vibration signal. The vibration signals obtained from sensors are modeled as both

autoregressive (AR) and autoregressive moving average (ARMA) time series. A new

damage sensitive feature (DSF) based on the autoregressive (AR) coefficients is

formulated. The relationship between the AR coefficients used in the DSF and the

physical parameters of the system are investigated. It is shown that the AR coefficients

are related to poles of the structural system and as expected, changes in stiffness are

manifested as changes in the AR coefficients. It will be shown that there is a difference in

the mean values of the DSF of the signals obtained from the damaged and undamaged

cases. From t-tests, it will be demonstrated that the difference in the means of the DSF's

of the damaged and undamaged signals is statistically significant. The algorithm is tested

on several data sets from the ASCE Benchmark Structure (Johnson et al., 2004). This

damage detection algorithm is valid for stationary signals obtained from linear systems.

In contrast to prior pattern classification and statistical signal processing algorithms that


9

have been able to identify primarily severe damage, the proposed algorithm is able to

identify minor to severe damage as defined for the Benchmark Structure.

In the context of vibration based structural health monitoring, time series modeling of

vibration signals has been investigated by Farrar and his colleagues (Sohn et al., 2001;

Sohn and Farrar, 2001). Sohn et al, 2001 propose a two-tier approach in which the

vibration signal is first fitted as an AR model. This is followed by fitting an

autoregressive model with exogenous inputs (ARX) with the output fitted to the same

vibration signal. The residuals extracted from the AR model are used as the exogenous

input. The main premise of this approach is that the residual error associated with the

AR-ARX model obtained from modeling a vibration signal from an undamaged structure

will be lower than that obtained from a damaged structure.

This chapter first summarizes the time series modeling aspects of the vibration signals.

Then, closed form expressions for the AR coefficients are formulated correlating them to

the parameters of the physical system. The variation of the damage sensitive feature

(DSF) for each damage pattern, as specified in the ASCE Benchmark Structure, is

discussed next. Hypothesis testing using the t-test is explained and then the results of the

applications of the algorithm on the ASCE Benchmark Structure are presented.

2.1 Description of the Damage Detection

Algorithm

Structural damage affects the dynamic properties of a structure, resulting in a change in

the statistical characteristics of the measured acceleration time histories. Thus, damage

detection can be performed using time series analysis of vibration signals measured from

a structure before and after damage. In this study, both AR and ARMA time series are

used to model the vibration data obtained from the sensor. The analysis is limited to


10

linear vibration data (before and after the event) and the assumption is made that after

damage has taken place, the structure still behaves linearly under normal every day loads

even though its properties may have changed. Thus, the present study is limited to linear

stationary signals.

2.1.1 Modeling of the Vibration Signals

A typical vibration signal from Sensor 2 is shown in Figure 2.1. This signal is used to

illustrate the model characteristics.

Figure 2.1: Plot of a typical raw acceleration time history from an undamaged case serving as the reference signal for subsequent damage detection (Sensor 2)


11

Before fitting a time series model to the sensor data, it is important to perform

standardization (or normalization) in order to compare acceleration time histories (at a

sensor location) that may have occurred due to different loading conditions (i.e., different

magnitudes and directions of loads) and/or different environmental conditions. After

normalization the features extracted from the signals from undamaged cases would have

similar statistical characteristics and can be compared.

Let xi(t) (i = 1,…,N) be the acceleration signal from sensor i, where N is the number of

sensors. This sensor signal is then partitioned into different streams xij(t) (i = 1,…,N and j

= 1,…,M), where i denotes the sensor number, j denotes the jth stream of data from the

sensor i and M is the number of streams. Then, the normalized signal txij~ is obtained as

follows:

ij

ijijij

txtx

~ (2.1)

where, ij and ij are the mean and standard deviation of the jth stream of sensor i

respectively. For notational convenience, xij(t) will be used instead of txij~ in the

subsequent development.

The next step is to check for trends and stationarity in the data (Brockwell and Davis,

2002), which can be achieved by observing the autocorrelation function (ACF). Figure

2.2 shows that the autocorrelation function of the normalized data has a cyclical trend

that will need to be removed. For detrending the data, three methods are used: (i)

harmonic regression, (ii) simple average window and (iii) moving average window

(Brockwell and Davis, 2002). It is found that harmonic regression could not remove the

trends and thus a combination of the simple average window and the moving average

window is used. The window sizes are chosen so that the residuals obtained from this

process are stationary. A review of the autocorrelation plot or the Ljung-Box statistic

provides further test that this condition is met. A more detailed explanation of the Ljung

Box statistic is provided later.


12

Once the initial data pre-processing is complete, the optimal ARMA model order and its

coefficients are estimated (Brockwell and Davis, 2002). The ARMA model is given by:

p

k

q

kijijkijkij tktktxtx

1 1

(2.2)

where, xij(t) is the normalized acceleration signal, k and k are the kth AR and MA

coefficient respectively; p and q are the model orders of the AR and MA processes

respectively and ij(t) is the residual term. Also, note that the AR model is an ARMA

model when the order of the MA terms is zero. The AR model is given by

p

kijijkij tktxtx

1

(2.3)

Figure 2.2: Autocorrelation function of the normalized signal


13

The Burg and Innovations algorithms are used for estimating the coefficients of the AR

and ARMA processes, respectively (Brockwell and Davis, 2002). The optimal model

order is obtained using the Akaike Information Criteria (AIC) (Brockwell and Davis,

2002). The AIC consists of two terms, one of which is a log-likelihood function and the

other term, which penalizes the number of terms in the time series model. AIC is defined

as:

12log2 qpMLAIC (2.4)

where, ML is the value of the maximum likelihood obtained.

Figure 2.3a shows the variation of the AIC values with the AR model order for different

MA orders. It is observed that, for as the MA model orders vary from q = 0 to 3, there is

very little difference in the AIC values of the AR and ARMA models at each model order

p. Since the AR process is the simpler model, it is chosen as the optimal time series

model that captures the characteristics of the signal. From the variation of the values of

AIC, it is observed that an AR model order of 5-8 is appropriate for the analysis. In

addition, a cross validation analysis is carried out to check the accuracy of the results.

This is performed as follows: for a particular data stream, the data set is split in two, one

is used for the analysis and the other is used for forecasting. In the analysis part, the

coefficients of the AR / ARMA model are calculated. Using these coefficients, the values

of the acceleration data are predicted. The residual error between the predicted values and

actual values are obtained. The root mean square (rms) value of the residual error is

plotted in Figure 2.3b. As expected, the rms value of the error decreases with the model

order and it is seen that model orders of 5-8 are appropriate for further analysis.

In order to obtain the AR coefficients, the Burg Algorithm is applied. Then the residuals

are tested to determine if they are normal, independent and identically distributed (i.i.d).

These tests are illustrated in Figure 2.4.


14

(a)

(b)

Figure 2.3: Determination of Optimal AR model order (a) Variation of AIC with AR model order for MA orders varying from 0 to 3 and (b) Variation of cross validation error

with AR model order

Figure 2.4a shows the normal probability plot of the residuals. The straight line variation

indicates a normal distribution of the data, which is violated only at the tails. Figure 2.4b


15

shows the variation of the residuals with time. It is seen that there is no trend, therefore,

indicating homoskedasticity. Figure 2.4c shows the autocorrelation function of the

residuals, from which it is observed that the values of the ACF at lags greater than one

are not statistically significant. The Ljung-Box statistic is also used to test the i.i.d

assumption of the AR residuals. The Ljung-Box statistic is defined as follows:

h

jLB jn

jnnQ

1

2

2 (2.5)

where n is the sample size, (j) is the autocorrelation function at lag j, and h is the

number of lags being tested. The null hypothesis of randomness is rejected if QLB>21-,h,

where is the level of significance of the hypothesis test and 21-,h is the (1-)th

percentile of the 2 distribution with h degrees of freedom. For this particular dataset, the

null hypothesis is not rejected. Thus, the assumptions made on the residuals are satisfied.

The total duration of the record xi(t) is 480 seconds. The record is divided into 80

segments, denoted by xij(t), j=1,2…80, each having 6 seconds duration sampled at 1000

Hz resulting in 6000 data points per segment. The AR coefficients are computed for each

six second segment of the acceleration data and the first three AR coefficients are used

for the calculation of the DSF. To determine the sensitivity of the coefficients to the

number of data points in the signal, analyses were performed in the range of 1000 to 6000

points in increments of 1000. The AR coefficients were found to reach stable values at

about 3000 points; however, 6000 points were used in the analysis presented in this

study. The stability of the first AR coefficients with the number of data points is

presented in Table 2.1. Both the mean and standard deviation of the coefficients are listed

in this table.


16

(a) (b)

(c)

Figure 2.4: Verification of the i.i.d characteristics and normality of residuals: (a) Variation of residuals with time. (b) Normal probability plot of the residuals. (c)

Variation of the autocorrelation function of the residuals with lag.


17

Table 2.1: Sensitivity of AR coefficients to the number of data points

Value of AR Coefficients

Number of Data Points 1000 2000 3000 4000 5000 6000

Mean of 1

(Std. Deviation of 1) 1.0441

(0.1947)1.0587

(0.1369)1.0566

(0.1088)1.0453

(0.0981) 1.0359

(0.0831) 1.0301

(0.0788)Mean of 2


(0.1966)1.0502

(0.1373)1.0517

(0.1002)1.0459

(0.0840) 1.0403

(0.0710) 1.0358

(0.0582)Mean of 3


(0.1366)1.2644

(0.0861)1.2772

(0.0608)1.2762

(0.0451) 1.2761

(0.0360) 1.2712

(0.0338)

2.1.2 Development of Damage Sensitive Feature (DSF)

In this section, the autoregressive coefficients are used to develop features that

discriminate between damaged and non-damaged states of a structure. Several damage

sensitive features (DSF) were investigated. Of the various DSFs considered, those

depending on the first three AR coefficients appeared to be most promising because these

coefficients are statistically the most significant among all the coefficients of the model.

After testing several different combinations with the first three coefficients (as is shown

in Section 2.1.3), it was found that the first AR coefficient normalized by the square root

of the sum of the squares of the first three AR coefficients provides the most robust

damage sensitive feature. Thus, the proposed damage sensitive feature (DSF) is defined

as follows:

23

22

21

1

DSF (2.6)

where 1, 2 and 3 are the first three AR coefficients. Variations of the DSF with the

record number for different damage patterns are illustrated in Figure 2.5. From these

figures it can be seen that for all damage patterns there is a significant difference in the


18

mean levels of the DSF’s of the damaged and the undamaged states. Thus, to test

statistical difference between the means of two groups of data, the standard t-test is used

(Rice, 1999).

2.1.3 Correlation of the AR Coefficients to the Structural

System

The AR coefficients generally contain information about the modal natural frequencies

and the damping ratios (Maia and Silva, 1998; Lynch, 2004). The ARMA model in the

context of the linear input vibration (assuming to be white noise) may then be treated as

an autoregressive model with exogenous input (ARX) time series, where the input is a

white noise excitation. This model can be examined in the complex z-domain by applying

the time-shifting property of the z-transform (Oppenheim and Schafer, 1986). The z-

transform of a function f(t), denoted by F(z), is defined as follows (Oppenheim and

Schafer, 1986):

k

kzkfzF (2.7)

For a signal shifted by a time units, f(t-a), the z-transform of f(t-a) is given as follows:

zF z atfZ -a (2.8)

This is known as the time shifting property of the z-transform.

Applying the z-transform to both sides of Equation (2.2) and ignoring the effect of the

error term, we get

p

k

q

kij

kkij

kkij zzzXzzX

1 1

(2.9)


19

where, Xij(z) and ij(z) are the z-transforms of the xij(t) and ij(t) respectively. Then, the

transfer function H(z) is derived as

p

p

qq

ij

ij

zzz

zzz

z

zXzH

...1

...2

21

1

22

11

(2.10)

The denominator of the transfer function [H(z)] is a polynomial equation of order p

known as the characteristic equation. The roots of the characteristic equation, known as

the poles of the system, are expressed as follows:

0...22

11

pppp zzz (2.11)

The poles, zpole, of the characteristic equation are a good indicator of the modal natural

frequencies and the damping ratios given by (Maia and Silva, 1998):

tjt

pole

nn

ez21

(2.12)

where, and n are the damping ratio and natural frequency of the particular mode and

t is the sampling time of the signal. Equation (2.12) may be rewritten as jpole rez ,

where the amplitude r and phase angle are expressed as

;tner (2.13)

tn 21 (2.14)

Using simple theory of polynomial roots, it can be shown that

1, i

ipolez (2.15)

2,

,, ji

jpoleipole zz (2.16)

3,,,

,, kpolekji

jpoleipole zzz (2.17)


20

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2.5: Variation of DSF with record number for different damage patterns for Sensor 2: (a) Damage Pattern 1, (b) Damage Pattern 2, (c) Damage Pattern 3, (d) Damage Pattern

4, (e) Damage Pattern 5 and (f) Damage Pattern 6.


21

Without loss of generality, p is assumed to be even and all the poles to be imaginary.

Thus, Equation (2.15) can be rewritten as follows:

2

11,1 cos2

p

iii

p

iipole rz (2.18)

Differentiating with respect to a parameter i, say an element of the stiffness matrix, we

get

2

1

11

p

j i

j

ji

k

k (2.19)

where, ki is the ith modal stiffness. Differentiating with respect to ki and assuming that the

damping ratio is a constant in each mode, we get the following

iiii

ii

i

i

iiii

i

i

i

km

tr

kr

k

r

k

sin1cos

sin2cos2

2

1

(2.20)

where n,i is the ith natural frequency, ti

inier , and tinii ,21 . Taking the

absolute value of the sensitivity ik

1 and since the sampling interval is generally small

and in the range of 0.005-0.02 seconds, we get

iiii

i

i km

t

km

tr

k

1 (2.21)

To obtain an approximation for i

ik

, the sensitivities of eigenvalues and eigenvectors

are used. These are discussed below.


22

The eigenvalues and eigenvectors of a N degree of freedom system is obtained by solving

the following eigenvalue problem

Nrrrr ,..,1 allfor 02 MvKv (2.22)

where vr is the rth eigenvector corresponding to the eigenvalue ωr

2, and K and M are the

mass and stiffness matrices respectively. Differentiating Equation (2.22) with respect to

parameter θi (say an element of the stiffness matrix), to get

i

rrr

i

rr

i

rr

i

v

MvMv

KvK 22 (2.23)

Pre-multiplying with the transpose of vs and simplifying to get

i

rTsrsr

i

Tsr

Ts

i

rr

v

MvvK

vvMv 222 (2.24)

Using the orthogonality property and r = s, we get

ri

Tr

rTrri

r vK

vvMv

2

1 (2.25)

It is also shown in (Fox and Kapoor, 1968; Nelson, 1976) that the eigenvector sensitivity

is a linear combination of the eigenvectors, i.e.,

N

jr

irj

i

r

1

vv

(2.26)

where irj can be derived as (Nelson, 1976)

jr

jr

ri

Tj

jr

ri

ri

Tj

irj

for 2

1

for 22

2

vM

v

vMK

v

(2.27)

Since θi is one of the coefficients of the stiffness matrix, we get


23

jr

jrjr

ri

Tj

irj

for 0

for 22

vK

v

(2.28)

To obtain the derivative of the rth modal stiffness with respect to θi, i

rk

, differentiate

the equation kr = ωr2 mr, we get

ri

Tr

ri

rrr

i

r mk

Mvv

222 (2.29)

Using Equations (2.25) and (2.28), Equation (2.29) can be expressed as

ri

Tr

N

rjj

jjr

ri

Tj

rri

Tr

i

rkMv

vv

vK

v

vK

v

1

2222 (2.30)

This can be further simplified by using the orthogonality principle

ri

Tr

i

rkv

Kv

(2.31)

Since vr is normalized, it can be concluded that

1

i

rk (2.32)

Thus we can conclude that

2

1

1

p

i iii km

t (2.33)


24

Similar equations can be derived for i

2 and i

3 . Thus, it can be concluded that as the

stiffness decreases due to damage, the response of the structure will change resulting in

changes of the AR coefficients. Consequently, the damage sensitive feature based on the

AR coefficients can capture this change in measurements from an undamaged to

damaged structural state.

2.2 Damage Detection Algorithm Synthesis

The damage detection algorithm is summarized in the following steps:

Obtain signals from an undamaged structure, from sensor i, denoted by xi(t) (i = 1,…,

N), where N is the number of sensors. Segment the signal xi into chunks, xij(t) (j =

1,…, M), where M is the number of chunks. Populate a database with these baseline

signals.

Standardize the signal xij(t) to remove all trends and environmental conditions to

obtain txij~ .

Obtain signals from a potentially damaged structure for the same sensor, denoted by

zi(t), (i = 1,…, N). As in the previous steps, zi(t) is segmented into zij(t) (j = 1,…, M)

and is standardize to obtain tzij~ .

Fit an AR model to the signals txij~ and tzij

~ for all i and j.

For each sensor i, define and compute the statistics of damage sensitive feature, DSF,

for each chunk in the pre- and post-event signals. Compute the mean and pooled

variance of the DSF for the pre- and post-event signals.


25

Determine the statistical significance in the differences of mean values of the pre- and

post- event data using the t-test to report the damage decision at sensor i. Report the

p-value and the confidence intervals of the differences in the mean of the DSF for the

pre- and post-event signals.

Report the damage decision.

The advantage of the damage detection algorithm presented in this section is that it

depends on signals obtained at a specified location on of a structure. With current smart

sensing capabilities that provide computational power at the sensor location, the

algorithm can be embedded and executed at the data collection site. Because of its

simplicity, the algorithm also can be executed rapidly and efficiently providing critical

information in a timely manner.

2.3 Application Results

In order to test the validity of the algorithm, vibration signals obtained from the

numerical simulation study of the ASCE Benchmark Structure are used. The structure is a

four story, two-bay by two-bay steel braced frame, illustrated in Figure 2.6 (Johnson et

al., 2004). There are 16 sensors (measuring acceleration) in the building, and their

placement and direction of the measured acceleration are shown in Figure 2.7 (Johnson et

al., 2004). Damage is simulated by removing braces in various combinations, resulting in

a loss of stiffness. Damage patterns include

Damage pattern 1: Removal of all braces on the first floor (near sensors 1-4)

Damage pattern 2: Removal of all braces on the first and third floors (near sensors 1-4

and 9-12)

Damage pattern 3: Removal of one brace on the first floor (near sensor 2)


26

Damage pattern 4: Removal of one brace on the first (near sensor 2) and third floors

(near sensor 9)

Damage pattern 5: Damage pattern 4 + loosening of bolts near sensor 3

Damage pattern 6: Partial reduction of stiffness of one brace on the first floor (near

sensor 2)

Damage patterns 1 and 2 are major damage patterns, whereas damage patterns 4 and 5 are

moderate damage patterns; and damage patterns 3 and 6 are minor damage patterns.

Figure 2.6: ASCE Benchmark Structure (Johnson et al., 2004)

In the numerical simulation study of the benchmark structure, two finite element models

were used to generate the simulated response data: a 12 degree of freedom (DOF) shear-

building model that constrains all motion except two horizontal translations and one

rotation per floor and the second is a 120-DOF model that requires that floor nodes have


27

the same horizontal translation and in-plane rotation. The columns and floor beams are

modeled as Euler–Bernoulli beams and the braces have no flexural stiffness. There are

two loading conditions on the ASCE Benchmark. The first excitation is a series of

independent filtered Gaussian white noise loads generated using a sixth - order low-pass

Butterworth filter with a 100 Hz cutoff and applied at each story of the structure. This

load is intended to model wind or ambient vibration forces. The second loading is a

random excitation generated by a shaker on the roof-top of the center column.

x y

Face 1

Face 2

Face 3

Face 4

Figure 2.7: Placement of sensors and direction of acceleration in the ASCE Benchmark Structure (http:// wusceel.cive.wustl.edu/ asce.shm/ benchmarks.htm)


28

2.3.1 Damage Detection

Figure 2.5 shows the results from the application of the proposed damage algorithm to

the numerically simulated datasets of the ASCE Benchmark structure. From Figure 2.5

(a) – (f), it can be observed that there is a significant difference between the mean values

of the DSF’s obtained from the damaged and undamaged cases. To test the significance

of the difference in means of the DSF’s, a t-test is used (Rice, 1999).

If DSF, damaged and DSF, undamaged are defined as the mean values of the DSF’s obtained

from the damaged and undamaged case, respectively, then a hypothesis test may be set up

as follows to determine if their differences are significant:

damagedDSFundamagedDSF

damagedDSFundamagedDSF

H

H

,,1

,,0

:

:

(2.34)

where H0 and H1 are the null and alternate hypothesis respectively. H0 represents the

undamaged condition and H1 represents the damaged condition. The significance level of

the test is set at 0.05. The hypothesis used in Equation (2.34) is called a two-sided

alternative. For testing the above hypothesis, the t-statistic is used (Rice, 1999). The t-

statistic is defined as follows:

mns

t damagedDSFundamagedDSF

11

,,

(2.35)

where, m and n are the number of samples obtained from DSF of the damaged and

undamaged signals respectively; and s is the pooled sample variance, given as

2

11 2,

2,2

nm

SmSns damagedDSFundamagedDSF

(2.36)

where S2

() is the sample variance of (). For H1, the rejection region is defined as


29

22mntt (2.37)

where tn+m-2(/2) is the value of the t-distribution with n+m-2 degrees of freedom

obtained at /2. Also, the confidence interval of the difference between DSF, undamaged -

DSF, damaged is given as:

mn

stCI nmdamagedDSFundamagedDSF

11

2ˆˆ 2,,

(2.38)

Tables 2.2 -2.6 show the results of the damage decision results for damage patterns 1-6

for the numerical simulation study. The p-value, the point estimate and confidence

intervals of the differences in the means of the undamaged and damaged signals are

presented. The p-value is the probability that the DSF does not predict damage, given in

fact that there is damage in the structure. The p-value is a preliminary indicator of

damage. However, the difference in the means, DSF, undamaged - DSF, damaged, also needs to

be high compared to other values obtained at other sensor locations. These are indicated

by its point estimate and the confidence intervals.


30

Table 2.2: Results of damage decision for damage pattern 1

Sensor No.

Damage Decision

p-value Point Estimate of DSF, undamaged - DSF, damaged

CI of DSF, undamaged - DSF, damaged

1 H1 0.0 -0.2656 [-0.2956, -0.2357] 2 H1 0.0 0.3628 [0.3131, 0.4126] 3 H1 1.376710-13 -0.1290 [-0.1577, -0.1003] 4 H1 0.0 0.4495 [0.4082, 0.4909] 5 H1 5.3951e-007 0.0795 [0.0506, 0.1085] 6 H1 0.0 0.5623 [0.5179, 0.6068] 7 H1 0.0097 0.0391 [0.0098, 0.0685] 8 H1 0.0 0.4816 [0.4318, 0.5314] 9 H1 0.0 -0.1769 [-0.2057, -0.1481]

10 H1 0.0 0.2139 [0.1768, 0.2510] 11 H1 0.0 -0.1708 [-0.2011, -0.1405] 12 H1 2.6302e-011 0.1325 [0.0985, 0.1664] 13 H1 8.8818e-016 -0.1539 [-0.1843, -0.1235] 14 H1 0.0389 0.0395 [0.0021, 0.0768] 15 H1 0.0 -0.2320 [-0.2660, -0.1980] 16 H0 0.8026 0.0039 [-0.0267, 0.0344]


Sensor No.

Damage Decision



1 H1 0.0 0.3999 [0.3457, 0.4540] 2 H1 0.0 0.9858 [0.9357, 1.0359] 3 H1 0.0 0.2592 [0.2080, 0.3104] 4 H1 0.0 0.8598 [0.8125, 0.9071] 5 H1 0.0 0.2958 [0.2503, 0.3414] 6 H1 0.0 0.4600 [0.4054, 0.5147] 7 H1 0.0 0.2736 [0.2295, 0.3176] 8 H1 0.0 0.5553 [0.5006, 0.6100] 9 H1 0.0 0.4339 [0.4071, 0.4607]

10 H1 0.0 0.2412 [0.2046, 0.2777] 11 H1 0.0 0.3826 [0.3544, 0.4108] 12 H1 0.0 0.2721 [0.2368, 0.3075] 13 H1 0.0 0.2698 [0.2346, 0.3049] 14 H1 0.0 0.3488 [0.2321, 0.4656] 15 H1 0.0 0.2689 [0.2321, 0.3058] 16 H1 0.0 0.2793 [0.2280, 0.3305]


31


Sensor No.

Damage Decision



1 H0 0.2732 0.0212 [-0.0533, 0.0152] 2 H1 1.174010-10 0.1455 [0.1065, 0.1845] 3 H0 0.2886 0.0182 [-0.0156, 0.0520] 4 H1 8.989510-4 0.0669 [0.0283, 0.1055] 5 H0 0.2478 -0.0163 [-0.0441, 0.0116] 6 H1 0.0 0.1784 [0.1477, 0.2091] 7 H1 0.0305 -0.0298 [-0.0567, -0.0029] 8 H1 0.0337 0.0356 [0.0028, 0.0685] 9 H1 0.0 -0.1817 [-0.2129, -0.1505]

10 H1 1.801310-7 0.0814 [0.0531, 0.1097] 11 H1 6.064710-9 -0.1122 [-0.1464, -0.0780] 12 H1 0.0115 0.0406 [0.0094, 0.0719] 13 H1 1.444810-12 -0.1595 [-0.1972, -0.1218] 14 H1 7.755310-4 -0.0528 [-0.0829, -0.0228] 15 H1 4.624610-8 -0.1242 [-0.1651, -0.0834] 16 H1 1.401810-8 0.0971 [0.0666, 0.1277]

Table 2.5: Results of damage decision for damage pattern 4 and 5

Sensor No.

Damage Decision



1 H1 0.0031 0.0475 [0.0166, 0.0785] 2 H1 1.032410-11 0.1528 [0.1147, 0.1909] 3 H1 4.440910-16 0.1567 [0.1262, 0.1871] 4 H1 6.390510-6 0.0902 [0.0531, 0.1273] 5 H1 6.149410-5 0.0556 [0.0295, 0.0818] 6 H1 1.974010-13 0.1360 [0.1055, 0.1666] 7 H1 2.961410-8 0.0855 [0.0578, 0.1131] 8 H0 0.6574 0.0074 [-0.0259, 0.0407] 9 H1 6.685610-7 0.0865 [0.0547, 0.1183]

10 H1 2.793710-9 0.0949 [0.0667, 0.1231] 11 H1 1.508210-10 0.1329 [0.0970, 0.1688] 12 H1 0.0023 0.0487 [0.0179, 0.0794] 13 H1 6.924210-7 -0.0841 [-0.1151, -0.0531] 14 H0 0.2571 -0.0170 [0.0467, 0.0127] 15 H1 8.215510-4 -0.0628 [-0.0987, -0.0269] 16 H1 2.723810-9 0.1002 [0.0705, 0.1299]


32


Sensor No.

Damage Decision

p-value Point Estimate of DSF, undamaged - DSF,

damaged


1 H0 0.6826 -0.0048 [-0.0280, 0.0184] 2 H1 2.853310-8 0.0942 [0.0674, 0.1211] 3 H0 0.1799 0.0221 [-0.0104, 0.0545] 4 H1 1.981010-5 0.0772 [0.0434, 0.1110] 5 H0 0.8578 -0.0024 [-0.0294, 0.0245] 6 H1 8.368810-9 0.0949 [0.0656, 0.1241] 7 H0 0.7112 -0.0051 [-0.0322, 0.0221] 8 H1 3.427410-4 0.0533 [0.0250, 0.0816] 9 H1 0.0295 0.0157 [0.0016, 0.0298] 10 H1 0.0010 0.0501 [0.0208, 0.0794] 11 H0 0.3203 -0.0166 [-0.0495, 0.0164] 12 H1 0.0163 0.0376 [0.0071, 0.0680] 13 H0 0.0967 -0.0291 [-0.0636, 0.0054] 14 H1 0.0085 0.0382 [0.0100, 0.0663] 15 H0 0.2768 -0.0211 [-0.0595, 0.0173] 16 H1 1.720510-9 0.0940 [0.0666, 0.1214]

The conclusions drawn from the damage detection results are as follows:

In the case of major damage patterns 1 and 2, it is observed that the difference in the

means of the damaged and undamaged signals is statistically significant. However, in

the case of damage pattern 1, the t-test indicates that there is no damage at sensor 16.

For sensors 5, 7 and 14, it is observed that the difference in the means is small as

compared to other sensors. For damage pattern 2, all sensors show a statistical

significant difference in the means, thus indicating damage.

Bolt loosening (near sensor 3) was not detected in damage pattern 5. The maximum

difference in the means is obtained at sensors 2, 3 and 6. Again note that sensors 8

and 14 do not indicate damage, whereas sensors 1 and 12 have a very minor

difference in the means. Here again, damage has been consistently detected for

moderate damage patterns 4 and 5.


33

Minor damage has been detected. However, confidence intervals are smaller as

compared to those of the minor and moderate damage patterns. In damage pattern 6, it

is observed that for sensors 2, 6 and 16 indicate a higher difference in the mean as

compared to other sensors. Since, damage occurred on Face 2 of the structure, it

would be expected that this would lead to detection of damage on sensors on Faces 2

and 4. However, this is not the case for sensors on the higher floors and this may be

attributed to the torsional modes of the structure.

Figure 2.8 and Figure 2.9 illustrate the spread of the values of the DSF’s for damage

patterns 6 (minor) and 2 (major) respectively. From the analysis, the following

observations are made:

It is observed that the dispersion of the DSF of the damaged signal is significantly

higher than that of the undamaged signal.

The differences in the mean values of the DSF is particularly higher for the major

damage patterns in comparison to the minor damage patterns and thus can be used as

an indicator of damage extent.

In the case of damage pattern 6, the confidence intervals of the difference in the

means of the DSF’s are not too high. Thus, a more sensitive feature / better

classification scheme is required for efficient damage detection.

2.4 Summary

In this chapter, a damage detection algorithm based on time series modeling is discussed.

A damage sensitive feature DSF, which is a function of the first three regressive (AR)

components, is presented. The time series modeling aspects of vibration signals is

discussed. Both AR and ARMA models have been used to fit the vibration signal. It is

shown that the AR model with 5-8 parameters is the optimal time series for the vibration


34

(a)

(b)

DS

F

DS

F

Figure 2.8: Dispersion of Values of DSF’s for Damage Pattern 6 sensors along (a) Face 1 and (b) Face 2


35

(a)

(b)

DS

F

DS

F

DS

F

DS

F

Figure 2.9: Dispersion of values of DSF’s for Damage pattern 2 sensors along (a) Face 1 and (b) Face 2


36

signals considered in the study. Subsequently, a closed form equation relating the AR

coefficients with the parameters of the physical system is derived.

Next, a hypothesis test involving the t-test is used to obtain a damage decision. The

damage detection methodology is tested on the analytical results of the ASCE Benchmark

Structure. The results of the application of this damage detection algorithm indicates that

the algorithm is able to detect the existence of all damage patterns in the ASCE

Benchmark simulation experiment where minor, moderate and severe damage

corresponds to removal of single brace in a storey, removal of a brace on two storeys and

removal of all braces in two storeys, respectively. These results are very encouraging, but

represent initial testing of the algorithm and further investigations will be needed to test

the validity of the damage detection method. More testing is needed to investigate various

scenarios and conditions that introduce other damage patterns, such as cracking at joints

or loosening of bolts. While it may be difficult to simulate such conditions numerically,

they can be reproduced in the laboratory. Thus, additional testing will be performed as

such data become available. Ultimately, these algorithms will need to be tested with field

data.

The advantage of the statistical signal processing approach combined with the pattern

classification framework is that it does not require any elaborate finite element modeling.

Such an approach is particularly suited for wireless sensor analysis, which is able to

process data at the sensor unit location through embedded algorithms. Such data can then

be transmitted to a global master for additional damage analysis using system

identification methods.

37

Chapter 3

A Time Series Based Structural Damage Detection Algorithm Using Gaussian Mixture Modeling

In Chapter 2, it was shown that the damage sensitive feature (DSF) was able to detect

minor damage for the ASCE Benchmark Structure. However, it was also observed that

the confidence intervals for minor damage patterns were relatively smaller in comparison

to major damage patterns. Thus, a more robust classification scheme is needed to identify

minor damage patterns. To this end, a time series based detection algorithm utilizing the

Gaussian Mixture Models (GMM’s) is presented in Chapter 3.

Two critical aspects of damage diagnosis, detection and extent, are investigated. As in the

previous chapter, the vibration signals obtained from the structure are modeled as auto-

regressive (AR) processes. The feature vector used consists of the first three

autoregressive coefficients obtained from the modeling of the vibration signals. It is

observed that there is a migration of the extracted AR coefficients with increasing levels

of damage. To detect these changes in the AR coefficients, a clustering scheme called the

Gaussian Mixture Model (GMM) is used. Damage is detected if there is more than one

CHAPTER 3. A Time Series Based Structural Damage Detection Algorithm using Gaussian Mixture Modeling

38

cluster‡ in a particular dataset. This is achieved using the gap statistic, which determines

the optimal number of mixtures in that dataset. The Mahalanobis distance between the

mixture in question and the baseline (undamaged) mixture is investigated as a candidate

for quantifying damage extent. Application cases from the ASCE Benchmark Structure

simulated data are used to test the efficacy of the algorithm.

3.1 Overview of the Damage Diagnosis Algorithm

As discussed in Chapter 2, structural damage is detected using time series analysis of the

vibration signals measured from the pre-damaged and post-damaged states of the

structure. Figure 3.1 illustrates the effect of damage on the first three AR coefficients 1,

2 and 3. It is seen that the clouds of AR coefficients obtained by modeling vibration

signals before damage would migrate with the onset of damage. AR coefficients

corresponding to damage patterns 1 and 2 are plotted in this figure. Thus, the main

premise in the proposed algorithm is that there is a migration of clusters of the feature

vectors (AR coefficients) with damage.

The proposed algorithm is as follows:

1 Obtain signals (vibration and strain if available) from an undamaged structure,

from sensor i, denoted by xi(t) (i = 1,…,P), where P is the number of sensors.

Segment the signal xi(t) into chunks of finite duration xij(t) (j = 1,…,Q ), where Q

is the number of chunks. Populate the database with these baseline signals.

2 Model the chunks of time series data from each sensor as described in Chapter 2.

Extract the damage sensitive features from the signals that define them as feature

vectors. In this algorithm, the first three AR coefficients of the signals are used to

define the feature vectors. Denote the feature vectors as αi,baseline which is of

‡ In this dissertation, the words ‘cluster’ and ‘mixture’ and used interchangeably.


39

dimension Q3. It should be noted that we use the signals obtained at each sensor

and local signal processing is performed at that sensor before and after damage.

Figure 3.1: Migration of feature vectors (defined by the first three AR coefficients) from an undamaged state to damage patterns 1 and 2 as defined by the ASCE Benchmark

Structure

3 Obtain signals from a potentially damaged structure for the same sensor, denoted

by zi(t), (i = 1,…, P). As in the previous steps, zi(t) is segmented into zij(t) (j =

1,…, Q). Again the first three AR coefficients are used to define feature vectors

for damage detection. Denote the new feature vectors as αi,new which is of

dimension Q3.

4 Define the feature vector Yi (i = 1,…, P), of dimension 2Q3, as follows


40

newi

baselineii

,

,

α

αY (3.1)

5 Fix the number of clusters as k. For a fixed value of k, fit the feature vectors Yi,

defined in Equation (3.1), using a Gaussian mixture model (GMM) and obtain the

parameters of the GMM using the Expectation-Maximization (EM) algorithm.

Using the parameters of the GMM, calculate the gap statistic (Tibshirani et al.,

2001).

6 Use the gap statistic to determine the optimal number of clusters. If the number of

clusters is greater than one, then it is hypothesized that some degree of damage

has taken place. If, however, the clusters are very close based on the gap statistic,

then it is concluded that there is no damage. Such signals would be stored in the

baseline database.

7 If damage has occurred, using the covariance matrices obtained by using the EM

algorithm, calculate the Mahalanobis distance for the new mixture and the

baseline mixture.

8 Repeat steps 2-7 with the baseline data and the subsequent signals identifying

damage and quantifying damage according to steps 5, 6 and 7.

3.2 Modeling of Vibration Signals

In this subsection, the time series modeling of vibration signals is summarized. Details of

the various steps discussed below are available in Chapter 2. The main steps in the

modeling of vibration signals are listed below:

Standardize the vibration signal as described by Equation (2.1).


41

Fit the autoregressive (AR) or the autoregressive moving average (ARMA)

models to the vibration signal.

Use the Akaike Information Criteria (AIC) to ascertain the optimal model orders

of the AR and MA coefficients. Figure 2.3(a) illustrates this step.

Check the assumption of the time series models fitted. These include the

homoskedasticity, the normality condition and the i.i.d assumptions of the

residuals. This is illustrated in Figure 2.4

With respect to the vibration signals obtained from the ASCE Benchmark Structure, it is

found that the AR model is the optimal time series with the AR model order being

between 5 and 8.

For the baseline and new signals, the total duration of each record is 480 seconds. Each

record is divided into 80 segments (i.e., Q = 80), denoted by xij(t), j=1,2…80, each having

6 seconds duration sampled at 1000 Hz resulting in 6000 data points per segment. The

AR coefficients are computed for each six second segment of the acceleration data and

the first three AR coefficients are used in a feature vector Yi, as given in Equation (3.1),

that has a size of 160 by 3.

3.3 Gaussian Mixture Modeling

Figure 3.1 illustrates the migration of the feature vectors at sensor 2 from its undamaged

state (2,undamaged) to damage patterns 1 (2,DP1) and 2 (2,DP2), resulting in the formation

of three unique clusters. It is recalled that damage pattern 1 represents braces removed

from the first floor and damage pattern 2 represents the removal of braces from the first

and third floor. Thus, an algorithm is required to ascertain the number of clusters in a

dataset. To this end, Gaussian mixture models are used. Gaussian mixture models

(GMM’s) are frequently used as clustering algorithms in pattern classification (Hastie et


42

al., 2002). A Gaussian mixture model with M classes (or mixtures or clusters) has the

following form:

M

iiiiNf

1:1 ;Yy (3.2)

where, Y is the collection of N feature vectors (in this study N = 160), iii N ,~ is a

Gaussian vector with mean vector i and covariance matrix i and i is the non-negative

mixture weight for each class. The unknown parameters of the GMM = {i, i, i i =

1,2,…M} can be estimated using the maximum likelihood principle. The direct

maximization of the likelihood function is quite difficult and analytically intractable. For

this purpose, the expectation maximization (EM) algorithm is used. The derivation of the

EM algorithm is given in the Section 3.8. Define a random variable Ii (i = 1,..,N) whose

realization is an M dimensional indicator row vector, whose (i, j) component is 1 (i.e., Iij

= 1) if yi corresponds to the jth mixture. The EM algorithm in the context of the GMM is

as follows:

Step 1: Initialize the values of E(Iij) (i=1,…,N and j = 1,…,M), the mixture weights, the

means and covariance matrices of the classes of the GMM.

Step 2: The update equations are as follows:

N

kkj

N

kkkj

j

IE

IE

1

1

y (3.3)

N

kkj

N

k

Tjkjkkj

j

IE

IE

1

1

μyμyΣ (3.4)

N

IEN

kkj

j

1

(3.5)


43

N

kkiki

jijiij

Ip

IpIE

1

;1|

;1|

y

y

(3.6)

Equations (3.3) - (3.5) constitute the maximization (M-step) and Equation (3.6)

constitutes the expectation (E-step) of the algorithm. It should be noted that

jijT

jijdiji Ip μyΣμyΣy 12

1

2/ 2

1exp

2

1;1| (3.7)

where d is the dimension of the feature vector xi.

Step 3 : Calculate the log-likelihood function given by Equation (3.17) at the tth and

(t+1)th time steps. The definition of the log-likelihood function is given in Section

3.8. At the end of each step, test for convergence using:

1

111

log

loglog

t

tt

L

LL (3.8)

where is assumed to be 0.001.

As discussed above, in Step 1 of the EM algorithm, some initial guesses of the

parameters has to be made. The k-means algorithm (Hastie et al., 2002) is frequently used

to obtain a first estimate of the means of the clusters and is described below:

Initialize the cluster means to M randomly chosen points (since there are M

clusters)

For each cluster mean, j, find the points in the dataset closest to j. Denote

these set of points as Mj and the number of points as nj.

Compute the new mean


44

ji M

ij

j n y

yμ1

(3.9)

Iterate the above two steps until convergence is obtained.

Figure 3.2: Variation of log-likelihood with number of mixtures in the dataset

Figure 3.2 illustrates the variation of the log-likelihood with the number of mixtures. As

expected, as the number of mixtures increase, the log-likelihood increases.


45

3.4 Damage Diagnosis using Gaussian Mixture

Models

As stated earlier, the main premise in the proposed algorithm is that there is a migration

of clouds of feature vectors obtained from sequential measurements as damage is

incurred to the structure. Damage is identified by determining that there is more than one

mixture in a particular dataset given by Equation (3.1). Damage extent is obtained using

the Mahalanobis distance between the centroid of the mixture distribution under question

with respect to the baseline mixture. The formulations required for estimating the number

of mixtures and the Mahalanobis distance between mixtures is described in the following

sections.

3.4.1 Damage Identification using the Gap Statistic

Figure 3.1 illustrates the migration of the feature vector clouds for the damage patterns

considered in the ASCE Benchmark structure. In order to discriminate between damage

and no damage in a given feature vector, the number of clusters or mixtures has to be

determined. The number of mixtures for a particular dataset is determined by using the

gap statistic (Tibshirani et al., 2001).

As before, we consider M clusters, C1, C2,…, CM. Let the number of observations in the

rth cluster be denoted by nr. Then the within cluster sum of squared distance of cluster r,

denoted as Dr is given as

22

,,

2

rrr Ci

irCji

jiCji

ijr ndD yyyy (3.10)

where dij is the sum of squared distance between the ith and jth observation. In most

situations, the Euclidean distance between observations is used. This is illustrated in

Figure 3.3. The dispersion measure is denoted as Wk and is defined by


46

MkDn

Wk

rr

rk ,...1

2

1

1

(3.11)

Cluster r

Figure 3.3: Illustration of within cluster distance

The gap statistic is then given as follows (Tibshirani et al., 2001):

kknn WWEkGap loglog (3.12)

where En denotes the expectation with respect to some reference distribution. In this

study, the uniform distribution for the range of the observed values for that feature is

used, as proposed by Tibshirani et al., 2001. The estimate of k which maximizes the

Gapn(k) is the number of clusters in the dataset. Thus, for the computation of the gap

statistic, the following algorithm is used (Tibshirani et al., 2001):

Fix the number of clusters and then use the GMM’s to cluster the observed

data. For each of these cases, calculate Wk : k = 1,.., M.

Generate B reference datasets according to the uniform distribution and

calculate the dispersion measure Wkb for all b = 1,…,B, and k=1,…,M.

Compute the mean and the standard deviation as follows


47

B

sdks

WB

ksd

WB

bckb

bkbc

11

log1

log1

2

1

2

(3.13)

Choose the number of clusters by using the rule given below

k

B

bkb

k

WWB

kGap

skGapkGapthatsuchksmallestk

loglog1

)1( ˆ

1

1

(3.14)

From the application of this algorithm, if the number of clusters k, is found to be greater

than one, it is hypothesized that the signals come from different states of the structure and

this change of state is most likely due to damage.

3.5 Damage Extent using the Mahalanobis Metric

The Mahalanobis distance is a metric frequently used in multivariate analysis to

determine the separation of two distributions (Mardia et al., 2003). The Mahalanobis

distance between two vectors a and b with a covariance matrix is defined as follows:

baΣbaΣba 1,, T (3.15)

In the present study, the Mahalanobis distance is used to define a measure of the damage

extent. More specifically, the damage metric DM used is defined as (undamaged, damaged;

undamaged) where undamaged is the covariance matrix of the undamaged dataset, undamaged

and damaged, are the means of the undamaged and damaged dataset respectively. These

values are obtained after modeling the feature vectors as a GMM. Mathematically, the

metric DM may be defined as:


48

damagedundamagedundamaged

Tdamagedundamaged

undamageddamagedundamagedDM

μμΣμμ

Σμμ

1

;, (3.16)

The above formulation is valid for a two mixture dataset. However if the number of

mixtures is greater than two, the value of DM is chosen as the maximum of the values of

this measure computed between the undamaged mixture and the other mixtures.

3.6 Application

In order to test the validity of the algorithm, results from the numerical simulation of the

ASCE Benchmark Structure are used (Johnson et al., 2004). Details on the ASCE

Benchmark Structure are discussed in Chapter 2.


Figures 3.4, 3.5 and 3.6 illustrate the migration of the clouds of feature vectors composed

of the first three AR coefficients with minor, moderate and major amounts of damage

respectively. Thus, the migration of the mixtures for each pair of data would give an

indication of damage. The gap statistic provides a means for tracking the number and

migration of mixtures. If two or more mixtures are identified through this statistic, then

there is high likelihood of damage occurrence.

Damage pattern 0 (DP0) represents the undamaged state and data from that feature vector

is taken as the baseline state. Feature vectors from progressively increasing damage are

compared to the baseline in order to identify the onset and increase in damage. The first

damage pattern considered for the comparison is damage pattern 6 because it is the

smallest damage imposed on the structure. Using the gap statistic, the number of mixtures

obtained is 2. This is illustrated in Figure 3.4a. Applying the algorithm, all damage

patterns 1-6 have been identified.


49

(a)

(b)

Figure 3.4: Migration of the feature vectors with damage for minor patterns (a) Damage pattern 6 and (b) Damage Pattern 3


50

(a)

(b)

Figure 3.5: Migration of the feature vectors with damage for moderate patterns (a) Damage pattern 4 and (b) Damage Pattern 5


51

(a)

(b)

Figure 3.6: Migration of the feature vectors with damage for major patterns (a) Damage pattern 1 and (b) Damage Pattern 2


52

Figure 3.7 illustrates the variation of the gap statistic with number of mixtures for a

damaged case. The feature vectors are plotted in Figure 3.7(a). Note that there is a clear

separation of clusters in this dataset, plotted in Figure 3.7(a). Figure 3.7(b) shows the

variation of the observed and expected value of log(Wk) with number of mixtures. The

variation of the gap statistic is plotted in Figure 3.7(c). It shows that the gap statistic

yields the highest value at the second mixture, thus showing that there are two mixtures,

indicating damage in the structure. Since the gap statistic gives an indication of the

dispersion of the dataset with respect to a uniformly distributed dataset, the optimal

number of mixtures is obtained at where the gap statistic is maximized. A similar

illustration of the gap statistic for an undamaged case is given in Figure 3.8, where there

is no clear separation of clusters in the dataset. The number of mixtures identified by the

gap statistic is one, indicating no damage. It is also noted that the value of the gap

statistic is very small implying that there is very little difference between the generated

uniform distribution and the feature vector.


53

0.40.6

0.81

1.21.4

-1

0

1

20

0.5

1

1.5

1

2

3

Undamaged

Damaged

1 2 3 4 52.5

3

3.5

4

4.5

5

5.5

Number of Mixtures

Log-L

ikelih

ood

ObservedExpected

1 2 3 4 50

0.5

1

1.5

2

Number of Mixtures

Gap

(a)

(b)

(c)

Figure 3.7: Illustration of the gap statistic for a damaged case (a) Distribution of AR coefficients (b) Variation of the observed and expected value of log(Wk) with number of

mixtures (c) Variation of the gap statistic with number of mixtures


54

0.7 0.8 0.9 1 1.1 1.2 1.3

0.81

1.21.4

1.61.1

1.2

1.3

1.4

1.5

1

2

3

Undamaged Set 1

Undamaged Set 2

1 2 3 4 51.5

2

2.5

3

Number of Mixtures

Log-

Like

lihoo

d

ObservedExpected

1 2 3 4 5-0.15

-0.1

-0.05

0

0.05

0.1

Number of Mixtures

Gap

(a)

(b)

(c)

Figure 3.8: Illustration of the gap statistic for an undamaged case (a) Distribution of AR coefficients (b) Variation of the observed and expected value of log(Wk) with number of

mixtures (b) Variation of the gap statistic with number of mixtures


55

Table 3.1 shows the results of the EM algorithm, which yields the weights, means and

autocorrelation matrices of the feature vectors obtained from sensor 2 from its pre and

post damage states. It is observed that the weight associated with each mixture is between

0.4-0.5 for the presence of two mixtures and can be used as a simple rule of thumb for the

existence of more than one mixture.

Table 3.1: Results from the EM Algorithm for various damage patterns (DP)

DP Means Autocorrelation Matrices Weights

1

3282.0

1758.0

6709.0

2758.1

0407.1

0323.1

0000.18040.07866.0

8040.00000.18327.0

7866.08327.00000.1

0000.18730.07466.0

8730.00000.17130.0

7466.07130.00000.1

5000.0

5000.0

2

1347.0

6519.0

2057.0

2758.1

0407.1

0323.1

0000.11589.08849.0

1589.00000.14517.0

8849.04517.00000.1

0000.18730.07466.0

8730.00000.17130.0

7466.07130.00000.1

5000.0

5000.0

3

1317.1

8931.0

8861.0

2753.1

0403.1

0322.1

0000.16636.07072.0

6636.00000.14682.0

7072.04682.00000.1

0000.18700.07382.0

8700.00000.17123.0

7382.07123.00000.1

5032.0

4968.0

4

1145.1

8525.0

8795.0

2757.1

0406.1

0323.1

0000.15082.06312.0

5082.00000.13898.0

6312.03898.00000.1

0000.18731.07445.0

8731.00000.17124.0

7445.07124.00000.1

5005.0

4995.0

5

1145.1

8525.0

8795.0

2757.1

0406.1

0323.1

0000.15082.06312.0

5082.00000.13898.0

6312.03898.00000.1

0000.18731.07445.0

8731.00000.17124.0

7445.07124.00000.1

5005.0

4995.0

6

2419.1

9924.0

9369.0

2647.1

0494.1

0358.1

0000.17495.08832.0

7495.00000.16295.0

8832.06295.00000.1

0000.18611.08173.0

8611.00000.16760.0

8173.06760.00000.1

4642.0

5358.0


56

3.6.2 Damage Extent

The damage extent is calculated using Equation (3.16) and the results of the DM

calculated for sensor 2 are shown in the semi-log plot in Figure 3.9. As can be observed

from this figure, the damage metric DM increases for damage patterns 6, 3, 4, 5, 1 and 2,

which corresponds to a progressive increase in damage. Also, DM varies from 2.95

(corresponding to damage pattern 6) to 4591.56 (corresponding to damage pattern 2).

Figure 3.9: Variation of the damage metric DM with damage pattern for sensor 2

The variation of the damage metric (DM) for various sensors and different damage

patterns is given in Table 3.2.


57

Table 3.2: Variation of DM for various sensors and different damage patterns

DP1 DP2 DP3 DP4 DP5 DP6

1 431.38 1415.53 NA 50.90 50.90 NA 2 1374.21 4591.56 22.54 25.85 25.85 2.95 3 454.63 1673.77 NA 34.46 34.46 NA 4 1096.92 5993.50 32.33 48.73 48.73 2.78 5 295.61 826.69 NA 25.67 25.67 NA 6 1350.03 4306.10 21.79 24.04 24.04 2.64 7 265.79 703.07 NA 20.73 20.73 NA 8 858.93 3224.43 9.52 10.75 10.75 1.59 9 263.74 1068.42 14.25 46.43 46.43 NA

10 822.55 727.49 4.08 6.74 6.743 NA 11 340.21 1360.80 11.97 48.51 48.51 NA 12 506.16 344.10 4.79 4.73 4.73 NA 13 1444.02 4485.12 15.33 12.21 12.21 NA 14 463.91 1282.90 2.82 3.15 3.15 2.389 15 694.23 2632.63 3.69 49.29 49.29 2.09 16 756.74 2691.54 10.31 9.50 9.50 2.32

In Table 3.2, NA is used in cases where only one mixture is identified, indicating no

damage. From the analysis of vibration signals obtained from the ASCE Benchmark

Structure, the following observations are made:

Damage patterns 1 and 2 are detected consistently at all sensor locations.

These damage patterns are characterized by large values of DM indicating that

the baseline and new mixtures are significantly apart.

Since damage patterns 4 and 5 have braces removed on Faces 1 and 2 (Figure

2.7), it is seen that damage is consistently detected at all sensor locations.

Although bolt loosening takes place near sensor 3, damage patterns 4 and 5

show no difference, thus indicating that bolt loosening has not been detected.

Damage patterns 3 and 6 have the reduction of a brace stiffness on Face 2

(Figure 2.7). Thus, on the first two floors of the structure, it is observed that

damage has not been detected at sensors that are located on Faces 1 and 3. On


58

the higher floors, damage has been detected at all sensors. Examination of the

structure and its vibration characteristics would point to potential torsional

motion that could result from the removal of braces. Such torsional motion

would particularly be severe at the higher stories resulting in larger

differences in vibration responses at these sensor locations.

3.6.3 Effect of Noise on the Damage Diagnosis

In this section, the effect of mean zero additive Gaussian white noise on the damage

detection algorithm is studied. The ratio of the root mean square (rms) value of the noise

to the rms value of the signal is defined as the noise to signal ratio, and is denoted as

NSR. The NSR is varied from 0.05-0.15 and the means of the damaged and undamaged

data sets for various damage patterns are given in Table 3.3.

Table 3.3: Variation of the means of the undamaged and the damaged data obtained from sensor 2 with different noise to signal ratios (NSR)

DP NSR Mean of undamaged dataset Mean of damaged dataset

1 0.05 (1.0158, 1.0162, 1.2565) (0.6701, 0.1750, 0.3270) 0.10 (0.9687, 0.9482, 1.2033) (0.6672, 0.1727, 0.3250) 0.15 (0.9074, 0.8570, 1.1319) (0.6598, 0.1639, 0.3242)

2

0.05 (1.0146, 1.0136, 1.2553) (0.2051, 0.6514, -0.1344) 0.10 (0.9683, 0.9456, 1.2021) (0.2042, 0.6503, -0.1341) 0.15 (0.9132, 0.8652, 1.1393) (0.2013, 0.6440, -0.1311)

3

0.05 (1.0158, 1.0150, 1.2547) (0.8748, 0.8781, 1.1201) 0.10 (0.9728, 0.9539, 1.2084) (0.8581, 0.8467, 1.0971) 0.15 (0.9128, 0.8562, 1.1197) (0.7820,0.7813, 1.0561)

4

0.05 (1.0143, 1.0130, 1.2546) (0.8705, 0.8404, 1.1055) 0.10 (0.9710, 0.9517, 1.2062) (0.8459, 0.8049, 1.0777) 0.15 (0.9054, 0.8375, 1.1109) (0.7764,0.7463, 1.0419)

5

0.05 (1.0180, 1.0180, 1.2576) (0.8717, 0.8410, 1.1058) 0.10 (0.9518, 0.9093, 1.1669) (0.8005, 0.7866, 1.0703) 0.15 (0.8908, 0.8242, 1.1003) (0.7474, 0.7372, 1.0350)

6

0.05 (1.0150, 1.0211, 1.2420) (0.9215, 0.9750, 1.2297) 0.10 (0.9274, 0.9388, 1.1907) NA 0.15 (0.8684, 0.8532, 1.1251) NA


59

A similar comparison of the damage extent metric DM with noise is shown in Table 3.4.

Table 3.4: Variation of the damage metric DM with noise to signal ratio (NSR)

Damage Pattern Damage Metric DM

NSR = 0.0 NSR = 0.05 NSR = 0.10 NSR = 0.15

1 1374.21 1264.12 929.19 269.81 2 4591.56 4162.47 3020.61 1520.04 3 22.54 17.57 9.57 4.16 4 25.85 23.23 18.79 3.72 5 25.85 23.23 18.79 3.72 6 2.95 2.59 NA NA

Some of the observations that can be made from Table 3.3 and Table 3.4 are as follows:

For noise levels of 0.10 or larger, minor damage patterns, particularly damage

pattern 6, do not appear to be discriminated since the gap statistic estimates

that there is only one mixture because there is a large overlap of the two

mixtures.

For all noise levels, major and moderate damage patterns are detected since

there is a large separation between the damaged and undamaged feature vector

clouds.

DM’s are directly affected by the mean distances of the clusters. The DM’s

obtained for all the NSR values indicate a similar pattern of increasing

damage for each progressively increasing damage pattern (from 6, 3, 4, 5, 1

and 2).


60

3.7 Summary

In this chapter, the Gaussian Mixture Model is used with the previously developed time

series model to provide a more robust damage detection. Two critical aspects of damage

diagnosis that are investigated are detection and extent. The vibration signals obtained

from the structure are modeled as auto-regressive (AR) processes. The feature vector

used consists of the first three autoregressive coefficients obtained from the modeling of

the vibration signals. It is observed that the AR coefficients migrate apart from each other

with damage. A Gaussian Mixture Model (GMM) is used to model the feature vector.

Damage is detected using the gap statistic, which determines the optimal number of

mixtures in a dataset, containing the damaged and undamaged feature vectors. The

Mahalanobis distance between the two clusters of data is shown to be a good indicator of

damage extent. Simulated data from the ASCE Benchmark Structure have been used to

test the efficacy of the algorithm. This algorithm has also been tested for various noise

levels that are introduced to the simulated data.

The proposed GMM-based algorithm is shown to be very effective in detecting damage.

Application of the algorithm to the ASCE Benchmark simulation experiment demonstrate

that the algorithm is able to consistently detect minor, moderate and major damage

patterns respectively corresponding to removal of two-thirds of the cross sectional area of

a brace, removal of a brace on two floors and removal of all the braces on two floors.

However, loosening of bolts cannot be distinguished when it occurs in conjunction with

damage pattern 4. No data were available for bolt loosening by itself, thus such a damage

pattern could not be tested.

The magnitude of the damage metric DM based on the Mahalanobis distance appears to

be highly correlated to the damage extent even under the presence of noise. It is

demonstrated that the magnitude of the DM metric increases with increasing damage.


61

The limitations of the algorithm are identified as follows: this algorithm is valid for linear

stationary signals; the initial measurement is assumed to be the undamaged state and

changes are identified relative to that state. Thus, if measurements commence after

damage has occurred, then only additional damage would be identified unless a method

for determining the current state of the structure is also implemented. Also, damage to the

structure is assumed to be related primarily with decrease in stiffness. Thus, knowledge

of the material properties of the structure and its behavior under dynamic loading

conditions is absolutely necessary before the method can be used reliably.

In comparison to the algorithm developed in Chapter 2, the present algorithm is more

consistent in detecting and quantifying damage. These results are encouraging, but

represent initial testing of the algorithm and further investigations will be needed to test

the validity of the damage detection method with other data and feature vectors.

3.8 Appendix: The EM Algorithm

This appendix shows how the EM algorithm is used to obtain the parameters of the

Gaussian mixture model (Bilmes, 1998). Instead of maximizing the log-likelihood

function of the observed data L(Y; ), the log-likelihood function of Y and I = { Ii; i =

1,..,M}, L(X, I; ), is maximized. It is recalled that Y is the collection of N feature

vectors and Θ are the parameters of the GMM to be determined.

jiji

N

i

M

jij

N

i

M

j

Iijiji

IpI

IpIpL ij

log;1|log

1;1|log;,

1 1

1 1

θy

θyΘIY

(3.17)


62

where

jijT

jijdiji Ip μyΣμyΣy 12

1

2/ 2

1exp

2

1;1| . Since the Ii are

not known, the expectation of Equation (3.17) with respect to the random variable Ii is

used.

jiji

N

i

M

jij IpIEL

log;1|log;,1 1

1 θyΘIY (3.18)

The M step maximizes the expected value of the log-likelihood function as defined in

Equation (3.18) and maximizes it with respect to the unknown parameters, i, i, and i.

It can be shown that 1

;1|log

j

Tji

j

iji Ipμy

μ

θy (Mardia et al., 2003). Thus,

N

kkj

N

kkkj

jj I

IL

1

11

0;,

yμ

μ

ΘIY (3.19)

Similarly differentiating Equation (3.17) with respect to the inverse of j, we get using

the fact that Tjijij

j

iji IpμyμyΣ

Σ

θy

2

1

2

1;1|log1

N

kkj

N

k

Tjkjkkj

jj IE

IEL

1

11

1

0;,

μyμyΣ

Σ

ΘIY (3.20)

In order to maximize the expected value of the log-likelihood function with respect to i,

the constraint that

M

ii

1

1 has to be used. Thus, we use the Lagrange multiplier to

obtain a new log-likelihood function which is given as:

1,,;,1

12M

iiLL ΘIYΘIY (3.21)


63

Differentiate (3.21) with respect to i, we get the following

NIEIE

MjIE

MjYLL

M

j

N

kkj

M

j

M

jj

N

kkj

j

N

kkj

jj

1 11 11

1

12

0

,...,1 0

,...,1 0;,;, ΘIΘIY

(3.22)

N

IEN

kkj

j

1

(3.23)

The E-step is now explained. It should be checked whether maximizing the log-

likelihood function L1(Y, I; ) is equivalent to maximizing the log-likelihood function of

the observed data, L(Y; ). This can be achieved by taking the expectation of Iij with

respect to the posterior probability density function of I, p(I| Y; ) (Bilmes, 1998).

M

kkiki

jiji

iijiijij

Ip

Ip

IpIpIE

1

;1

;1

;00;11

θy

θy

θyθy

(3.24)

The proof of the above is given as follows.

IIIIX

II

II

I

I

I

ΘIY

II

ΘIYIIΘIYΘY

II

dffL

df

Ypf

f

YpE

f

pE

df

pfdpp

ff

log;,

;,

log

Inequality sJensen' ;,

log;,

log

;,log;,log;log

1

(3.25)


64

where Ef(I) is the expectation with respect to f(I). The second term in Equation (3.25) does

not depend on , and thus it suffices to maximizing L1(Y, I; ), which is what is done in

the M-step. As in the E-step, use f(I) = p(I| Y; ),

ΘYIΘYΘYI

IΘYI

ΘIYΘYII

I

ΘIYI

;log;log;

;

;,log;

;,log

pdpp

dp

ppd

f

pf

(3.26)

Thus, the lower bound in Equation (3.25) becomes an equality and thus validates both the

E and M steps.

65

Chapter 4

Damage Feature Extraction from Wavelet Transform of Vibration Signals

Chapters 2 and 3 presented damage detection algorithms that utilized autoregressive (AR)

coefficients as feature vectors. These algorithms are valid for stationary signals obtained

from linear systems. In order to enable damage detection with non-stationary signals, the

wavelet transform is used in this chapter to formulate a feature vector.

In the context of SHM, earlier work was carried out in wavelet based system

identification of non-linear structures by Staszewski (1998), Ghanem and Romeo (2000)

and Kijewski and Kareem (2003). From a signal processing viewpoint, initial work done

was by Hou et al. (2000), where the discrete wavelet transform is used to study the

transient phenomenon when the stiffness of the structure is abruptly changed. Sun and

Chang (2002) have used the wavelet packet transform for decomposition of the signals,

where the wavelet packet component energies are utilized to detect damage and are

inputs to a neural network for damage assessment.

CHAPTER 4. Damage Feature Extraction from Wavelet Transform of Vibration Signals

66

In this chapter, a damage sensitive feature based on the energies of the Haar and Morlet

wavelet transforms of the vibration signal is derived. The reason these wavelet bases are

selected is because these bases have closed form expressions and aid in the ease of

mathematical derivation. However any appropriate wavelet basis can be used. For

example, in the next chapter, the Daubechies wavelet with four filter coefficients (DB4)

wavelet coefficients is used for feature extraction.

In the first part of this chapter, the theoretical aspects of wavelet decomposition of

vibration signals are discussed. The continuous wavelet transform of a signal is written in

terms of the Fourier transform of the signal and the wavelet basis. In the second part, this

framework is used to derive a closed form expression for the energies of the Haar and

Morlet wavelet for a single degree and multiple degree of freedom systems. The

relationship of the damage sensitive feature to physical parameters of the structure such

as mode shapes, stiffnesses and damping ratios is demonstrated. In order to illustrate this

wavelet based method for damage detection, it is applied to the ASCE Benchmark

Structure. The effect of noise on these datasets is also studied.

4.1 Properties of the Continuous Wavelet

Transform

A wavelet is a function (t)L2() (the space of square integrable functions) with the

following properties (Mallat, 1999):

1

0

dtt (4.1)

The mother wavelet function (t)L2() that is dilated/scaled by a and translated by b,

denoted by a,b(t) and is given as


67

a

bt

atba

1, (4.2)

where 1, ba . Then the continuous wavelet transform (CWT) of a function f(t)L2()

is given as:

dta

bt

atfbaWf

*

1, (4.3)

where * represents the complex conjugate. It should be noted that Equation (4.3) is a

convolution integral.

The main advantage of wavelet analysis over conventional spectral methods such as

Fourier methods is that data are localized in both time and scale domains (Mallat, 1999).

At lower scales, the wavelet basis function has a smaller support and thus is better able to

localize transient phenomena such as discontinuities in the dataset. Similarly, at higher

scales, the wavelet basis function has a wider support which helps in identifying long

range phenomena.

In the context of this study, it is useful to develop the wavelet transform in terms of the

Fourier Transform (FT) of a function rather than the function itself. Thus we define the

Fourier Transform of f(x) L2() as (Bracewell, 2000):

dtjtsxpetfsF

(4.4)

where, j is the square root of -1. The Inverse Fourier Transform (IFT) of F(s) is obtained

from

dsjtsxpesFtf

2

1 (4.5)

The Power Theorem is applied to obtain: (Bracewell, 2000)


68

dsssFdtttf baba

,, 2

1 (4.6)

where (s) is the FT of (t).

By substituting Equation (4.6) in Equation (4.3), it follows that

dsssFbaWf ba

,2

1, (4.7)

The Fourier Transform a,b(s) of the wavelet function a,b(t), is obtained as follows

asjsbadtjtsa

bt

a

dtjtsts baba

expexp1

exp,,

(4.8)

Thus, substituting Equation (4.8) in Equation (4.7), Equation (4.7) is rewritten as:

dsasjsbasFbaWf

exp

2

1, (4.9)

In this study, the Haar wavelet and the Morlet wavelet bases are considered. Close

investigation of these wavelets will provide a physical understanding of wavelet

coefficients of an acceleration signal and will provide a basis for correlating these

coefficients to damage sensitive features in a pattern classification scheme. The reason

why the Haar and Morlet bases are chosen is because these wavelets have a closed form

expression and are thus more convenient for mathematical derivations.

For the purposes of applying a statistical pattern classification method for damage

detection, we define the energy of the wavelet coefficients at appropriate scales as the

damage sensitive feature. Thus, the energy of the wavelet coefficients at scale a, Ea, is

defined as follows:

K

ba baxWE

1

2, (4.10)


69

where, baxW , is the wavelet coefficient of the acceleration signal at the ath scale and bth

time step, K is the number of data points in the signal and || is the absolute value of the

quantity.

In this study, damage detection is carried out by using the energy of the coefficients of

the sixth dyadic scale for the Haar wavelet, and the seventh dyadic scale for the Morlet

wavelet basis. Selection of these scales has to do with the support of the scaled wavelet

basis. The higher scales have larger support of wavelet basis, thus increasing the

likelihood of detecting long term changes. This implies that the wavelet coefficients at

higher scales would contain information about vibration modes at lower natural

frequencies. Since damage generally affects lower modes of vibration, these wavelet

coefficients at higher scales would be useful in damage detection. Also, at higher scales,

the wavelet coefficients do not pick up transient phenomenon such as spikes and jumps,

thus eliminating problems with noisy data.

In the following subsections we use the Fourier Transform of the Haar and Morlet

wavelets bases to derive the wavelet coefficients of a function f(t).

4.1.1 Haar Wavelet

The Haar wavelet is defined as follows:

0.15.0for 1

5.00for 1

t

tt (4.11)

The FT of the Haar wavelet can be shown to be equal to:

2

2exp1

1

js

jss (4.12)


70

Thus, from Equations (4.9) and (4.12), the wavelet coefficients using the Haar basis is

given as:

dsjas

jsbs

sF

a

jbafWH

2

2exp1exp

2,

(4.13)

The Haar wavelet and its Fourier transform are shown in Figure 4.1.

4.1.2 Morlet Wavelet

The Morlet wavelet is defined as follows:

2expexp

2

0

ttjt (4.14)

The FT of the Morlet wavelet can be shown to be equal to:

2

02

1exp2 ss (4.15)

In general, the value of 0 is chosen to be 5, which satisfies the admissibility condition

(Mallat, 1999). Thus, from Equations (4.9) and (4.15), the wavelet coefficients using the

Morlet basis are given as:

dsasjsbasFbafWM

252

1expexp

2

1, (4.16)

The Morlet wavelet and its Fourier transform are shown in Figure 4.2.


71

4.2 Derivation of the Damage Sensitive Feature

using Wavelet Coefficients of Acceleration

Signals for a SDOF System

Consider a single degree of freedom (SDOF) system, with mass m, damping coefficient c

and undamaged stiffness coefficient k, subject to a forcing function g(t) whose equation

of motion is given as:

tgkxxcxm (4.17)

Taking the Fourier transform of Equation (4.17) and applying the derivative rule

sFjsdt

tfdF n

n

n

the following expression is obtained:

sGsXkjcsms 2 (4.18)

where X(s) and G(s) are the FT’s of the displacement and forcing function respectively.

Here, we have made the assumption that the system is linear and the forcing function is

stationary. The first assumption is valid, since when we compare the damage with

undamaged system we are looking at an equivalent linear system with reduced stiffness.

The second assumption is not always valid in practice; however we can segment the

signal so that each signal is quasi-stationary.

With these assumptions, we can estimate the FT of the acceleration as:

kjcsms

sGssXstxFTsX

2

22 (4.19)

Using the framework developed in Sections 4.1.1, 4.1.2 and Equation (4.19), expressions

for the wavelet transform of acceleration signals is derived next.


72

t

ψ(t

)

(a)

(b)

Figure 4.1: Haar Wavelet (a) Haar Basis Function and (b) its Fourier Transform


73

t

Ψ(t

)

Figure 4.2: Morlet Wavelet (a) Morlet Basis Function and (b) its Fourier Transform


74

4.2.1 Wavelet Transform of Acceleration Signals

From Equations (4.9) and (4.19), it can be shown that

dsasjsba

kjcsms

sGsbaxW

exp

2

1,

2

2

(4.20)

4.2.1.1 Haar Wavelet Coefficients of Acceleration Signals

In particular, Haar wavelet coefficients of the acceleration signal can be derived as:

2exp1exp

2,

2

2

2

dsjas

jsbskjcsms

sGs

a

jbaxWH

(4.21)

In order to solve the above integral, the residue theorem and contour integration is used

(Kreyzig, 1994). The integral IH is defined as:

ds

jasjsb

kjcsms

ssGI H

2

2 2exp1exp

(4.22)

For an underdamped system, i.e., the damping ratio nm

c

2is less than 1, the damped

natural frequency 21 nd , n is the natural frequency of the SDOF system, and

the poles of Equation (4.22), p and q are calculated as

21, nnjqp (4.23)

The residues of the integral in Equation (4.22) are calculated as follows. A function h(z)

in the complex variable z is first defined as

2

2exp1exp

jazjzb

qzpz

zzGzh (4.24)


75

It can be proved that h(z) is analytic everywhere in the complex plane except at z = p and

z = q. Note that h(z) is analytic at z =0, since

2exp1

jaz is analytic at z =0. Using the

residue theorem, it can be shown that

qpH RRjI 2 (4.25)

where Rp and Rq are the residues of h(z) evaluated at p and q respectively. This proof of

Equation (4.25) is included in Section 4.6. Rp is given as:

;2

exp1exp

lim

2

qp

japjpbppG

zhpzR pzp

(4.26)

Similarly Rq is derived as

pq

jaqjqbqqG

Rq

2

2exp1exp

(4.27)

Here we assume that the G(s) is defined on the complex plane. Using the residue

theorem, it can be shown that

qpH RRa

baxW 1

, (4.28)

It is noted the residues Rp and Rq are related to the physical parameters of the system n

and ξ; and also on the loading on the SDOF system.

4.2.1.2 Morlet Wavelet Coefficients of Acceleration Signals

Using Equations (4.16) and (4.20), the Morlet wavelet transform of acceleration signals

may be derived as:


76

2

1expexp

2, 2

02

2

dsasjsbkjcsms

sGsabaxWM

(4.29)

In a similar fashion to the previous subsection, the integral IM is defined as follows:

dsasjsb

kjcsms

sGsI M

202

2

2

1expexp (4.30)

The residues of the integral IM are given in Equation (4.23). A function h(z) in the

complex variable z is first defined as

2

0

2

2

1expexp azjzb

qzpz

zGzzh (4.31)

It can be proved that h(z) is analytic everywhere in the complex plane except at z = p and

z = q. Using the residue theorem, it can be shown that IM = 2j(Rp+Rq), where Rp and Rq

are the residues of h(z) evaluated at p and q respectively. Rp is given as:

;2

1expexp

lim

20

2

qp

apjpbpGpzhpzR pzp

(4.32)

Similarly Rq is derived as

pq

aqjqbqGqRq

20

2

2

1expexp

(4.33)

Here we assume that the G(s) is defined on the complex plane. Using the residue

theorem, it can be shown that

qpM RRjabaxW 2, (4.34)

With Equation (4.34), it is demonstrated that the wavelet coefficients of the acceleration

contain information about the physical parameters of the system. From Equations (4.28)


77

and (4.34), we can derive the energy of the wavelet coefficients as defined by Equation

(4.10).

4.2.2 Damage Sensitive Feature

In this section, the damage sensitive feature is derived for a SDOF system using the Haar

and Morlet bases.

4.2.2.1 Haar Basis

Using the relationship

*22*2Re2

11, qpqpqpqpH RRRR

aRRRR

abaxW (4.35)

where Re() is the real part of the complex quantity, we can conclude that

*2222

1, qpqpH RRRR

abaxW (4.36)

It is noted that |p| = |q| = n. Also, we observe that

n

nnd

bjqb

bbjbjpb

expexp

expexpexpexp (4.37)

and

2

1

2

22

2exp1

2exp

2exp1

2exp1

2exp1

dnd jac

aja

japjap

(4.38)

Similarly, it is observed that


78

2

2

22

2exp1

2exp

2exp1

2exp1

dnd ja

cajajaq

(4.39)

where

2

exp1

21

na

cc .

The coefficient c1 is a decreasing term in a, whereas c2 is an increasing term in a. Since

we consider higher scales to compute the energies, Equations (4.38) and (4.39) can be

approximated as follows:

22

2

2

1121

2

1

12

exp1

2cos21

2cos21

2exp1

cja

c

ac

acc

jac

d

ddd

(4.40)

Thus, using Equations (4.37) and (4.40) we can conclude that

2

222

22

1

2

22

14

12

cos212exp

cqGa

cpGb

RR

dn

qp (4.41)

In a similar fashion, we can show that

2

2

22

*

*

14

1

cqGpGRR qp (4.42)

The expression in Equation (4.42) is a constant (i.e., not a function of b) and thus is not

included in Equation (4.43).

2

222

22

1

2

2

14

12

cos212exp

,

a

cqGa

cpGb

baxW

dn

H (4.43)


79

where

2

exp1

21

na

cc . For an acceleration signal sampled at 1/t Hz with K data

points, the energy at scale a for the Haar basis, EaHaar, is derived using Equation (4.43)

t

tKta

ba

cqGa

cpG

E

n

nnHaar

K

bn

d

Haara

2exp1

2exp12exp

2exp14

12

cos21

12

222

22

1

2

(4.44)

where

2

222

22

1

2

14

12

cos21

a

cqGa

cpG

a

d

Haar .

Equation (4.44) consists of Haar, which is a function of c1 and c2. Since the damage

sensitive feature is obtained at higher scales (i.e., the value of a increases) the value of c1

is not significant. As the extent of damage increases, the values of c1 increases and c2

decreases. In such cases, c2 will dominate, thus EaHaar is sensitive to damage at higher

scales.

4.2.2.2 Morlet Basis

Using similar principles as given in Equations (4.34) and (4.35),

*222

22, qpqpM RRRRabaxW (4.45)

It is noted that |p| = |q| = n. Also, we observe that

22222 21

2

1exp

2

1exp napa

dapa 00 expexp

(4.46)


80

From Equation (4.46), we obtain

dn aa

papaap

020

222

020

2220

exp2

1exp21

2

1exp

exp2

1exp

2

1exp

2

1exp

(4.47)

Similarly, it is observed that

22222 21

2

1exp

2

1exp naqa

daqa 00 expexp

(4.48)

From Equation (4.48), we obtain

dn aa

qaqaaq

020

222

020

2220

exp2

1exp21

2

1exp

exp2

1exp

2

1exp

2

1exp

(4.49)

Thus, using Equations (4.32), (4.33), (4.47) and (4.49) and we can conclude that

2

1

2

122

02222

22

14

21exp2exp

d

qGdpGab

RR

nnn

qp (4.50)

where, d1 = exp(2a0d). In a similar fashion, it is shown that

2

20

2222*

*

14

21exp

nn

qp

aqGpGRR (4.51)

Since Equation (4.51) is not a function of b,


81

2

1

2

122

02222

2

12

21exp2exp

,

d

qGdpGab

abaxW

nnn

M (4.52)

Thus, the energy of the Morlet based wavelet coefficients of the acceleration signal at

scale a, EaMorlet, is derived in a similar fashion and is given as follows:

t

tKtaE

n

nnMorl

Morleta

2exp1

2exp12exp (4.53)

where, 0 = 5,

2

1

2

122

02222

12

21exp

d

qGdpGa

aa

nn

Morl and d1=

exp(2a0d).

Assuming a constant value of (say, = 0.1), it is observed that d1 increases with

increases in the value of a. Also d1, which is a function of the damped natural frequency,

decreases with increase in the extent of damage. Since EaMorlet is a function of d1 and n,

both of which decrease with an increase in damage extent, EaMorlet is a good indicator of

damage. It should be noted that Ea is also a function of the loading.

4.3 Derivation of the Damage Sensitive Feature

using Wavelet Coefficients of Acceleration

Signals for a MDOF System

We next consider a structural system with N degrees of freedom (dof) with M, C and K

defined as the mass, damping and stiffness matrices (of size NN), respectively. The

forcing function is denoted by g(t). For a proportionally damped system and assuming

that the damping ratio in each mode is equal to , the transfer function of the


82

displacement at the kth dof xk(t) and the forcing function at the lth dof, Hkl(s), can be

derived as (Maia and Silva, 1998):

N

r rr

lrkr

l

kkl sjssG

sXsH

122 2

(4.54)

where Xk(s) and Gl(s) is the Fourier transform of the displacement at the kth dof xk(t) and

the forcing function at the lth dof gl(t). Thus, the FT of the acceleration txk is given as:

N

r rr

llrkrk sjs

sGssX

122

2

2 (4.55)

where r is the rth modal natural frequency, kr and lr are the kth and lth elements of the

mass normalized rth mode shape vector r. Using similar principles utilized in Section

4.2, the wavelet coefficients of a MDOF system are derived in the next subsection.

4.3.1 Wavelet Coefficients of Acceleration Signals

With respect to the Haar wavelet, the wavelet coefficient of the acceleration txk is

derived as:

ds

jasjsb

sjs

sGs

a

jbaxW

N

r rr

llrkrkH

2

122 2

exp1exp22

,

(4.56)

In order to calculate the above integral, we will use principles of contour integration as is

done for SDOF systems presented in the previous section. Again, the integral Ir for the rth

vibration mode, is defined as:

dsjas

jsbsjs

sGsI

rr

llrkrr

2

22 2exp1exp

2 (4.57)

The poles of Equation (4.57), p1,r and p2,r are calculated as


83

2

,2

,1 1

rr

r

r jp (4.58)

The function t(z) in the complex variable z defined as

2

21 2exp1exp

jaz

jzbpzpz

zzGzt llrkr (4.59)

It can be proven that t(z) is analytic everywhere in the complex plane except at z = p1,r

and z = p2,r. Using the residue theorem, it can be shown that Ir = 2j(R1,r+R2,r), where R1,r

and R2,r are the residues of t(z) evaluated at p1,r and p2,r respectively. R1,r and R2,r are

evaluated as:

;2

exp1exp

lim,2,1

2

,1,1,1,1

,1,1 ,1

rr

rrrlrlrkr

rpzr pp

japbjppGp

ztpzRr

(4.60)

Similarly R2,r can be derived as

rr

rrrllrkr

r pp

japbjppGp

R,1,2

2

,2,2,22

,2

2exp1exp

(4.61)

Using the residue theorem, it can be shown that

N

rrrH RR

abaxW

1,2,1

1, (4.62)

Again, we approximate the energy of the Haar wavelet coefficients of acceleration txk

at scale a as

N

rrr

Haarka RR

aE

1

2

,2

2

,1,

1. The rth damped natural frequency d,r is

given as: 2, 1 rrd . We define

2

exp1

,2,1

r

rr

a

cc .


84

Using similar approximations as in the previous section, we obtain the following

expression for HaarkaE , as:

K

b

N

r

rlrd

rlrlrkr

Haara a

cpGa

cpGb

E1 1

2

222

2

,2

2

,1

2

,122

14

12

cos212exp

(4.63)

Interchanging the summations, we can derive that

N

r r

rrlrkrHaar

Haara t

tKtraE

1

22

2exp1

2exp12exp, (4.64)

where

2

22,2

2

,2

2

,,1

2

,1

14

12

cos21

,

a

cpGa

cpG

ra

rrlrd

rrl

Haar .

The energy of the Morlet based wavelet coefficients of acceleration txk at scale a,

EaMorlet, is derived in a similar fashion and is given as follows:

N

r r

rrlrkrMorl

Morleta t

tKtraE

1

22

2exp1

2exp12exp, (4.65)

where 0 = 5,

2

,1

2

,122

02222

12

21exp

,

r

rrr

Morl

d

qGdpGa

ara and d1,r =

exp(2a0d,r).

It is again observed that the energies of the wavelet coefficients for both the Haar and

Morlet wavelet basis contain information of the physical system. Ea contains modal

information of the system through the kth and lth mass normalized eigen vectors k and

l. In the case of EaHaar, as the extent of damage increases and which mode is excited due

to the increase in damage, c1,r increases and c2,r decreases. As the value of a increases, the


85

value of c1,r is not significant. In such cases, c2,r will dominate. In the case of EaMorlet,

assuming a constant value of the damping ratio = 0.1, it is observed that d1,r increases

with an increase in the value of a. Also d1,r reduces with increase in the extent of damage

depending on which mode is excited due to the damage. Thus Ea can be used as an

indicator of damage.

Thus, it can be concluded that as the stiffness decreases due to damage, the response of

the structure will change resulting in changes of energies of the wavelet coefficients.

Consequently, the damage sensitive feature based on the wavelet coefficients can capture

this change in measurements from an undamaged to damaged structural state.

4.4 Application

In order to test the validity of the above derived damage sensitive feature, numerically

simulated datasets from the ASCE Benchmark Structure have been used. The description

of the ASCE Benchmark Structure was provided in Chapter 2. Damage detection analysis

is performed with both the Haar and Morlet wavelet bases.


Damage detection is performed under the premise that the damage sensitive feature will

migrate with the onset of damage. In this study, the damage sensitive feature is defined as

the energy of the wavelet coefficients at the seventh dyadic scale for the Morlet wavelet

(denoted by E7) and the sixth dyadic scale for the Haar wavelet (denoted by E6). The

reason for choosing the seventh dyadic scale for the Morlet wavelet is because this scale

is optimal for capturing important characteristics of the signal which are sensitive to

damage. It is noted that similar results are obtained for the fifth and sixth dyadic scales.


86

The results of the studies for sensors 2, 3, and 9 are presented. First, results using the

Morlet wavelet basis are presented, followed by the Haar basis.

4.4.1.1 Sensor 2

Figure 4.3 show the migration of the features extracted from signals from sensor 2, from

an undamaged state to the damaged state for damage patterns 6 and 3. In the case of

damage pattern 6 (Figure 4.3(a)), where the stiffness of a brace is partially reduced on the

first floor, it is observed that there is a small separation between the means of E7.

However, this separation is larger for damage pattern 3 (Figure 4.3(b)), where a brace is

removed on the first floor. This difference in the means is significantly larger in the case

of major damage patterns 1 and 2 (Figure 4.4). It is also noted that the variance of the

clouds increase with increase in damage.

4.4.1.2 Sensor 3

Figure 4.5 illustrates the migration of E7, extracted from sensor 3 for damage patterns 4

and 5. The means of the damage sensitive feature E7 for the damaged clouds of damage

patterns 4 and 5 are 21.39 and 21.45 respectively. These results indicate that the change

due to bolt loosening was not detected.

4.4.1.3 Sensor 9

Figure 4.6 illustrates the feature clouds, extracted from sensor 9, for damage patterns 3

and 4. For damage pattern 3, it is observed that there is very little separation between the

clouds because the brace has been removed in the x direction and sensor 9 measures the

acceleration in the y direction (Figure 2.7). For damage pattern 4, there are two distinct

clouds of feature vectors with damage since a brace has been removed from the third

storey in the y direction near sensor 9.


87

Similar results are obtained with the Haar wavelet for major damage patterns, but the

clouds do not show separation for minor damage patterns. The reason for this is that the

Haar wavelet is a simple wavelet (Figure 4.1) and thus may not be able to capture

important details of the vibration signal. Figure 4.7 - 4.10 illustrates the effectiveness of

the Haar wavelet in damage detection. It is noted that similar trends were observed for the

Haar wavelet at the fifth dyadic scale.

In the previous chapter, a damage extent measure was derived using the Mahalanobis

distance. The same measure DM is used in damage extent calculations and is defined as:

damagedundamagedundamagedT

damagedundamagedDM μμΣμμ 1 (4.66)

where undamaged is the covariance matrix of the undamaged dataset, undamaged and damaged,

are the means of the undamaged and damaged dataset respectively. Table 4.1 shows the

variation of DM (for the Morlet wavelet) for all sensors and various damage patterns.


88

(a)

(b)

Figure 4.3: Migration of the Morlet wavelet based damage sensitive feature E7 for sensor 2 with damage for minor patterns (a) Damage pattern 6 and (b) Damage Pattern 3


89

(a)

(b)

Figure 4.4: Migration of Morlet wavelet based damage sensitive feature E7 for sensor 2 with damage for major patterns (a) Damage pattern 1 and (b) Damage Pattern 2


90

(a)

(b)

Figure 4.5: Migration of Morlet wavelet based damage sensitive feature E7 for sensor 3 with damage for (a) Damage pattern 4 and (b) Damage Pattern 5 (Undamaged ;

Damaged +)


91

(a)

(b)

Figure 4.6: Migration of Morlet wavelet based damage sensitive feature E7 for sensor 9 with damage for (a) Damage pattern 3 and (b) Damage Pattern 4 (Undamaged ;

Damaged +)


92

(a)

(b)

Figure 4.7: Migration of the Haar wavelet based damage sensitive feature E6 for sensor 2 with damage for minor patterns (a) Damage pattern 6 and (b) Damage Pattern 3


93

(a)

(b)

Figure 4.8: Migration of the Haar wavelet based damage sensitive feature E6 for sensor 2 with damage for major patterns (a) Damage pattern 1 and (b) Damage Pattern 2


94

(a)

(b)

Figure 4.9: Migration of the Haar wavelet based damage sensitive feature E6 for sensor 3 with damage for (a) Damage pattern 4 and (b) Damage Pattern 5 (Undamaged ;

Damaged +)


95

(a)

(b)

Figure 4.10: Migration of the Haar wavelet based damage sensitive feature E6 for sensor 9 with damage for (a) Damage pattern 3 and (b) Damage Pattern 4 (Undamaged ;

Damaged +)


96

Table 4.1: Variation of DM for the Morlet wavelet based damage sensitive feature for various sensors and different damage patterns

Sensor Damage Metric DM


1 9.68 21.52 0.85 0.95 0.95 0.33 2 99.63 146.00 7.71 7.49 7.49 1.63 3 9.12 18.43 0.98 2.57 2.57 0.35 4 97.10 123.32 5.89 5.58 5.58 1.33 5 5.85 10.96 1.27 1.49 1.49 0.39 6 62.01 72.05 6.70 6.60 6.60 1.46 7 7.13 12.65 1.87 2.13 2.13 0.39 8 63.94 80.00 4.66 4.55 4.55 1.08 9 6.18 11.35 0.95 6.65 6.65 0.25

10 50.89 43.29 5.46 5.49 5.49 1.52 11 5.54 9.51 1.22 5.49 5.49 0.25 12 53.05 43.02 5.02 5.06 5.06 1.11 13 3.12 8.12 0.97 1.60 1.60 0.27 14 48.64 43.83 5.74 5.52 5.52 1.29 15 4.53 10.89 1.06 1.48 1.48 0.24 16 40.76 41.20 3.87 3.90 3.90 0.88

From the analysis, the following observations are made

The values of DM are correlated to the amount of damage. As can be observed

from Table 4.1, the general trend for the damage metric DM is that it increases

for damage patterns 6, 3, 4, 5, 1 and 2, which corresponds to a progressive

increase in damage.

With regard to DP4 and DP5, there is no change in the values of the damage

measure, thus indicating that bolt loosening could not be detected.

4.4.1.4 Effect of Noise

The effect of zero mean additive Gaussian white noise on the damage detection algorithm

is studied. The ratio of the root mean square (rms) value of the noise to the rms value of


97

the signal is defined as the noise to signal ratio, and is denoted as NSR. The NSR is

varied from 0.05 to 0.15. The values of DM with these noise levels for sensor 2 are

presented in Table 4.2.

Table 4.2: Variation of DM for sensor 2 with different noise to signal ratios (NSR) for damage patterns DP 1-6

NSR Damage Metric DM


0.0 103.26 151.25 7.87 7.66 7.66 1.68 0.5 57.00 78.17 2.11 1.84 1.84 0.36

0.10 20.11 35.16 0.88 0.53 0.53 0.00 0.15 10.49 17.13 0.22 0.46 0.46 0.00

From the analysis, the following observations are made:

The values of DM are correlated to the amount of damage. For damage

patterns 3 and 4, there is very slight difference between these values since at

sensor 2, DP 4 is DP 3 + removal of a brace in a direction perpendicular to the

direction of acceleration measured at sensor 2. Similar conclusions can be

made for DP 4 and DP 5.

The values of DM are sensitive to noise and only major damage patterns 1 and

2 can be detected at high ranges of noise levels

As expected, the separation of the clouds decreases with noise.


98

4.5 Summary

In this chapter, a damage sensitive feature based on the wavelet transform of the vibration

signal is derived. The damage sensitive feature is defined as the energy of the wavelet

coefficients at higher scales. Theoretical aspects of wavelet decomposition of vibration

signals are presented. Expressions for the energies of wavelet coefficients using the Haar

and Morlet wavelet bases are derived for a single degree and multi-degree of freedom

system. The derived damage sensitive feature is applied to various datasets for the ASCE

Benchmark Structure using the Haar and Morlet wavelets. The effect of noise on these

datasets is also studied.

It is observed that the Haar wavelet is able to detect only major damage patterns. The

reason for this is that the Haar wavelet is a simple wavelet (Figure 4.1) and thus may not

be able to capture important details of the vibration signal. In comparison, the Morlet

wavelet performs much better and is able to detect minor damage patterns. Thus for

applications, it would be advisable to use energies of the Morlet wavelet coefficients at

higher scales.

The Morlet wavelet based energies at the seventh dyadic scale were able to detect

damage for major damage patterns at relatively high levels of noise. However, at higher

noise levels, minor damage patterns are not detected. The reason for this might be

because that the sensitivity of the loading at these scales might have dominated the

sensitivity of the damage on the damage sensitive feature. Thus, it would be critical to

perform normalization step of the damage diagnosis algorithm. This is discussed in

Chapter 5.

The proposed damage sensitive feature is shown to be effective in detecting damage for

numerically simulated datasets obtained from the ASCE Benchmark Structure.

Application of the damage sensitive feature to the ASCE Benchmark simulation

experiment demonstrates that the algorithm is able to detect minor, moderate and major


99

damage patterns. However, loosening of bolts cannot be distinguished when it occurs in

conjunction with damage pattern 4. In general it would be difficult to detect bolt

loosening, unless a signal is obtained at the bolt itself and is used to identify damage.

With the acceleration measurements presented in the simulation experiment, highly

localized damage would be difficult to capture. It is also shown that the damage decision

for minor damage patterns is effected by high levels of noise.

4.6 Appendix: Derivation of the Integral IH

In this appendix, the derivation for the computation of Equation (4.22) is given.

p q

(R,0) A

B

C

(-R,0)

1R

2,R

Figure 4.11: Illustration of the Proof of the Contour Integration Formula


100

Let the path R = 1,R + 2,R, where Figure 4.11 shows that 1,R is the arc ABC and 2,R is

the arc CA. If R is sufficiently large, the both poles p and q are within R. By the residue

theorem,

qp RRjdzzhR

2 (4.67)

where Rp and Rq are poles of the function h(z) and are given by Equation (4.26) and

Equation (4.27) respectively. In this section, we will show that 0lim,2

R

dzzhR

. For this

purpose, we prove the following Lemma.

Lemma: Let d > 0. Then, 0sinexplim0

ddRR

.

Proof:

2/

2/

00

sinexpsinexpsinexp ddRddRddR (4.68)

To compute the second integral, make the substitution = – . Thus,

2/

02/

sinexpsinexp

ddRddR (4.69)

Thus,

2/

00

sinexp2sinexp

ddRddR (4.70)

Now, choose an arbitrarily small > 0

2/

0

2/

0

sinexpsinexpsinexp

ddRddRddR (4.71)


101

Since the function sinexp dR is decreasing in in [0,/2] and

sinexpsinexp dRdR in [ε,/2], we obtain

sinexp2

sinexp2/

dRddR (4.72)

If R is sufficiently large enough, i.e.,

2log

sin

1

dR , then

2/

sinexp ddR .

Thus, for any > 0, there exists R, such that if R > R,

2sinexp2/

0

ddR . Hence,

0sinexplim2sinexplim2/

00

ddRddRRR

(4.73)

Now on 2,R, z = Rexp(j), and [0,] and for any real valued a,

sinexpsinexpcosexpexp aRaRiRjaz (4.74)

Furthermore,

sinexpsin2

exp21

exp2

exp212

exp12

aRRa

jazjazjaz

(4.75)

Using the fact that dz = jRexp(j)d, we obtain

expexp

sinexpsin2

exp21sinexp

0,2

dqRjpRj

aRRa

bRzG

RdzzhR

(4.76)

Here again, we can approximate


102

R

qRqRj

R

pRpRj 1exp;1exp (4.77)

Thus, for sufficiently large R, and say for some fixed constant , e.g., = 0.5, we get

4expexp

2

qRjpRj

R

(4.78)

Thus, Equation (4.76) maybe rewritten as

0

00

sinexp

sin2

exp2sinexp

exp4

,2 dRba

dRba

dbR

RjGR

dzzhR

(4.79)

By the Lemma, each of these integrals on the RHS of Equation (4.79) has to converge to

zero.

Thus, 0lim,2

R

dzzhR

Then,

R

R

HRR

IdzzhdzzhR

limlim

(4.80)

Thus, IH = 2j(Rp + Rq).

103

Chapter 5

A Wavelet Based Damage Detection Algorithm

In Chapter 4, a damage sensitive feature based on the energies of the wavelet transform

of the vibration signal was derived. In this chapter, a damage detection algorithm based

on this feature is presented. This algorithm requires the creation of a database of

normalized baseline signals. A methodology is developed for obtaining the best signal in

the database, closest to the new signal, using the lower singular values of the energies of

the wavelet coefficients at the first dyadic scale. To obtain the damage sensitive feature

vector, the energies of the wavelet coefficients at the fifth, sixth and seventh dyadic

scales are used. Principal components are used in this regard. Damage detection is

performed by using the k-means algorithm and the gap statistic. The k-means algorithm

estimates the cluster centers in a dataset and the gap statistic is used to determine the

optimal number of clusters in the dataset. It is hypothesized that more than one cluster is

an indication of change and this change is most likely due to damage. Finally, this

algorithm is tested using datasets from the ASCE Benchmark Structure.

CHAPTER 5. A Wavelet Based Damage Detection Algorithm

104

5.1 Overview of Algorithm

This algorithm requires creating a database of baseline measurements (which can include

accelerations and strains) and computing the coefficients of the continuous Daubechies

wavelet of order four (DB4) at appropriate scales. The DB4 wavelet is chosen because it

is similar to the Morlet wavelet and has a discrete wavelet counterpart which can be used

when embedding the algorithm at the sensor level. Following this, principal components

analysis is performed on the energies of the DB4 wavelet coefficients at the first dyadic

scale. This step helps in obtaining the closest signal in the database, which describes the

loading condition of the new signal. Once the closest baseline signal in the database is

chosen, feature vectors are calculated using the DB4 wavelet coefficients at the fifth,

sixth and seventh dyadic scales. Finally the k-means algorithm and the gap statistic are

used to discriminate between damaged and undamaged states in the structure. Damage is

detected when the algorithm predicts more than one cluster in the feature vectors under

question. The proposed detection algorithm is as follows:

i Obtain signals from an undamaged structure, from sensor i, denoted by xi(t) (i

= 1,…,P), where P is the number of sensors. Segment the signal xi(t) into

chunks of finite duration xij(t) (j = 1,…,Q ), where Q is the number of chunks.

Normalize these signals to obtain a mean zero and standard deviation one

signal as described in Equation (2.1). Populate the database with these

normalized baseline signals.

ii Compute the DB4 wavelet coefficients of the baseline signals at the first, fifth,

sixth and seventh dyadic scales. Calculate the norm of the wavelet coefficients

within a window size of size d = 20 data points. These are represented by

Qjjbaselinejbaselinejbaselinejbaselinejbaseline ,...,1 ,,7,,6,,5,,1, EEEEE (5.1)

where Ei,baseline,j is the energy vector of the wavelet coefficients of the jth

baseline signal at the ith dyadic scale. It is noted that Ei,baseline,j (i = 1, 5, 6, 7; j


105

= 1,…,Q) is a vector of dimension K/d 1, where each baseline signal is of

length K.

iii Obtain the new signal at a later time, say t. This new signal could be obtained

from a potentially damaged location.

iv Compute the DB4 wavelet coefficients of the new signals at the first, fourth,

fifth and sixth dyadic scales. Energies of these coefficients are calculated in a

similar fashion as performed in step ii. These are represented by

Qjjnewjnewjnewjnewnew ,...,1 ,,7,,6,,5,,1 EEEEE (5.2)

where Ei,new,j is the energy of the wavelet coefficients of the jth chunk of the new

signal at the ith dyadic scale.

v Selection of closest baseline signal step: Search for the best baseline signal

(obtained in Step i) with similar environmental and loading conditions. This is

performed as follows:

For each baseline signal, form a matrix Z0,j = [E1,baseline,j E1,new,j] (j =

1,…,Q) of dimension K/d 2.

Perform a principal components (PC) analysis on Z0, j.

Obtain the principal direction, v1j (j = 1,…,Q). Also, obtain the minimum

variance (square of the lowest singular value) and denote it as 0j.

Choose the signal closest to the new signal as that signal with the value of

1,21

,11 j

jj v

v and the lowest value of 0j, where v11,j and v21,j are the

values of the components of vector v1j.


106

vi Feature extraction step: Extract features by comparing the energies of the new

signal with that of best signal. This is performed as follows:

Form a matrix Zj,k = [Ei,best,j Ei,new,j] (i = 5,6,7; j = 1,…,Q; k = 1,2,3), where

Ei,best,j is the energy of the wavelet coefficients of the jth chunk of the best

signal in the database at the ith dyadic scale.

Perform a principal component analysis on Zj,k (j = 1,…,Q; k = 1,2,3).

Obtain the minimum variance (square of the lowest singular value) and

denote it as j,k (j = 1,…,Q; k = 1,2,3).

Choose the damage sensitive feature vector as = [1 2 3] where 1 is

the vector whose jth component is j,1.

vii Classification step: Use the k-means clustering algorithm with the gap statistic

to discriminate between a damaged state and an undamaged state.

Fix the number of clusters as k. For a fixed value of k, obtain the cluster

centers of the grouped feature vectors X defined in Equation (5.3), using

the k-means algorithm.

new

undamaged

κ

κX (5.3)

where undamaged and new are the feature vectors obtained from an undamaged

and new signal respectively. Calculate the gap statistic (Tibshirani et al.,

2001).

Use the gap statistic to determine the optimal number of clusters. If the

number of clusters is greater than one, then it is hypothesized that some

degree of damage has taken place. If, however, the clusters are very close


107

based on the gap statistic, then it is concluded that there is no damage.

Such signals would be stored in the baseline database.

viii If there is damage, calculate the extent of damage by using the Euclidean

distance between the means of the damaged and undamaged clusters.

ix Go to step iii.

5.2 Application of Principal Components Analysis

in Optimal Selection of Baseline Signal and

Feature Extraction

In order to perform steps (v) and (vi), principal components analysis will be used. The

theory of principal components analysis is explained below.

5.2.1 Principal Components Analysis

Principal components analysis is a linear transformation used in multivariate statistical

analysis, generally used to reduce the dimension of the dataset and to find patterns (or

feature extraction) in the dataset (Mardia et al., 2001). The mathematical principle behind

PCA is explained as follows: Consider a mean centered matrix Y of dimension, say Np.

The sample covariance matrix is given by S = YTY/N-1. Then the eigen decomposition of

YTY is given as

p

p

TT

ddddiag ,...,,

...

21

21

2

D

vvvV

VVDYY

(5.4)


108

where, V is a set of eigenvectors vi (i =1,..,p) and D is the singular value matrix. The

eigenvectors vj are called the principal components directions of Y. The first principal

component z1 = Yv1 has the largest sample variance amongst all the linear combinations

of the columns of Y (Mardia et al., 2001). In this study, principal components (PC’s) are

not used to reduce dimensionality but are used for feature extraction.

5.2.2 Selection of the Closest Baseline Signal

In practice, vibration data are collected under different operational conditions (loading

amplitude and direction; and environmental conditions such as temperature and

humidity). In order to compare ambient (or linear) vibration signals under various

operational conditions, we will use the energies of the wavelet coefficients at the 1st

dyadic scales to determine whether there is any difference in the loading conditions of the

signal or not. The reason why lower scales are chosen is that the wavelet coefficients at

these scales will be able to take into account the transient phenomenon such as jumps and

spikes and thus describe the loading conditions better.

Figure 5.1 illustrates the plot of the energies of the wavelet coefficients at the first dyadic

scale, for a similar and dissimilar vibration signal. The vector v1 is plotted at 45o to the x

axis. In the case when we obtain acceleration datasets from similar loading conditions, it

is noticed that the clouds cluster along the direction of v1. The reason for this is that these

values of E1,baseline,j and E1,new,j should be similar and thus cluster around v1, implying a

low variance in the direction of v2. The best signal in the database closest to the new

signal is selected using the following procedure:

Form the matrix Z0,j = [E1,baseline,j E1,new,j] (j = 1,…,Q). Perform a principal

component analysis on Z0,j.


109

jj

j

jjj

Tjjjj

Tj

dddiag

Qj

21

21

2,0,0

,

,...,1

D

vvV

VDVZZ

(5.5)

where, Vj is a set of eigenvectors and Dj is the singular value matrix, obtained from

the decomposition of Z0,j.

Obtain the principal directions v1j to calculate the ratio j

jj v

v

,21

,11 and 22,0j

j d

(j=1,…,Q). Among all the signals in the database, find that signal in the database

with the values of the ratio j approximately equal to one and having the lowest

value of 0,j.

E 1, baseline,1

Similar signal

Dissimilar signal

E 1 , new,1

v 1

v 1 ’

v 2

v2 ’

Figure 5.1: Illustration of a similar and dissimilar cloud by comparing E1,baseline and E1,new


110

In the case when the signals are not similar, the value of E1,new,1 and E1,baseline,1 would be

different and thus deviate away from the vector v1. Also, the variance (spread) in the

direction v2 would give an indication of similarity. To this end, the principal directions

will be used to obtain the directions of highest variances. The variance in the direction of

the second principal direction is used to show how dissimilar the signals are. This is

denoted as 0, whose jth component is 0,j. Thus, the lower the value of 0,j, more similar

the signals are. The reason why the first scale is chosen is because the coefficients at the

first scale will be able to detect transient phenomenon, which is a good descriptor of

loading conditions. This assumes that damage does not the affect the higher frequency

modes of vibration.

Figure 5.2 illustrates the comparison of two loading conditions as defined in the ASCE

Benchmark Structure Experiment. The first excitation is a series of independent filtered

Gaussian white noise loads generated using a sixth - order low-pass Butterworth filter

with a 100 Hz cutoff and applied at each story of the structure. The second loading is a

random excitation generated by a shaker on the roof-top of the center column.

Figure 5.2(a) shows the histogram of when comparing an acceleration signal from

sensor 2 for damage pattern 2 to baseline signals recorded for the same loading

conditions. The values of are in the range of 0.8-1.6, indicating that loading conditions

are similar. Also it is noted that even though damage pattern 2 is the most severe of the

damage patterns, it does not affect the values of .

Similarly, Figure 5.2(b) shows the histogram of when comparing an acceleration signal

from sensor 2 for an undamaged signal with baseline signals recorded for different

loading conditions. The values of are much higher than one and are in the range of 4.0-

6.0, indicating that loading conditions are not similar.


111

(b)

(a)

Figure 5.2: Histogram of for sensor 2 for (a) similar loading condition with DP2 and (b) dissimilar loading conditions for undamaged cases


112

Number of records

κ 0

Figure 5.3: Variation of 0 for similar loading conditions comparing undamaged case and damage pattern 2

Figure 5.3 shows the variation of 0 for similar loading conditions while comparing a

signal from sensor 2 and damage pattern 2 to a database of eighty similar baseline

signals. The lowest value of 0 is chosen as the best signal in the database, closest to the

new signal.

5.2.3 Feature Extraction

Figure 5.4 illustrates the feature extraction procedure. Values of the energies of the

wavelet coefficients at the fifth dyadic scale of the best baseline signal E5,best and the new

signal E5,new are compared. Principal components analysis (PCA) is performed on this

dataset and the variance in the second principal direction v2, denoted as 1, is chosen as


113

the first element of the feature vector . Similarly 2 and 3 (second and third elements of

the feature vector) are derived using the energies at the sixth and seventh dyadic scales

respectively. Intuitively, it can be understood a large value of is a good indicator of

damage.

E5, best

Undamaged cloud

Damaged cloud

v3

v2

v4

E 5 , damaged

v 1

Figure 5.4: Illustration of damaged and undamaged cloud using principal components analysis

It should be noted that the fifth, sixth and seventh dyadic scales are chosen by trial and

error. Thus, when using this algorithm with a real structure, a finite element model of the

structure under question should be developed. Damage should then be induced on the

finite element model and the most sensitive scales can be chosen from a similar analysis

as stated in this section.


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Figure 5.5: Variation of the damage sensitive feature vectors for damage patterns (DP) 0, 1 and 6 as defined in the ASCE Benchmark Experiment

Figure 5.5 illustrates the variation of the damage sensitive feature for an undamaged case,

major damage pattern 1 and minor damage pattern 6. It is noted that the values of the

damage sensitive feature for an undamaged case are close to zero. However, for damage

patterns 1 and 6, the values of the feature vector are significantly greater than zero and

form separate clouds with respect to the undamaged cloud, thus indicating damage. From

Figure 5.5, it is also observed that as the level of damage increases the distance of the

cloud from the origin increases. In the next section, a classification methodology for

damage detection using the k-means algorithm and the gap statistic is explained. Also, a

damage measure DM using the Euclidean distance between the means of the undamaged

and damaged feature vectors is developed.


115

5.3 Damage Diagnosis

The k-means algorithm is a commonly used hard clustering scheme. In this particular

study, the k-means algorithm is used instead of the Gaussian mixtures modeling, since

there is a larger separation of clouds while using wavelet based feature vectors (Figure

5.5) as in comparison to the same study with the AR coefficients as feature vectors

(Figure 3.4a). A damage measure DM using the Euclidean distance between the means of

the undamaged and damaged feature vectors is developed.

5.3.1 Damage Detection using the k-means Algorithm and

the Gap Statistic

Figure 5.5 shows the results from the application of the proposed damage algorithm to

the numerically simulated datasets of the ASCE Benchmark structure. From Figure 5.5, it

can be observed that there is a distinct separation in the clouds of damage sensitive

feature vector for damage patterns 1 and 6 with the respect to the undamaged case. The

dataset used for damage detection is given by the grouped feature vector X as defined in

Equation (5.3). The damage detection using the k-means algorithm and the gap statistic is

performed as follows:

For k = 1 to M (where M is the number of clusters)

o Initialize the means of the dataset to k randomly chosen points

o For each cluster mean, j, (j= 1,…,k), find the points in the dataset

closest to j. Denote these set of points as Cj and the number of points

as nj.

o Compute the new mean


116

ji C

ij

j n x

xμ1

(5.6)

where, xi is a vector belonging to cluster j, Cj.

o Iterate the above two steps until convergence is obtained.

o After computing the means of the dataset, the gap statistic is computed

by using Equation (3.13).

Using the rules in Equation (3.14), find the optimal number of clusters in the

dataset.

Figure 5.6 illustrates the migration of clusters for an undamaged feature vector and

damage pattern (DP) 6. It is observed that even though DP 6 is a minor damage pattern,

there is a large separation between the feature vectors before and after damage. The k-

means algorithm is used to obtain the cluster centers in the dataset. The gap statistic

predicts that there are 2 clusters in the dataset, indicating that there is damage.

Figure 5.7 and Figure 5.8 illustrates a similar trend in the migration of the feature vectors.

In all these cases, the gap statistic predicts that there are two clusters in the dataset, thus

indicating damage. The results for all sensors in the ASCE Benchmark Experiment are

given in the next subsection, where the damage extent measure is developed. It should be

noted that results presented in Section 5.3 would have similar trends if the Morlet wavelet

were used.


117

(a)

(b)

Figure 5.6: Migration of the feature vectors κ with damage for minor patterns (a) Damage pattern 6 and (b) a zoom in of the undamaged cloud (Undamaged ; Damaged +)


118

(a)

(b)

Figure 5.7: Migration of the feature vectors with damage for damage patterns (a) Damage pattern 3 and (b) Damage Pattern 4 (Undamaged ; Damaged +)


119

(a)

(b)

Figure 5.8: Migration of the feature vectors with damage for major patterns (a) Damage pattern 1 and (b) Damage Pattern 2 (Undamaged ; Damaged +)


120

5.3.2 Damage Extent Measure

The damage sensitive feature is defined as follows:

undamagednewT

undamagednewDM ,,,, ˆˆˆˆ μμμμ (5.7)

where, new,ˆ μ and undamaged,ˆ μ are the sample means of new and undamaged respectively.

Table 5.1 shows the variation of DM for all sensors and damage patterns as defined in the

ASCE Benchmark Experiment.

Table 5.1: Variation of DM for the DB4 wavelet based damage sensitive feature for all sensors and different damage patterns

Sensor DP1 DP2 DP3 DP4 DP5 DP6

1 98.07 100.35 11.20 72.14 72.14 NA 2 129.31 119.52 109.27 108.54 108.54 63.36 3 105.89 105.51 13.62 76.74 76.74 NA 4 144.37 127.67 123.77 123.16 123.16 71.53 5 91.81 96.54 14.31 71.37 71.37 NA 6 109.14 136.85 118.73 118.81 118.81 79.37 7 83.77 88.08 14.53 63.39 63.39 NA 8 106.05 128.91 105.65 105.22 105.22 71.21 9 102.21 99.38 19.18 68.82 68.82 NA

10 122.73 123.77 139.95 139.75 139.75 98.04 11 108.8023 103.50 22.41 72.94 72.94 NA 12 111.39 112.44 122.09 122.07 122.07 88.20 13 110.60 108.19 19.33 82.20 82.20 NA 14 123.01 108.23 126.89 126.21 126.21 83.69 15 98.18 97.01 17.89 71.56 71.56 NA 16 126.27 113.72 124.56 124.42 124.42 88.26

From Table 5.1, the following observations are made

The values of DM are correlated to the amount of damage. For damage

patterns 1 and 2, the values of DM are similar. A similar observation is made


121

for damage patterns 3 and 4 for sensors placed on Faces 2 and 4 of the

structure (Figure 2.7).

For minor damage patterns 3 and 6, where the stiffness of brace is reduced on

Face 2 of the ASCE Benchmark Structure (Figure 2.7), the values of DM are

high for sensors on Faces 2 and 4 in comparison to Faces 1 and 3. It is also

observed that no damage is detected at sensors on Faces 1 and 3 for damage

pattern 6. These are denoted as NA in Table 5.1.

For sensors instrumented on Faces 1 and 3 of the ASCE Benchmark Structure

(Figure 2.7), the values of DM are significantly higher for damage pattern 4 in

comparison to damage pattern 3. For example, in the case of sensor 1, the

value of DM increases from 11.20 for DP 3 to 72.14 for DP 4. The reason for

this trend is because DP 3 involves the removal of a brace on Face 2, whereas

DP 4 is achieved by removing a brace on Face 1 and Face 2 of the structure.

All damage patterns are detected consistently using the wavelet based feature

vector.

For all damage patterns, the values of DM are higher for sensors on Faces 2

and 4. The reason for this behavior may be attributed to the fact that this is the

weak direction of the structure.

5.4 Summary

This chapter describes the development of a wavelet based damage detection algorithm.

The feature vector is based on the energies of wavelet coefficients, where the theoretical

formulation was developed in Chapter 4. This algorithm requires the creation of a

database of baseline measurements and computing the continuous Daubechies wavelet of

order four (DB4) wavelet coefficients at appropriate scales. Following this, principal


122

components analyses are performed on the energies of the DB4 wavelet coefficients at

the first dyadic scale for comparing the new signal to the closest one in the database.

Once the optimal baseline signal in the database is chosen, feature vectors are calculated

using the energies of the DB4 wavelet coefficients at the fifth, sixth and seventh dyadic

scales. The k-means algorithm is used to obtain the means of the clusters and is used for

damage detection along with the gap statistic. A damage measure based on the Euclidean

distance between the means of the damaged and undamaged datasets is also described.

The k-means algorithm and Gap statistic works consistently well for detecting damage

patterns as defined for the ASCE Benchmark Structure. The values of DM also correlate

well with the extent of damage.

Although this algorithm show promise in identifying and quantifying damage, the

algorithm has to be validated using experimental and field data. These should be for

varying degrees of damage, loading conditions, environmental conditions such as

temperature and humidity, as well as different sequences of damage occurrences.

Different damage locations on the structure should be considered and damage sequences

should also be investigated. It is only after extensive experimentation and field testing

with calibration that these models can be widely applied. Never-the-less, the results

presented here are encouraging and represent a good initial step towards achieving this

goal.

123

Chapter 6

Summary, Conclusions and Future Work

Recent research efforts in wireless structural health monitoring have resulted in an

explosion in the development of new sensors. Little attention, however, has been focused

on the efficient and effective use of the data collected by these sensors. While these

wireless sensor networks enable dense instrumentation, the amount of data that needs to

be transmitted can prove to be prohibitive. The main difficulty arises from the low data

rates associated with low power ad-hoc wireless sensor networks. Thus, data transmission

over the wireless network is demanding, time consuming and can significantly reduce

power source life. Typically these data are required because current damage detection

algorithms perform global system level analysis rather than local sensor level analysis. In

this dissertation, three local sensor based damage diagnosis algorithms using statistical

signal processing and pattern classification techniques have been developed. The main

features of these algorithms are that they are simple, robust and computationally efficient.

The main contributions of this dissertation are as follows:

Demonstrated the use of statistical signal processing techniques to model the

vibration signal and extracting damage sensitive features for classification.

CHAPTER 6. Summary, Conclusions and Future Work

124

Derived closed form equations between the autoregressive and wavelet based

feature vectors and the physical characteristics of the structure. This provides a

quantitative insight into the working of these algorithms.

Used pattern classification schemes to discriminate features obtained from signals

of an undamaged structure and a damaged structure.

Derived damage metrics using the feature vectors to model damage extent.

6.1 Summary

The main premise in the algorithms developed is that the vibration signals are affected by

damage and these changes can be tracked to detect damage. Three damage detection

algorithms have been developed and are summarized in Table 6.1. In addition, a damage

extent measure has been formulated using an appropriate metric.

The first algorithm developed in Chapter 2 uses time series to model the vibration signal

and defines a damage sensitive feature DSF using the first three AR coefficients. A t-test

on the DSF’s is used to discriminate between an undamaged state and a damaged state.

This algorithm is valid for linear and stationary signals. The validation of the algorithm

with datasets obtained from the ASCE Benchmark Structure shows that all damage

patterns were identified. However, for minor damage patterns, it is noted that the

difference between the means of the damage sensitive features was lower in comparison

to that of major patterns.

The second algorithm, developed in Chapter 3, uses the first three AR coefficients as the

feature vector. Damage detection is performed using the Gaussian Mixture Models

(GMM’s) and the gap statistic. This algorithm is more robust than the algorithm 1 and is

again valid for linear, stationary signals. The damage measure has been developed using

the Mahalanobis distance between the means of the damaged and undamaged datasets.


125

All damage patterns defined for the ASCE Benchmark Structure was consistently

identified and the proposed damage metric DM is well correlated to the extent of damage.

The third algorithm, developed in Chapters 4 and 5, uses the wavelet energies at the fifth,

sixth and seventh dyadic scales as feature vectors. This algorithm allows the use of non-

stationary signals. This algorithm requires a creation of a database of baseline signals for

comparison. The first part of the algorithm is to find that signal in the database closest to

the new signal. The second part of the algorithm is to obtain feature vectors. Both of

these steps are performed using principal components analysis. Damage detection is

performed using the k-means algorithm in conjunction with the gap statistic. A damage

measure is developed using the Euclidean distance between the means of the damaged

and undamaged feature vector. All damage patterns are consistently identified and the

proposed damage metric DM is well correlated to the extent of damage.

Table 6.1: Summary of damage detection algorithms developed in this dissertation

Algorithm Modeling of

Vibration Signal Feature Vector

Classification Algorithm

Damage Metric

1 AR/ARMA time

series DSF

Hypothesis Testing

NA

2 AR/ARMA time

series First three AR

coefficients GMM and the Gap Statistic

Mahalanobis Distance

3 Wavelet

Transform

Energies of the wavelet coefficients

at higher dyadic scales

k-means algorithm and the

gap statistic

Euclidean Distance

Some of the features of the above developed algorithms are as follows:

(i) simple since these algorithms do not require intensive finite element modeling

and updating in comparison to conventional system identification algorithms


126

(ii) robust since these algorithms are able to detect and quantify minor damage

patterns

(iii) computationally efficient since these algorithms use only processing of signals

at the sensor level and lead to significant saving in computation time.

6.2 Conclusions

The results of the application of the damage detection algorithm to vibration signals from

the ASCE Benchmark Structure lead to the following main conclusions:

All three algorithms are able to detect the existence of minor, moderate and severe

damage patterns in the ASCE Benchmark simulation experiment where minor,

moderate and severe damage corresponds to removal of single brace in a storey,

removal of a brace on two storeys and removal of all braces in two storeys,

respectively. It is also observed that the damage extent metric is well correlated to

the level of damage.

In Algorithm 1, it is shown that the AR model with 5-8 parameters is the optimal

time series for the vibration signals considered in the study. The differences in the

mean values of the damage sensitive feature DSF, as defined in Equation (2.6), is

higher for the major damage patterns in comparison to the minor damage patterns

and thus can be used as an indicator of damage extent. In the case of the minor

damage pattern (DP6), the confidence intervals of the difference in the means of

the DSF’s are not too high. Thus, a more sensitive feature / better classification

scheme is required for efficient damage detection.

Using Algorithm 2, it is seen that minor damage patterns are more consistently

identified using a more robust classification scheme. For all noise levels, major

and moderate damage patterns are detected since there is a large separation


127

between the damaged and undamaged feature vector clouds. However for large

noise levels, minor damage patterns, particularly damage pattern 6, do not appear

to be discriminated. The magnitude of the damage metric DM based on the

Mahalanobis distance appears to be highly correlated to the damage extent even

under the presence of noise. It is demonstrated that the magnitude of the DM

metric increases with increasing damage.

It is observed that the Haar wavelet is able to detect only major damage patterns.

In comparison, the Morlet wavelet performs much better and is able to detect

minor damage patterns. Thus for applications, it would be advisable to use

energies of the Morlet wavelet coefficients at higher scales. Algorithm 3 is more

sensitive to damage since the separation between the clouds of the Morlet wavelet

energies of the undamaged and damaged structure is much higher in comparison

to Algorithms 1 and 2.

6.3 Future Work and Research Needs

Although damage detection methodologies are getting more robust, no algorithm predicts

the location of damage accurately. Once damage diagnosis is complete, it is important to

perform damage prognosis. Almost no progress has been made in the field of damage

prognosis.

6.3.1 Damage Diagnosis

Damage detection is performed under various operational conditions, such as

environmental conditions such as temperature and humidity and loading conditions which

include the direction and magnitude of loading. Thus, algorithms are required to compare

feature vectors obtained from various loading conditions. Most of the currently developed


128

algorithms have not been tested on field data. Although field data are not readily

available, some of the currently available experimental data obtained are from a four

story steel frame used by the National Taiwan University (Lynch et al., 2006) and field

data obtained from a bridge in Switzerland (Wenzel, 2006). To gain wide adoption,

damage detection algorithms need to be tested with field data.

Damage localization is still an open problem in structural health monitoring. The damage

extent metric (DM) developed in this dissertation does not show any spatial pattern

(Figure 2.9). Thus, damage localization could not be achieved. One of the main reasons

for this is because acceleration measurements are global in nature and thus cannot capture

local effects. Thus, using strain data, which is a local measurement, could help in

localization of damage (Noh et al., 2007). Measurement data near cracks is also lacking

and simple experiments need to be generated on small samples simulating the initiation

and propagation of cracks. Measurements (vibration and strain) need to be obtained at

various stages of the crack propagation and at different distances from the crack. Such

measurements will also aid in the localization of the damage.

Furthermore, more research needs to be carried out in the field of data fusion, where

information from various sensors is statistically combined to obtain a more robust

decision (Wald, 1998). In addition, it would be ideal to fuse results from model based

methodologies (using physical models) and non model based algorithms (as presented in

this dissertation) to obtain more robust and meaningful results.

6.3.2 Damage Prognosis

The area of damage prognosis is still in its infancy. Damage prognosis deals with the

calculation of residual strength of the structure and the prediction of the residual life

capacity of the structure (Rytter, 1993). To this end, algorithms for local and global

damage prognosis have to be developed. These algorithms would involve the use of

modeling and simulation. These include physics based models, surrogate models, coupled


129

models and knowledge-based models (Farrar et al., 2003). Physics based models

generally involve finite element modeling where the model has to be updated at each

stage of progressive damage. These models are computationally very expensive and thus

require surrogate models to reduce the computational complexity of the problem.

Surrogate models generally include machine learning algorithms such as neural networks,

self organizing maps etc. Also calibration and validation of these models are required.

Uncertainty quantification becomes an important issue in these models. Modeling,

loading and measurement uncertainties have to be evaluated. For prediction of the

residual life of the structure, time variant reliability based methods can be used. The

challenges of these tasks are explained by Farrar et al., 2003.

130

Bibliography

J. A. Bilmes. A Gentle Tutorial of the EM Algorithm and its Application to Parameter

Estimation for Gaussian Mixture and Hidden Markov Models, Technical Report 97-021,

International Computer Science Institute, University of California. Berkeley, 1998.

R. N. Bracewell. The Fourier Transform and its Applications, McGraw Hill, Third

Edition, New York, 1999.

P. J. Brockwell and R. A. Davis. Introduction to Time Series and Forecasting, Springer-

Verlag, Second Edition, New York, 2002.

F- K. Chang. Proceedings of the 1st, 2nd and 3rd International Workshops on Structural

Health Monitoring, Stanford University, Stanford, CA. CRC Press, New York, 1999,

2001, 2003 and 2005.

S. W. Doebling, C. R. Farrar, M. B. Prime and D. W. Shevitz. Damage Identification and

Health Monitoring of Structural and Mechanical Systems from Changes in Their

Vibration Characteristics: A Literature Review, Los Alamos National Laboratory Report

LA-13070-MS, Los Alamos National Laboratory, Los Alamos, NM, 1996.

R. O. Duda, P. E. Hart and D. G. Stork. Pattern Classification, Wiley-Interscience

Publications, New York, 2001.

C. R. Farrar and T. A. Duffey. “Vibration based Damage Detection in Rotating

Machinery and Comparison to Civil Engineering Applications,” Proceedings of Damage

Assessment of Structures, Dublin, Ireland, 1999.

Bibliography

131

C. R. Farrar, H. Sohn, F. M. Hemez, M. C. Anderson, M. T. Bement, P. J. Cornwell, S.

W. Doebling, J. F. Schultze, N. Lieven and A. N. Robertson. Damage Prognosis: Current

Status and Future Needs, Los Alamos National Laboratory Report LA-14501-MS, Los

Alamos National Laboratory, Los Alamos, NM, 2003

R. Fox and M. Kapoor. “Rate of Change of Eigenvalues and Eigenvectors,” AIAA

Journal 6:2426-2429, 1968.

R. Ghanem and F. Romeo. “A Wavelet Based Approach for Identification of Linear Time

Varying Dynamical Systems,” Journal of Sound and Vibration, 234(5): 555-576.

T. Hastie, R. Tibshirani and J. Freidman. Elements of Statistical Learning: Data Mining,

Inference and Prediction, Springer Verlag, First edition, New York, 2001.

Z. K. Hou, M. Noori and R. St. Amand. “Wavelet-based Approach for Structural Damage

Detection,” ASCE Journal of Engineering Mechanics, 126(7): 677-683, 2000.

E. A. Johnson, H. F. Lam, L. S. Katafygiotis and J. L. Beck. “A Benchmark Problem for

Structural Health Monitoring and Damage Detection,” Proceedings of 14th Engineering

Mechanics Conference, Austin, TX, USA, 2000.

E. A. Johnson, H. F. Lam, L. S. Katafygiotis and J. L. Beck. “Phase I IASC-ASCE

Structural Health Monitoring Benchmark Problem Using Simulated Data,” Journal of

Engineering Mechanics, 130(1):3-15, 2004.

T. Kijewski and A. Kareem. “Wavelet Transforms for System Identification:

Considerations for Civil Engineering Applications,” Journal of Computer-Aided Civil

and Infrastructure Engineering, 18: 341-357, 2003.

E. Kreyzig. Advanced Engineering Mathematics, John Wiley and Sons, Eighth Edition,

New York, 1998.

Bibliography

132

J. P. Lynch. “Linear Classification of System Poles for Structural Damage Detection

Using Piezoelectric Active Sensors,” Proceedings of SPIE 11th Annual International

Symposium on Smart Structures and Materials, San Diego, CA, USA, 2004.

J. P. Lynch, Y. Wang, K-C. Lu, T-C. Hou and C-H. Loh. “Post-Seismic Damage

Assessment of Steel Structures Instrumented with Self-Interrogating Wireless Sensors,”

Proceedings of the 8th National Conference on Earthquake Engineering (8NCEE), San

Francisco, CA, 2006.

J. P. Lynch, A. Sundararajan, K. H. Law and A. S. Kiremidjian. “Embedding Algorithms

in a Wireless Structural Monitoring System,” Proceedings of the International

Conference on Advances and New Challenges in Earthquake Engineering Research

(ICANCEER02), Hong Kong, China, 2002.

J. P. Lynch, A. Sundararajan, K. H. Law, A. S. Kiremidjian and E. Carryer. “Embedding

Damage Detection Algorithms in a Wireless Sensing Unit for Attainment of Operational

Power Efficiency,” Smart Materials and Structures, 13(4):800-810, 2004.

N. M. M Maia and J. M. M Silva. Theoretical and Experimental Modal Analysis,

Research Studies Press, Hertfordshire, England, 1998.

S. Mallat. A Wavelet Tour of Signal Processing, Academic Press, New York, 1999.

K. V. Mardia, J. T Kent and J. M. Bibby. Multivariate Analysis, Academic Press,

London, 2003.

K. K. Nair and A. S. Kiremidjian. “Time Series Based Structural Damage Detection

Algorithm Using Gaussian Mixtures Modeling,” ASME Journal of Dynamic Systems,

Measurement and Control, 129(3): 285-293, 2007.

K. K. Nair, A. S. Kiremidjian and K. H. Law. “Time Series Based Damage Detection and

Localization Algorithm with Application to the ASCE benchmark Structure,” Journal of

Sound and Vibration, 291 (2): 349-368, 2006.

Bibliography

133

R. B. Nelson. “Simplified Calculations of Eigenvector Derivatives for Large Dynamic

Systems,” AIAA Journal, 14: 1201-1205, 1976.

H. Noh, K. K. Nair, A. S. Kiremidjian and C-H. Loh. “Application of a Time Series

Based Damage Detection Algorithm to the Taiwanese Benchmark Experiment,”

Proceedings of the 10th International Conference on Application of Statistics and

Probability in Civil Engineering (ICASP10), University of Tokyo, Tokyo, 2007.

A. V. Oppenheim and R. W. Schafer. Digital Signal Processing, First Edition, Pearson

Higher Education, 1986.

J. A. Rice. Mathematical Statistics and Data Analysis, Second Edition, Duxbury Press,

Second Edition, 1999.

A. Rytter. Vibrational Based Inspection of Civil Engineering Structures, PhD

Dissertation, Department of Building Technology and Structure Engineering, Aalborg

University, Denmark, 1993.

H. Sohn, M. Dzwonczyk, E. G. Straser, A. S. Kiremidjian, K. H. Law and T. Meng. “An

Experimental Study of Temperature Effect on Modal Parameters of the Alamosa Canyon

Bridge,” Earthquake Engineering and Structural Dynamics, John Wiley & Sons, 28(9):

879-897, 1999.

H. Sohn and C. R. Farrar. “Damage Diagnosis Using Time Series Analysis of Vibration

Signals,” Smart Materials and Structures, 10(3): 446-451, 2001.

H. Sohn, C. R. Farrar, H. F. Hunter and K. Worden. Applying the LANL Statistical

Pattern Recognition Paradigm for Structural Health Monitoring to Data from a Surface-

Effect Fast Patrol Boat, Los Alamos National Laboratory Report LA-13761-MS, Los

Alamos National Laboratory, Los Alamos, NM, 2001.

Bibliography

134

W. J. Staszewski. “Identification of Non-Linear Systems Using Multi-Scale Ridges and

Skeletons of the Wavelet Transform,” Journal of Sound and Vibration, 214(4): 639-658,

2000.

E. G. Straser and A. S. Kiremidjian. Modular Wireless Damage Monitoring System for

Structures, Technical Report No. 128, John A. Blume Earthquake Engineering Center,

Department of Civil and Environmental Engineering, Stanford University, Stanford, CA,

1998.

Z. Sun, and C. C. Chang. “Statistical Wavelet Based Method for Structural Health

Monitoring,” ASCE Journal of Structural Engineering, 130(7): 1055-1062, 2004.

R. Tibshirani, G. Walther and T. Hastie. “Estimating the Number of Clusters in a Dataset

via the Gap Statistic,” Journal of the Royal Statistical Society: Series B, 63(2): 411-423,

2001.

The ASCE Benchmark Group Webpage, http:// wusceel.cive.wustl.edu/ asce.shm/

benchmarks.htm, 2000.

L. Wald. “Data Fusion: A Conceptual Approach for an Efficient Exploitation of Remote

Sensing Images,” Proceedings of EARSeL Conference on Fusion of Earth Data, 17-23,

1998.

H. Wenzel, Personal Communication, 2006.

damage diagnosis of algorithms for wireless structural

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