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A MECHANICAL IMPEDANCE APPROACH FOR

STRUCTURAL IDENTIFICATION, HEALTH MONITORING

AND NON-DESTRUCTIVE EVALUATION USING

PIEZO-IMPEDANCE TRANSDUCERS

SURESH BHALLA

SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING

NANYANG TECHNOLOGICAL UNIVERSITY

2004

A Mechanical Impedance Approach for Structural

Identification, Health Monitoring and Non-Destructive

Evaluation Using Piezo-Impedance Transducers

Suresh Bhalla

School of Civil and Environmental Engineering

A thesis submitted to the Nanyang Technological Universityin fulfillment of the requirements for the degree of

Doctor of Philosophy

2004

i

ACKNOWLEDGEMENTS

First and foremost, I would like to extend my sincere thanks and gratitude

towards my supervisor, Professor Soh Chee-Kiong, for his continuous guidance,

encouragement and strong support during the course of my Ph.D. research. I am forever

grateful for his kindness and contributions, not only towards my research, but towards

my professional growth as well.

I am also grateful to other members of Professor Soh’s research team, namely

Prof Yang Yaowen, Xu Jianfeng, Akshay Naidu, Ong Chin Wee, Jin Zhanli and Wang

Chao, for giving numerous suggestions during the weekly research meetings. Often,

during presentations, the team members would pose questions that would immensely

help in improving my work. My special thanks go towards Prof Lu Yong, who not only

provided extremely useful suggestions as the examiner of my first year report, but also

personally oversaw the execution of many critical experiments. Thanks are also due to

Mr Lim Say Ian and Mr Goo Kian Tiong (Jimmy), who provided assistance in

performing many of the experiments as a part of their final year projects.

I express my special thanks to Mrs Koh, Mrs Ho, Ms May Sim, Mr Subhas, Mr

Tan and other technicians, who provided their technical support generously during the

lab work. Without their support and practical tips as well as the good work

environment in the Structures Lab, it would not have been possible to finish the

experimental work so smoothly. I also express my gratitude towards my colleagues and

other supporting staff at the School of Civil & Environmental Engineering, who

directly or indirectly contributed towards my research.

I am very thankful to my parents for their encouragement and sacrifices, and I

wish to mention a very special acknowledgement to Rupali, my wife, for her continued

support and co-operation. She maintained amazing home in-spite of her own research

programme.

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TABLE OF CONTENTS

PAGE

ACKNOWLEDGEMENTS...……………………………………………………….i

TABLE OF CONTENTS…………………………………………………………...ii

SUMMARY…...…………………………………………………………………….x

LIST OF TABLES…………………..…………………………………………….xiii

LIST OF FIGURES…………………………………………………………...…...xiv

LIST OF SYMBOLS……………………………………………………………....xxi

LIST OF ACRONYMS…………………………………………….……………...xxv

CHAPTER 1: INTRODUCTION…………………………………………………1

1.1 Structural Damages and Failures…………..…………………………….1

1.2 An Overview of Recent Structural Failures……………………………..2

1.3 Structural Health Monitoring ………………………………………..….6

1.4 Requirements for any SHM System…………………………………….7

1.5 SHM by Electro-mechanical Impedance (EMI) Technique…………….9

1.6 Research Objectives …………………………………...………………11

1.7 Research Originality and Contributions..……………………………...11

1.8 Thesis Organisation...………………………………………………….12

CHAPTER 2: ELECTRO-MECHANICAL IMPEDANCE (EMI)…………...14

TECHNIQUE FOR SHM AND NDE

2.1 State of the Art in SHM/ NDE………………………………………...14

2.1.1 Global SHM Techniques………………………………………14

2.1.2 Local SHM Techniques……………………………………..…18

2.1.3 Advent of Smart Materials, Structures and Systems for.………21

SHM and NDE

2.2 Smart Systems/ Structures………………….………………………….22

2.2.1 Definition of Smart Systems/ Structures………………………22

2.2.2 Smart Materials…...……………………………………………24

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2.2.3 Active and Passive Smart Materials………………..………..….25

2.2.4 Applications of Piezoelectric Materials ……………..………....26

2.2.5 Smart Materials: Future Applications…………………………...27

2.3 Piezoelectricity and Piezoelectric Materials………………………...…..27

2.3.1 Constitutive Relations…………………………………………...28

2.3.2 Second Order Effects………………………….....……………...32

2.3.3 Pyroelectricity and Ferroelectricity.………………………….…33

2.3.4 Commercial Piezoelectric Materials………………………...…..33

2.4 Piezoelectric Materials as Mechatronic Impedance Transducers…….…36

(MITs) for SHM

2.4.1 Physical Principles…………………..…………………………..37

2.4.2 Method of Application…………………………..……………...42

2.4.3 Major Technological Developments During Last Nine Years.…42

2.4.4 Details of PZT Patches …………………………..…………….44

2.4.5 Selection of Frequency Range………………………….…..…...45

2.4.6 Sensing Zone of Piezo-Impedance Transducers………………...46

2.4.7 Modes of Wave Propagation………………………..…………...47

2.4.8 Effects of Temperature…………………………………...……..48

2.4.9 Effects of Noise and Other Miscellaneous Factors…....………...49

2.4.10 Thermal Stresses in Piezo-Impedance Transducers………….….50

2.4.11 Multiple Sensor Requirements…..……………………………...50

2.4.12 Signal Processing Techniques and Damage Quantification…….51

2.5 Advantages of EMI Technique………………………………………….54

2.6 Limitations of EMI Technique………………………………………….56

2.7 Needs for Further Research in EMI Technique…………………..….….57

2.7.1 Theoretical and Data Processing Considerations..………..……..57

2.7.2 Hardware/ Technology Considerations….………………..…….58

2.8 Concluding Remarks………………………………………..………......59

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CHAPTER 3: PZT-STRUCTURE ELECTRO-MECHANICAL……………..60

INTERACTION

3.1 Introduction………………………………………….……………..…60

3.2 Mechanical Impedance of Structures….……...………………………60

3.3 Mechanical Impedance of PZT Patches.……...………………………62

3.4 Electro-Mechanical Interaction in Single Degree of Freedom………..65

(SDOF) Systems

3.5 Structure-PZT Interaction in Complex Systems...……………………80

3.6 Implications of Structure-PZT Interaction……………………………84

3.7 Decomposition of Coupled Electro-Mechanical Admittance…………84

3.8 Concluding Remarks………………………………………...…….…..88

CHAPTER 4: DAMAGE ASSESSMENT OF SKELETAL STRUCTURES…89

VIA EXTRACTED MECHANICAL IMPEDANCE

4.1 Introduction………………………………………………………...…89

4.2 Analogy Between Electrical and Mechanical Systems…...……....….89

4.3 Measurement of Mechanical Impedance………………………….…..91

4.4 Decomposition of Admittance Signatures……………….. …………..92

4.5 Extraction of Structural Mechanical Impedance of Skeletal…..……...94

Structures

4.5.1 Computational Procedure………………………………………94

4.5.2 Determination of (tan κl/ κl)……………………………..…….96

4.5.3 Physical Interpretation of Drive Point Impedance……………..97

4.6 Definition of Damage Metric Based on Extracted Structural ….….....98

Impedance

4.7 Proof of Concept Application: Diagnosis of Vibration Induced..…….98

Damages

4.7.1 Flexural Damage Prediction by PZT Patch #2……..…..….…100

4.7.2 Shear Damage Prediction by PZT Patch #1……..……..…….103

4.7.3 Damage Sensitivity of the Proposed Methodology…………..104

4.8 Discussions………………………………………..………………...106

v

4.9 Concluding Remarks………………………………………...……...106

CHAPTER 5: GENERALIZED ELECTRO-MECHANICAL ……………..107

IMPEDANCE FORMULATIONS: THEORETICAL DEVELOPMENT

AND SHM APPLICATIONS

5.1 Introduction…………………...……………………………….…...107

5.2 Existing PZT-Structure Interaction Models…….……………….…107

5.3 Limitations of Existing Modelling Approaches...……………….…110

5.4 Definition of Effective Mechanical Impedance……………....….…111

5.5 Electro-Mechanical Formulations Based on Effective Impedance....113

5.6 Experimental Verification……..…….……….……………………..117

5.6.1 Details of Experimental Set-up…………………………..117

5.6.2 Determination of Structural EDP Impedance by FEM…..118

5.6.3 Modelling of Structural Damping………………………...121

5.6.4 Wavelength Analysis and Convergence Test………….…122

5.6.5 Comparison Between Theoretical and Experimental….….122

Signatures

5.7 Refining the Model of PZT Sensor-Actuator Patch………………...126

5.8 Decomposition of Coupled Electro-Mechanical Admittance…….…134

5.9 Extraction of Structural Mechanical Impedance………………….…136

5.10 System Parameter Identification from Extracted Impedance Spectra.138

5.11 Damage Diagnosis in Aerospace and Mechanical Systems………...143

5.12 Extension to Damage Diagnosis in Civil- Structural Systems……...151

5.13 Concluding Remarks………………………………………………...153

CHAPTER 6: CALIBRATION OF PIEZO-IMPEDANCE ………………….155

TRANSDUCERS FOR STRENGTH PREDICTION AND DAMAGE

ASSESSMENT OF CONCRETE

6.1 Introduction………………………………………………………..…...155

6.2 Conventional NDE Methods in Concrete……………………………...155

6.2.1 Surface Hardness Methods………………………………….156

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6.2.2 Rebound Method……………………………………………156

6.2.3 Penetration Techniques……………………………………...157

6.2.4 Pullout Test………………………………………………….157

6.2.5 Resonant Frequency Method………………………………..157

6.2.6 Ultrasonic Pulse Velocity Method…………………………..158

6.3 Concrete Strength Evaluation Using EMI Technique………………….160

6.4 Extraction of Damage Sensitive Concrete Parameters from……………164

Admittance Signatures

6.5 Monitoring Concrete Curing Using Extracted Impedance…….………..169

Parameters

6.6 Establishment of Impedance-Based Damage Model for Concrete……...173

6.6.1 Definition of Damage Variable…………………..……..……173

6.6.2 Theory of Statistics and Probability…………..……………..174

6.6.3 Theory of Fuzzy Sets……………….………………………..176

6.6.4 Statistical Analysis of Damage Variable for Concrete……….178

6.6.5 Fuzzy Probabilistic Damage Calibration of Piezo-…………..178

Impedance Transducers

6.7 Discussions………….…………………………………………………..183

6.8 Concluding Remarks…………………………………………………....185

CHAPTER 7: INCLUSION OF INTERFACIAL SHEAR LAG EFFECT…..186

IN IMPEDANCE MODELS

7.1 Introduction……………………………………………………….….186

7.2 Shear Lag Effect……………………………………………………...186

7.2.1 PZT Patch as Sensor……………………………………....188

7.2.2 PZT Patch as Actuator….…………………………………192

7.3 Integration of Shear Lag Effect into Impedance Models…………....194

7.4 Inclusion of Shear Lag Effect in 1D Impedance Model..…………....196

7.5 Extension to 2D-Effective Impedance Based Model….…………….201

7.6 Experimental Verification…………………………………………...203

vii

7.7 Parametric Study on Adhesive Layer Induced Admittance………...207

Signatures

7.7.1 Influence of Bond Layer Shear Modulus (Gs)………..…..207

7.7.2 Influence of Bond Layer Thickness (ts)…………………..209

7.7.3 Influence of Damping of Adhesive Layer (η′ )…………...210

7.7.4 Overall Influence of Parameter effp ……………………...211

7.7.5 Overall Influence of Parameter qeff……………………….212

7.7.6 Influence of Sensor Length (l)……………………………213

7.7.7 Quantification of Overall Influence of Bond Layer………214

7.8 Summary and Concluding Remarks…………………………….…..214

CHAPTER 8: PRACTICAL ISSUES RELATED TO EMI TECHNIQUE….215

8.1 Introduction……………………………………………………….….215

8.2 Evaluation of Long term Repeatability of Signatures…….………… 215

8.3 Protection of PZT Transducers Against Environment………………. 216

8.4 Multiplexing of Signals from PZT Arrays………………………...…220

8.5 Concluding Remarks…………………………………………………222

CHAPTER 9: CONCLUSIONS AND RECOMMENDATIONS………….…223

9.1 Introduction……………………………………………………….…223

9.2 Research Conclusions and Contributions………..……………….….223

9.3 Recommendations for Future Work………………………………....228

AUTHOR’S PUBLICATIONS………………………………………..….…….230

REFERENCES…………………………………………………………..…..…..234

viii

APPENDICES

Appendix A Visual Basic program to derive conductance and

susceptance plots from ANSYS output. This program is

based on 1D impedance model of Liang et al. (1994), Eq.

(2.24)

252

Appendix B Visual Basic program to derive real and imaginary

components of structural impedance from admittance

signatures. This program is based on 1D impedance model

of Liang et al. (1994), Eq. 2.24

254

Appendix C MATLAB program to derive electro-mechanical

admittance signatures from ANSYS output. The program is

based on the new 2D model based on effective impedance,

covered in Chapter 5 (Eq. 5.30).

256

Appendix D MATLAB program to derive electro-mechanical

admittance signatures from ANSYS output, using updated

PZT model (twin-peak). The program is based on the new

2D model based on effective impedance, covered in

Chapter 5 (Eq. 5.56).

258

Appendix E MATLAB program to derive structural mechanical

impedance from experimental admittance signatures, using

updated PZT model (twin-peak). The program is based on

the new 2D model based on effective impedance, covered

in Chapter 5 (Eq. 5.56).

260

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Appendix F MATLAB program to compute fuzzy failure probability. 262

Appendix G MATLAB program to derive electro-mechanical

admittance signatures from ANSYS output, taking shear

lag in the adhesive layer into account. The program is

based on the new 2D model based on effective impedance,

covered in Chapter 5 (Eq. 5.30).

263

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SUMMARY

The last few decades have witnessed construction of vast infrastructural

facilities in Singapore and other parts of the world. Now, the ageing of these structures

is creating maintenance problems and increasingly prompting the development of

automated structural health monitoring (SHM) and non-destructive evaluation (NDE)

systems, which can provide cost-effective alternative to traditional visual inspection.

Similar necessity is increasingly felt for civil and military aircraft, spaceships, heavy

machinery, trains, and so on, where long endurance combined with intensive usage

causes gradual but unnoticed deterioration, often leading to unexpected disasters, such

the as the Columbia Shuttle breakdown.

The recent advent of ‘smart’ or ‘intelligent’ materials and structures concept

and technologies has ushered a new avenue for the development SHM/ NDE systems.

Smart piezoelectric-ceramic (PZT) materials, for example, have emerged as high

frequency mechatronic impedance transducers (MITs) for SHM and NDE. As MIT, the

PZT patches are not only robust, cost-effective, and show high damage sensitivity, but

are also ideal for already constructed infrastructures and currently operational

machinery because they only require non-intrusive external installation. The piezo-

impedance transducers, acting as collocated actuators and sensors, employ ultrasonic

vibrations (typically in 30-400 kHz range) to read the characteristic ‘signature’ of the

structure, which contains vital information governing the phenomenological nature of

the structure, and can be analysed to predict the onset of structural damages. High

operational frequency ensures a sensitivity high enough to capture any damage at the

incipient stage itself, much before it acquires detectable macroscopic dimensions. This

new SHM/ NDE technique is popularly called the electro-mechanical impedance

(EMI) technique in the literature.

In spite of enormous potential due to its low-cost and high sensitivity, the EMI

technique is still in the infancy stage as far as damage severity assessment or access to

xi

the inherent damage mechanism is concerned. Changes in the diagnostic signature and

the nature, severity and type of damage are not well correlated. Till date, all the

existing approaches are non-parametric and statistical in nature and are able to utilize

only the real part of signature. The information concerning damage carried by the

imaginary part is therefore lost. Besides, no attempt has been made to extract the

mechanical impedance of the interrogated structure from the electro-mechanical

signatures, partly due to the non-existence of suitable impedance models.

This research has focused on utilizing the underlying PZT-structure electro-

mechanical interaction for an impedance based structural identification and SHM/ NDE

using the EMI technique. A new concept of active signatures has been introduced to

extract the damage-sensitive information from the raw signatures and a new PZT-

structure interaction model has been developed based on the concept of ‘effective

impedance’. The proposed formulations can be conveniently employed to extract the

hidden damage sensitive structural parameters for any ‘unknown’ structure by means of

surface-bonded PZT patches. A new experimental technique has been developed to

‘update’ the model of the PZT patch, so as to enable it extract the host structure’s

impedance information much more accurately. A unified impedance approach has been

developed to ‘identify’ the host structure from the extracted mechanical impedance

spectra and carry out quantitative and parametric damage prediction. This has made

possible greater information about the nature of damage in terms of stiffness, damping

and mass changes, which was so far lacking. As proof-of-concept, the new diagnostic

approach has been applied on representative aerospace and civil structural components.

Further, in order to rigorously calibrate the piezo-impedance transducers for

damage assessment, comprehensive tests were carried out on concrete specimens. An

empirical fuzzy probabilistic damage model has been proposed for predicting damage

level in concrete using piezo-impedance transducers. In addition, a new experimental

technique has been developed to predict in situ concrete strength non-destructively

using the EMI technique, thereby imparting it further edge over the contemporary NDE

techniques. Finally, the intermediate bond layer between the PZT patch and the

xii

structure has been integrated into the impedance models, thereby enabling a rigorous

analysis of the shear lag effect associated with the bond layer.

It is hoped that this research will make significant contributions in the field of

SHM and NDE and will enable the maintenance engineers to make much more timely

and accurate prediction of damages in any structural component.

xiii

LIST OF TABLES

Page

Table 2.1 Sensitivities of common local NDE techniques 21

(Boller, 2003).

Table 3.1 Key parameters of PZT patch . 66

Table 3.2 Key material properties of structure. 82

Table 4.1 Key properties of PZT patches (PI Ceramic, 2003). 100

Table 4.2 Typical base motions and time-histories to which test 101

frame was subjected.

Table 5.1 Physical properties of Al 6061-T6. 117

Table 5.2 Details of modes of vibration of test structure. 123

Table 5.3 Mechanical impedance of combinations of spring, mass 139

and damper.

Table 6.1 Averaged parameters of test sample of PZT patches. 165

Table 6.2 Common probability distributions. 175

xiv

LIST OF FIGURES

Page

Fig. 1.1 Accident involving Aloha Airlines (LAMSS, 2003)….………….…2

Fig. 1.2 Accident involving American Airlines Airbus A300-600 ………….2

(LAMSS, 2003).

Fig. 1.3 Image of Columbia about a minute before it broke apart……..……3

(AWST, 2003).

Fig. 1.4 Shuttle left wing cutaway diagrams (NASA, 2003)………….……..4

Fig. 1.5 Damage identified on RCC panel 8 in Discovery after a mission…..5

in 2000 (CAIB, 2003).

Fig. 1.6 The Mianus river bridge collapse (USDT, 2003)…………………...6

Fig. 1.7 Illustrating the components and operation of typical SHM system…7

(Boller, 2002).

Fig. 2.1 Classification of smart structures (Rogers, 1990)…..…………..….24

Fig. 2.2 Common smart materials and associated stimulus-response………25

Fig. 2.3 Centro-symmetric crystals: the act of stretching does not cause…..28

any dipole moment (µ = dipole moment).

Fig. 2.4 Noncentro-symmetric crystals: the act of stretching causes dipole..28

moment in the crystal (µ = dipole moment)

Fig. 2.5 A piezoelectric material sheet with conventional 1, 2 and 3 axes...30

Fig. 2.6 Strain vs electric field for PZT (piezoelectric) and………….……..32

PMN (electrostrictive).

Fig. 2.7 Polarization vs electric field for ferroelectric crystals………...…...33

Fig. 2.8 Modelling PZT-structure interaction…………………………...….37

Fig. 2.9 Conductance and susceptance plots of a PZT patch bonded to…....41

bottom flange of a steel beam.

Fig. 2.10 A typical commercially available PZT patch…………….………..45

xv

Fig. 2.11 Modes of wave propagation associated with PZT patch…………..48

(Giurgiutiu and Rogers, 1997).

Fig. 3.1 Representation of harmonic force and velocity by rotating phasors.61

Fig. 3.2 Determination of mechanical impedance of a PZT patch...…..……62

Fig. 3.3 Variation of actuator impedance with frequency………....…..……65

Fig. 3.4 A PZT patch coupled to a spring-mass-damper system...………….66

Fig. 3.5 Signatures for SDOF-Case I, m = 2.0 kg, k = 1.974x107N/m,

c = 125.7Ns/m……………………….…………………………..…68

Fig. 3.6 Signatures for SDOF-CaseII, m = 200 kg, k = 1.974x109N/m,

c = 12566.4Ns/m…………………….……………………………..71

Fig. 3.7 Signatures for SDOF-CaseIII, m = 0.2 kg, k = 1.974x106N/m,

c = 12.57Ns/m……………………….………………………..……73

Fig. 3.8 Signatures for SDOF-CaseIV, m = 2500 kg, k = 2.46x1010N/m,

c = 3927Ns/m..……………………..…………………………...….74

Fig. 3.9 Signatures for caseV..………..…….…………………………...….76

Fig. 3.10 Signatures for SDOF-caseVI, m = 0.0002 kg, k = 197.4N/m,

c = 0.01257Ns/m…………………….………………………….….78

Fig. 3.11 Appearance of large number of ‘false’ peaks.……………………...79

Fig. 3.12 A MDOF system considered for PZT-structure interaction………..81

Fig. 3.13 Graphical representation of Mode 48 (f = 162.46 kHz)…..………..82

Fig. 3.14 Signatures for MDOF system considered in Fig. 3.12…..……….…83

Fig. 3.15 Active-conductance and active-susceptance (modified…………….87

signatures after filtering out the PZT contribution).

Fig. 3.16 Active-susceptance plot for Case-II………………………………..87

Fig. 4.1 (a) A SDOF system under dynamic excitation……………….....….90

(b) Phasor representation of spring force (Fs), damping force (Fd)

and inertial force (Fi)

Fig. 4.2 (a) Details of test frame………………………………..……………99

(b) Test frame just before applying loads

Fig. 4.3 Raw-signatures of PZT patch #2 at various damage states (1,2..6).102

Fig. 4.4 Damage prediction by patch #2…………………………….….….102

xvi

Fig. 4.5 Raw-signatures of PZT patch #1 at various damage states (1,2..8).104

Fig. 4.6 Damage prediction by patch #1…………………………….….….104

Fig. 4.7 (a) Natural frequency of vibration of floor #2 beam at various…...105

damage states.

(b) Evaluation of damage based on natural frequency, raw-

conductance and extracted mechanical impedance.

Fig. 5.1 Modelling of PZT-structure interaction by static approach………108

Fig. 5.2 Modelling PZT-structure 2D physical coupling by …………….109

impedance approach (Zhou et al., 1996).

Fig. 5.3 A PZT patch bonded to an ‘unknown’ host structure…………….112

Fig. 5.4 A square PZT patch under 2D interaction with host structure…...113

Fig. 5.5 Experimental set-up to verify effective impedance based new….117

electro-mechanical formulations.

Fig. 5.6 Finite element model of one-quarter of test structure……………119

Fig. 5.7 Examination of mode 24 to check adequacy of mesh size ………124

of 1mm.

Fig. 5.8 Comparison between experimental and theoretical signatures…..125

Fig. 5.9 Plots of quasi-static admittance functions of free PZT patches….127

to obtain electric permittivity and dielectric loss factor.

Fig. 5.10 Experimental and analytical plots of free PZT signatures…….…129

Fig. 5.11 Plots of free-PZT admittance signatures using an updated………131

PZT model.

Fig. 5.12 Comparison between experimental and theoretical signatures…..133

based on updated PZT model.

Fig. 5.13 (a) PZT effective impedance, based on idealised and updated…. 134

Models.

(b) Error in extracted structural impedance in the absence of

updated PZT model.

(c) Relative magnitudes of structure and PZT impedances.

Fig. 5.14 Comparison between |Zeff|-1 obtained experimentally and………..137

numerically.

xvii

Fig. 5.15 Impedance plots of basic structural elements- spring, damper……138

and mass.

Fig. 5.16 Mechanical impedance of aluminium block in 25-40 kHz………..140

frequency range.

Fig. 5.17 Mechanical impedance of aluminium block in 180-200 kHz…….142

frequency range. The equivalent system plots are obtained for

system 11 (Table 5.3).

Fig. 5.18 Refinement of equivalent system by introduction of………….….142

additional spring K* and additional damper C*.

Fig. 5.19 Mechanical impedance of aluminium block in 180-200 kHz…….143

frequency range for refined equivalent system ( shown in Fig. 5.18).

Fig. 5.20 Levels of damage induced on test specimen (aluminium block)…144

Fig. 5.21 Effect of damage on extracted mechanical impedance…………...145

in 25-40 kHz range.

Fig. 5.22 Effect of damage on equivalent system parameters………………145

in 25-40kHz range

Fig. 5.23 Effect of damage on extracted mechanical impedance…………...147

in 180-200kHz range.

Fig. 5.24 Plot of mechanical impedance of aluminium block in 180-200….148

for various damage states.

Fig. 5.25 Effect of damage on equivalent system parameters………………149

in 180-200kHz range

Fig. 5.26 Plot of residual specimen area versus equivalent spring constant..150

Fig. 5.27 Damage diagnosis of a prototype RC bridge using proposed…….151

methodology.

Fig. 5.28 Mechanical impedance of RC bridge in 120-140 kHz frequency..152

range. The equivalent system plots are obtained for a parallel

spring damper combination.

Fig. 5.29 Effect of damage on equivalent system parameters of RC bridge..153

Fig. 6.1 (a) Determining natural frequency of specimen using sonometer..158

xviii

(b) Correlation between dynamic modulus and concrete strength.

(Source: Malhotra, 1976)

Fig. 6.2 (a) Determining velocity of sound in concrete using PUNDIT…...159

(b) Correlation between ultrasonic pulse velocity and strength.

(Source: Malhotra, 1976)

Fig. 6.3 Admittance spectra for free and fully clamped PZT patches…….160

Fig. 6.4 (a) Optical fibre pieces laid on concrete surface before applying..161

adhesive.

(b) Bonded PZT patch.

Fig. 6.5 Effect of concrete strength on first resonant frequency of PZT….162

patch.

Fig. 6.6 Correlation between concrete strength and first resonant………..163

frequency.

Fig. 6.7 Concrete cube to be ‘identified’ by piezo-impedance……………164

transducer.

Fig. 6.8 Equivalent system ‘identified’ by PZT patch………………….…165

Fig. 6.9 Impedance plots for concrete cube C43.…………..…….…….….166

Fig. 6.10 Experimental set-up for inducing damage on concrete cubes……167

Fig. 6.11 Load histories of four concrete cubes…..…………..………….…167

Fig. 6.12 Correlation between loss in secant modulus and loss in ..…….…168

equivalent spring stiffness with damage progression.

Fig. 6.13 Changes in equivalent damping and equivalent stiffness for….…169

concrete cube C43.

Fig. 6.14 Monitoring concrete curing using EMI technique………….……170

Fig. 6.15 Short-term effect of concrete curing on conductance signatures...171

Fig. 6.16 Long-term effect of concrete curing on conductance signatures...171

Fig. 6.17 Effect of concrete curing on equivalent spring stiffness……..….172

Fig. 6.18 Different types of membership functions for fuzzy sets…..…….177

Fig. 6.19 Effect of damage on equivalent spring stiffness………….……..179

Fig. 6.20 Theoretical and empirical probability density functions….…….180

near failure.

xix

Fig. 6.21 Fuzzy failure probabilities of concrete cubes at incipient………182

damage level and at failure stage

Fig. 6.22 Fuzzy failure probabilities of concrete cubes at various..………182

load levels.

Fig. 6.23 Typical stress-strain plot for PZT (Cheng and Reece, 2001)..….183

Fig. 6.24 Cubes after the test……………………………………..……….184

Fig. 7.1 A PZT patch bonded to a beam using adhesive bond layer….….186

Fig. 7.2 Deformation in bonding layer and PZT patch…..…………..….187

Fig. 7.3 Strain distribution across the length of PZT patch……………..190

for various values of Γ.

Fig. 7.4 Variation of effective length with shear lag factor…………..…191

Fig. 7.5 Distribution of piezoelectric and beam strains for various……..193

values of Γ.

Fig. 7.6 Modified impedance model of Xu and Liu (2002) including ….194

bond layer.

Fig. 7.7 Stresses acting on an infinitesimal PZT element…………...….201

Fig. 7.8 Theoretical normalized conductance…………...…………..….204

Fig. 7.9 Experimental normalized conductance for ts/tp = 0.417……….205

and ts/tp = 0.838.

Fig. 7.10 Theoretical normalized susceptance……………..………….….205

Fig. 7.11 Experimental normalized susceptance for ts/tp =0.417………....205

and ts/tp = 0.838.

Fig. 7.12 Analytical and experimental plots for ts/tp equal to 1.5……….. 206

Fig. 7.13 Influence of shear modulus of elasticity of bond layer..……….208

Fig. 7.14 Influence of bond layer thickness……………….…….………..209

Fig. 7.15 Influence of damping of bond layer…………………………….210

Fig. 7.16 Influence of Parameter effp ……….………………..…....……..211

Fig. 7.17 Influence of Parameter qeff…………………..………………………...212

Fig. 7.18 Influence of sensor length……….………………………….......213

Fig. 8.1 Test specimen for evaluating repeatability of ………………….217

xx

admittance signatures.

Fig. 8.2 A set of conductance signatures of PZT patch #1spanning over…217

two months.

Fig. 8.3 A set of susceptance signatures of PZT patch #1spanning over….217

two months.

Fig. 8.4 Effect of humidity on signature………………………………..…219

Fig. 8.5 Effect of damage on signatures………………………………..…219

Fig. 8.6 Test specimen for evaluating signature multiplexing………....….220

Fig. 8.7 Experimental set-up consisting of impedance analyzer,….. ....….221

controller PC and multiplexer.

Fig. 8.8 Effect of damage on collective signature of 20 PZT patches….…222

xxi

LIST OF SYMBOLS

A Area

B Raw susceptance

BA, Active susceptance

BP Passive susceptance

C, C1, C2 Correction factor(s) to update model of PZT

c Damping constant

[C] Damping matrix

D1, D2, D3 Electric displacement across surfaces normal to 1, 2, 3 axes

respectively

[D] Electric displacement vector

d31 (dik) Piezoelectric strain coefficient of PZT patch corresponding to

axes 3(i) and 1(k)

Di Damage variable at ith frequency point

Dc Critical value of damage variable

DU, DL Upper and lower limits of damage variable in the fuzzy interval

E3 (Ei) Electric field along axis 3 (i) of PZT patch

[E] Electric field vector

f Frequency

f Boundary traction (per unit length)

F (Effective) Force

fm Membership function of a fuzzy set

F̂ Empirical cumulative distribution function

G Raw conductance

GA Active conductance

GP Passive conductance

xxii

Gs Shear modulus of elasticity of bond layer

h Thickness of PZT patch

I Complex Electric current

j 1−

k Spring constant

[K] Stiffness matrix

l Half-length of PZT patch

m Mass

M Bending moment

Mmn Electrostriction coefficient

[M] Mass matrix

po Perimeter of PZT patch in undeformed condition

p(D) Probability density function of damage variable D

S1 (Si) Mechanical strain along axis 1 (i)Ekms An element of the elastic compliance matrix at constant electric field

T1 (Ti) Mechanical stress along axis 1 (i) of PZT patch

T Complex tangent function

tp Thickness of PZT patch

ts Thickness of bond layer

u Displacement

V Complex electric voltage

wp Width of PZT patch

x Real part of the mechanical impedance of structure

xa Real part of the mechanical impedance of PZT patch

y Imaginary part of the mechanical impedance of structure

ya Imaginary part of the mechanical impedance of PZT patch

Y Complex electro-mechanical admittance

EY Complex Young’s modulus of elasticity at constant electric field

AY Active component of complex admittance

xxiii

PY Passive components of complex admittance

Z Complex mechanical impedance of structure (Z = x + yj)

Za Complex Mechanical impedance of PZT patch (Za = xa + yaj)T33ε Complex permitivity of PZT patch along axis 3 at constant stress

ω Angular frequency (rad/s)

δ Dielectric loss factor

ρ Material density

η Mechanical loss factor of PZT patch

η′ Mechanical loss factor of adhesive

τ Shear stress

α Mass damping factor

β Stiffness damping factor

φ Phase lag

Γ Shear lag parameter

ξ Strain lag ratio

ξd Damping ratio

ν Poisson’s ratio

Λ Free piezoelectric strain (= E3d31)

ψ Product of beam to PZT modulus and thickness ratios

Structural mechanical impedance correction factor

κ Wave number

γ Shear strain

µ Mean

σ Standard deviation

φ Phase lag {=tan-1(y/x)}oiG , 1

iG Pre-damage and post-damage raw conductance respectively for ith

frequency pointoG , 1G Mean value of pre-damage and post-damage raw conductance

xxiv

Subscripts

A Active

eff Effective

o Amplitude of a quantity

eq Equivalent; Equilibrium

f, free Free

i Imaginary

P Passive

p Relevant to PZT patch

qs Quasi-static

r Real

res Resultant

s Under static conditions

1,2,3 or x,y,z Coordinate axes

Superscripts

T Quantity at constant stress

E Quantity at constant electric field

xxv

LIST OF ACRONYMS

ACS Active Conductance Signature

ASS Active Susceptance Signature

ASTM American Society for Testing and Materials

ATM Adaptive Template Matching

AWST Aviation Week and Space Technology

CC Correlation Coefficient

CAIB Columbia Accident Investigation Board

EC Eddy Currents

EDP Effective Drive Point

ELODS Equivalent Level of Degradation System

EMI Electro-Mechanical Impedance

ER Electro-Rheological (Fluid)

FFP Fuzzy Failure Probability

FEM Finite Element Method

IDT Inter Digital Transducers

LAMSS Laboratory for Active Materials and Smart

Structures

LCR Inductance (L) Capacitance (C) and Resistor (R)

(Circuit)

MAPD Mean Absolute Percent Deviation

MDOF Multiple Degree of Freedom (System)

MEMS Micro-Electro Mechanical Systems

MIT Mechatronic Impedance Transducer

NASA National Astronautics and Space Administration

NDE Non-Destructive Evaluation

xxvi

NDT Non-Destructive Testing

PC Personal Computer

PCS Passive Conductance Signature

PSS Passive Susceptance Signature

PUNDIT Portable Ultrasonic Non-Destructive Digital

Indicating Tester

PVDF Polyvinvylidene Fluoride

PZT Lead (Pb) Zirconate Titanate

RC Reinforced Concrete

RCC Reinforced Carbon Carbon

RCS Raw Conductance Signature

RD Relative Deviation

RMS Root Mean Square

RMSD Root Mean Square Deviation

SAC Signature Assurance Criteria

RSS Raw Susceptance Signature

SDOF Single Degree of Freedom (System)

SHM Structural Health Monitoring

SMA Shape Memory Alloy

USDT United States Department of Transport

UTM Universal Testing Machine

WCC Waveform Chain Code (Technique)

Chapter 1: Introduction

1

Chapter 1

INTRODUCTION

1.1 STRUCTURAL DAMAGES AND FAILURES

Structures are assemblies of load carrying members capable of safely

transferring the superimposed loads to the foundations. They are constructed (e.g.

buildings, bridges, dams, transmission towers, etc.) or manufactured (e.g. machines,

trains, ships, aircraft, etc.) to serve specific functions during their design lives. Each

structure forms an integral component of civil, mechanical or aerospace systems. In

order to serve their designated functions, the structures must satisfy both strength

and serviceability criteria throughout their stipulated design lives. However, with

the passage of time, some amount of deterioration and damages are bound to occur,

due to a variety of factors; such as environmental degradation, fatigue, excessive

loads, natural calamities or simply due to long endurance combined with intensive

usage. Even the best designed structures, constructed from advanced high strength

materials, are not 100% immune from damage.

According to Yao (1985), ‘damage’ is defined as a deficiency or deterioration

in the strength of a structure, caused by external loads, environmental conditions,

or human errors. Physically, a damage may be visible as a crack, delamination,

debonding, reduction in thickness/ cross-section, or exfoliation. The term ‘damage’

carries much different meaning from the term ‘failure’. In most general terms,

‘failure’ refers to any action leading to an inability on the part of a structure or

machine to function in the intended manner (Ugural and Fenster, 1995). Fracture,

permanent deformation, buckling and even excessive linear elastic deformation may

be regarded as modes of failure. Failure results when a particular type of damage

exceeds its threshold value, thereby impairing the safety and/ or the functioning of

the structure seriously.


2

1.2 AN OVERVIEW OF RECENT STRUCTURAL FAILURES

On April 28, 1988, Boeing 737 of Aloha Airlines met with a severe mid-flight

accident in which entire fuselage panels were ripped apart from the main body, as

shown in Fig. 1.1. Fortunately, the passengers remained held against air pressure by

their safety belts. The underlying cause of this accident was later found to be the

appearance of multi-site cracks in the skin joints, which led to the unzipping of

large portions of the fuselage (LAMSS, 2003). However, these cracks could not be

detected during the routine pre-flight inspections.

Similarly, on November 12, 2001, the mid air crashing of the American

Airlines Airbus A 300-600 (Flight 587) was one of the deadliest accidents in the

American aviation history. From preliminary investigations, it was found that the

tail (vertical stabilizer) broke off during take off, right from the root of the

connection to the main body, as shown in Fig. 1.2(a). The investigators found the

Fig. 1.1 Accident involving Aloha Airlines (LAMSS, 2003).

Fig. 1.2 Accident involving American Airlines Airbus A300-600 (LAMSS, 2003).

(a) Breakaway tail component. (b) Close-up view of breakaway composite joint.


3

existence of an undetected damage in the tail, caused by previous mid air events

involving severe loading, which had resulted in the weakening of the composite

joint. Surprisingly, the conventional NDE techniques, including visual inspections,

had failed to detect the presence of the previous damages. This incipient damage

was further aggravated by the aerodynamic loads and the tail finally broke apart, as

shown in Fig. 1.2(b).

Another recent aerospace disaster, which attracted worldwide attention, was

the crashing of the NASA space shuttle Columbia, on February 1, 2003, during its

re-entry into earth’s atmosphere. Fig. 1.3 shows the US Air Force image of

Columbia taken about a minute before it broke apart (AWST, 2003). This image

shows that the left inboard wing was jagged near the location where it begins to

intersect the fuselage. This location houses reinforced carbon-carbon (RCC)

composites, which constitute critical structural and thermal protection components

of any shuttle. The right wing, on the other hand, can be seen to be smooth along its

entire length. The ragged edge on the left leading wing indicated that either a

structural breach occurred there, or that a small portion of the leading edge fell off,

allowing the 2000oF re-entry heat to erode the additional structure there.

Comprehensive investigation into the disaster was carried out by Columbia

Accident Investigation Board (CAIB, 2003) and the findings were made public on

August 26, 2003. The CAIB report confirmed that the physical cause of the loss of

Columbia was a breach in its thermal protection system on the leading edge of the

left wing. This breach was initiated by a piece of insulating foam, separated from

the left bipod ramp, that struck the left wing in the vicinity of the lower half of RCC

Fig. 1.3 Image of Columbia about a minute before it broke apart. (AWST, 2003).

Right wing

Left wing

Wing distortion

Flow distortion


4

panel 8, 81.9 seconds after the launch (Chapter 3, page 49 of CAIB report). As

shown in Fig. 1.4, each wing’s leading edge consists of 22 RCC panels. RCC is a

hard structural material characterized by high strength over extreme temperatures

ranging from –250oF to 3000oF. During re-entry, this breach allowed the

superheated air to penetrate into and melt the aluminium structure (melting point:

1200oF) of the left wing, thereby weakening it, until the aerodynamic forces caused

failure of the wing and total break-up of the orbiter.

Ironically, although the event of foam striking the left wing had caught the

attention of the ground team, the space shuttle was not equipped with any NDE

system on-board to assess the level of damage caused. Conclusions of ground team

based on computational analysis that the impact was not so severe proved wrong.

Following additional findings of CAIB are worth taking note of:

(i) The RCC is vulnerable to damage due to oxidation if oxygen penetrates the

microscopic fissures of the silicon-carbide protective coating. The loss of

mass due to oxidation reduces the load capacity of the structure. Currently,

the mass loss cannot be directly measured (Finding F 3.3-4, page 58 CAIB

report). This weakening can eventually lead to significant deterioration, for

example, as shown in Fig. 1.5 for panel 8 of space shuttle Discovery, after a

mission in January 2000.

Fig. 1.4 Shuttle left wing cutaway diagrams (NASA, 2003).

(a) Complete view of spaceship Columbia. (b) Left-wing showing RCC panels.

(a) (b)

(1-10) (16-17)

RCC Panels (1-10 and 16-17)


5

(ii) During manufacturing, the integrity of production composites used in the

RCC system is checked by physical tap, ultrasonic, radiographic, eddy

currents, visual tests and also by limited number of destructive tests.

However, no rigorous test plan is followed after assembly in the shuttle. Post

flight inspection is primarily visual and tactile (poking with finger). The

board noted that the current inspection techniques are not adequate to assess

structural integrity of RCC, the supporting structure and the attached

hardware. (Findings F 3.2-2 and F 3.2-3, page 58 of CAIB report).

(iii) There are no qualified NDE techniques to determine the characteristics of

the foam in the as-installed condition before flight (finding F 3.2-2, page 55

of CAIB report).

In view of the above findings, the CAIB recommended NASA to develop

and implement a comprehensive inspection plan to determine the structural integrity

of all RCC system components, taking advantage of the “advanced NDE

technology” (recommendation R 3.3.1, page 59).

Military aircraft also suffer similar mid flight accidents due to damages. In

the past 10 years, the Indian Air force has lost more than 100 MiG fighter aircraft

with over 80 pilots dead. This amounts to billions of dollars worth of equipment and

human resources. During the past 3 years alone, 52 such fighter planes have been

lost (based on Defence Minister’s statement in parliament on 25 July 2003). No

Fig. 1.5 Damage identified on RCC panel 8 in Discovery after a mission in 2000

(CAIB, 2003).

Damage


6

sophisticated SHM system is presently in place to monitor the planes during flight

and prevent loss of the aircraft and the pilot.

Besides the above aerospace failures, numerous instances of civil-structural

failures have occurred. Many buildings and bridges constructed during the

economical boom of the eighties are now showing the problems of ageing, for

which the maintenance engineers are not logistically prepared. The Mianus river

bridge collapse (see Fig. 1.6), in Greenwich, during June 1983, resulted from a

hangar pin connection failure due to excessive corrosion accumulation (USDT,

2003). This failure emphasized that special inspection techniques are necessary for

civil-structures also, since visual inspection is likely to miss out many critical

incipient damages. There are a total of 127,154 railway bridges in India, taking

freight traffic of over 550 million tonnes and passenger load of more than 500

billion passenger kilometers every year. Out of these bridges, 56,169 (44.17%) are

more than 80 years old and hence prone to disaster any time (Hindustan Times, 20

July 2003). Hence, a rigorous inspection and test plan is necessary to ensure

passenger safety and prevent unexpected losses.

1.3 STRUCTURAL HEALTH MONITORING

The brief overview of recent catastrophic accidents in the preceding section has

clearly shown the destructive power of any structural damage when it starts to grow

from the incipient level. Hence, even a minor damage of incipient nature should not

Fig. 1.6 The Mianus River Bridge collapse (USDT, 2003).


7

be ignored since it carries the potential to grow and cause failure, either leading to

wide scale loss of life and property or halting some revenue earning activity or both.

It is this possibility which calls for a rigorous inspection of the structures on a

regular basis or in other words, structural health monitoring (SHM).

SHM is defined as the acquisition, validation and analysis of technical data to

facilitate life cycle management decisions. SHM denotes a reliable system with the

ability to detect and interpret adverse ‘changes’ in a structure due to damage or

normal operations (Kessler et al., 2002). The idea of SHM is pictorially illustrated

in Fig. 1.7 (Boller, 2002). Such a system typically consists of sensors, actuators,

amplifiers and signal conditioning circuits. While sensors are employed to predict

damage, the actuators serve to excite the structure or decelerate/ arrest the damage.

1.4 REQUIREMENTS FOR ANY SHM SYSTEM

In the aviation sector, the aircraft are designed for specific number of flight

hours based on a specified usage under predefined load spectrum. However, often

the airline or the air force continues to fly the aircraft much beyond their initial

design life. Presently, the average age of the US air force fleet stands at 22 years. It

is expected to increase to 25 years in 2007 and 30 years in 2020 (Boller, 2002).

Since the US Air Force cannot boost its purchases by 170 aircraft per year, this

problem is expected to be more severe in the long run. The same holds true for the

Fig. 1.7 Illustrating the components and operation of typical

SHM system (Boller, 2002).

SignalAnalyzerFilterAmplifierSensorActuator

StructureImpator

Signal Generator


8

civil aircraft as well. In general, aircraft demand large amount of inspection at well

defined intervals, ranging from daily checks to over 120 months, especially when

they are highly loaded or when they reach older days.

Till date, visual inspection supplemented with magnifying glass, tap test and

some primitive non-destructive tests (dye-penetrant, magnetic particle etc.) has been

the most prevalent method of pre-emptive structural inspections for the aircraft.

Usually, trained personnel conduct these inspections, and the procedure is not only

very tedious and time consuming, but also characterised by high implementation

costs. It is estimated that about 27% of an aircraft’s life cycle cost is spent on

inspections and repair, excluding the opportunity cost associated with the time it

remains grounded (Kessler et al., 2002). This is not due to a large effort in detecting

damage via non-destructive testing (NDT) equipment, but owing to the fact that

many critical components such as the main lading gear fitting need to be dismantled

before inspections and reassembled afterwards. Rather, this process (dismantling

and reassembling) eats up to 45% of the entire inspection time (Boller, 2002).

Hence, an unobtrusive automated inspection mechanism to detect the onset of

damages in such inaccessible components can significantly enhance flight safety

besides reducing the operating costs.

Similarly, in civil-structures, often the critical parts are not be readily

accessible and demand removal of the existing finishes (such as false ceilings),

which makes the inspection process extremely laborious as well as costly. Most of

the existing non-destructive evaluation (NDE) techniques (such as ultrasonic,

penetrant dye testing, acoustic emission etc.) demand physically moving a probe,

which proves impractical for the large-sized civil structures. These considerations

call for a means of SHM that should avoid the dismantling and reassembling

process or removal of the finishes and should also avoid physically moving heavy

equipment. Such a system can achieve a significant reduction in the inspection

time, effort and cost.

The need to develop this kind of SHM system has recently attracted a large

number of academic and industrial researchers from various disciplines. The

ultimate goal of all SHM related research is to enable systems and structures

monitor their own integrity while in operation and throughout their design lives.


9

Such system should preferably be real time and online. By real-time, it is implied

that the level of responsiveness of such a system should be immediate or quick

enough to enable appropriate remedial action or evacuation. By on-line, it is implied

that the alerting system must use user friendly on-screen imaging and audible

alarms.

The application areas for SHM techniques are aerospace systems, mechanical

and chemical pressure vessels, nuclear power plants, dams, bridges and buildings.

In general, adoption of automated SHM is highly justified in the case of

components for which the loads are less predictable and maintenance is restricted

and costly. It may be unwarranted for low-cost components or if the loading and

component’s behaviour are well understood and do not show significant variation.

Although SHM has been shown feasible by numerous researchers, it has still

not developed to the stage of being generally recognized as an element of the

overall engineering system. The main reasons for this, according to Boller (2002)

are:

(i) Benefits resulting from such system have not been carefully quantified.

(ii) This is still not statutory requirement.

(iii) Validation and certification needs to be done on a broader basis.

(iv) Rapid emergence of new technologies and obsolescence of the old ones,

leading to confusion in general.

In general, SHM can enable taking greater advantage of structural material

potential, thereby saving natural as well as financial resources.

1.5 SHM BY ELECTRO-MECHANICAL IMPEDANCE (EMI) TECHNIQUE

The recent developments in the area of smart materials and systems have

ushered new openings for SHM and NDE. Smart materials, such as the

piezoceramics, the shape memory alloys and the fibre-optic materials can facilitate

the development of non-obtrusive miniaturized systems with higher resolution,

faster response and far greater reliability than the conventional NDE techniques.

Especially, the so-called ‘active’ smart materials possess immense capabilities of

damage diagnosis because of their inherent stimulus-response and energy


10

transduction capabilities. These materials can be easily embedded or bonded

unobtrusively on locations inaccessible for physical inspection. Hence, they meet

the requirements outlined in the previous section for any viable SHM system.

Among the so many smart materials available today, the piezoelectric-ceramic

(PZT) materials have emerged as high frequency mechatronic impedance

transducers (MITs) for SHM during the last nine years (Sun et al., 1995; Ayres et

al., 1998; Soh et al., 2000; Park, 2000; Bhalla, 2001). In this application, a PZT

patch is bonded to the structure to be monitored and its electro-mechanical

conductance signature across a high frequency band serves as a diagnostic-signature

of the structure. The technique is popularly called as the electro-mechanical

impedance (EMI) technique. The EMI technique has been shown to be extremely

sensitive to incipient damages, is practically immune to mechanical noise and

demands a low implementation cost (Park et al., 2000a). The PZT patches can be

easily bonded to inaccessible locations of structures and aircraft and can be

interrogated as and when required, without necessitating the structures to be placed

out of service or any dismantling/ re-assembling of the critical components. All

these features definitely give an edge to the EMI technique over other existing

passive sensor systems.

However, the EMI technique is presently in the developmental stage as far as

understanding the underlying damage mechanism or quantitative damage prediction

are concerned. The changes in the electro-mechanical signatures are not well

correlated with the changes in the underlying structural parameters. Till date, all the

methods utilize raw signatures alone and make use of statistical indicators to

quantify damage, which is rather a crude way of analysis. Hence, no structural

parameter based damage quantification and damage severity prediction approach is

presently available.

This research was carried out with the objective of upgrading the EMI technique

from its present state-of-the-art and expanding its NDE capabilities. The following

sections highlight the objectives and contributions to the EMI technique by this

research.


11

1.6 RESEARCH OBJECTIVES

The primary objective of this research was to investigate and suitably model the

key electro-mechanical interaction between the PZT transducer, the intermediate

bonding layer and the host structure in PZT-based smart systems. This was pursued

to enable an impedance-based structural identification and extraction of damage

sensitive structural parameters for any ‘unknown’ system from the interrogation of

the bonded PZT patch alone, without warranting any information a priori. These

parameters are expected to govern the phenomenological nature and behavior of the

structure. Hence, this process is expected to enable a more rigorous and quantitative

evaluation of structural damages, besides providing a greater insight into the

underlying damage mechanism. Further, this research aimed at rigorously

calibrating the impedance parameters with damage and extending the technique for

more meaningful applications such as in situ material strength assessment.

1.7 RESEARCH ORIGINALITY AND CONTRIBUTIONS

This research programme aimed to expand the present capabilities of the

EMI technique for experimental structural identification as well as NDE/ SHM.

This research has attempted to balance theoretical developments with practical

applications in order to maximize the potential benefits of the EMI technique. The

original contributions of this research can be summarized as follows.

(i) A new concept of active-signature has been introduced to facilitate the

extraction of damage sensitive signature component using signature

decomposition.

(ii) A new PZT-structure interaction model has been developed based on the

concept of ‘effective impedance’. The new impedance formulations can be

conveniently employed to extract the 2D mechanical impedance of any

‘unknown’ structure from the admittance signatures of a surface-bonded

PZT patch. The hidden structural parameters governing the

phenomenological nature of the structure can thus be identified by this

process.


12

(iii) A new experimental technique has been developed to ‘update’ the model of

the PZT patch to enable it extract the impedance information of the host

structure much more accurately. The new impedance formulations are

employed in conjunction with the ‘updated’ PZT model to ‘identify’ the host

structure and to carry out a parametric damage assessment, thereby

revealing more information about the associated damage mechanism. Many

proof-of-concept applications of the proposed methodology, ranging from

precision machine and aerospace components to civil-structures, are

presented.

(iv) An empirical fuzzy probabilistic damage model has been proposed to

calibrate the identified damage-sensitive structural parameters with damage

progression for concrete. Besides, a new experimental technique has been

developed to predict in situ concrete strength non-destructively.

(v) Inclusion and rigorous analysis of the adhesive bond layer (between the PZT

and the host structure) into impedance formulations and its implications on

the accuracy of structural identification have been rigorously dealt with.

(vi) Practical issues in the widespread application of the EMI technique, such as

signature repeatability, sensor protection and sensor multiplexing have been

duly addressed.

The findings of the present research work have been published in many

international refereed journals and conferences, as detailed on page 230.

1.8 THESIS ORGANISATION

This thesis consists of a total of nine chapters including this introductory

chapter. Chapter 2 presents a detailed review of state-of-the art in SHM,

introduction to the concept of smart systems and materials, description of the EMI

technique and the current challenges facing the effective implementation of the

technique on real-life structures. Chapter 3 deals with the important issues of

structure-transducer electro-mechanical interaction, which is key to effective

implementation of the technique for structural identification as well as NDE/ SHM.

It also provides a rigorous mathematical analysis of the coupling between the PZT


13

patch and the host structure and motivations for signature decomposition.

Significant deductions are made from this interaction and utilized in the subsequent

chapters. Chapter 4 presents a mathematical analysis to extract the real and

imaginary parts of the structural impedance of skeletal structures from the measured

admittance signatures. Based on these parameters, a new methodology is developed

for parametric quantification of the damage. Proof-of-concept application of the

methodology on a model RC frame is presented. Chapter 5 presents the theoretical

derivation, experimental verification and NDE applications of new generalized

impedance formulations based on the concept of ‘effective impedance’. Chapter 6

presents the results from comprehensive tests conducted on concrete cubes to

calibrate the extracted structural parameters with damage severity. Chapter 7 deals

with modelling the behaviour of interfacial bond layer and its implications on the

admittance signatures. Chapter 8 deals with key practical issues governing the

application of the EMI technique. Finally, conclusions and recommendations are

presented in Chapter 9, which is followed by a list of author’s publications, a

comprehensive list of references, and appendices.

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

14

Chapter 2

ELECTRO-MECHANICAL IMPEDANCE TECHNIQUE

FOR SHM AND NDE

2.1 STATE-OF-THE ART IN SHM/ NDE

The prime motivations behind the ongoing research on SHM and NDE were

elaborately covered in Chapter 1. This chapter primarily deals with a critical review

of the various available SHM/ NDE techniques with regard to the EMI technique.

For any critical structure under service, it is very important to monitor (a) load

spectrum; and/ or (b) occurrence of damages. Whereas monitoring the load

spectrum and the corresponding deflections/ strains helps in validating key design

considerations, monitoring the occurrence of damages is key to ensure safety by

preventing catastrophic failures. This thesis is concerned with part (b) only, by

means of the EMI technique.

In a broad sense, the SHM/ NDE methodologies can be classified as global and

local. The global techniques rely on global structural response for damage

identification whereas the local techniques employ localized structural interrogation

for this purpose.

2.1.1 Global SHM Techniques

The global SHM techniques can be further divided into two categories-

dynamic and static. In global dynamic techniques, the test-structure is subjected to

low-frequency excitations, either harmonic or impulse, and the resulting vibration

responses (displacements, velocities or accelerations) are picked up at specified

locations along the structure. The vibration pick-up data is processed to extract the

first few mode shapes and the corresponding natural frequencies of the structure,

which, when compared with the corresponding data for the healthy state, yield

information pertaining to the locations and the severity of the damages. In this


15

connection, the impulse excitation technique is much more expedient than harmonic

excitation (which is however much more accurate) and hence preferred for quick

estimates (Giurgiutiu and Zagrai, 2002).

Application of this principle for damage detection can be found as early as

in the 1970’s (e.g. Adams et al., 1978). Subsequently, this concept was employed

for structural system identification, which is to establish a mathematical model of

the structure from the experimental input-output data (e.g. Yao, 1985; Oreta and

Tanabe, 1994; Loh and Tou, 1995). It may be mentioned that many of these

techniques consist of ‘updating’ a numerical model of the structure from the test

measurements. In the 1990’s, with the development of improved sensors, testing

hardware and data acquisition and processing techniques, many researchers

developed ‘quick’ SHM algorithms (mainly for bridge type structures), such as the

change in curvature mode shapes method (Pandey et al., 1991), the change in

stiffness method (Zimmerman and Kaouk, 1994), the change in flexibility method

(Pandey and Biswas, 1994) and the damage index method (Stubbs and Kim, 1994).

A comparative evaluation of these algorithms on an actual bridge structure, by

Farrar and Jauregui (1998), showed the damage index method to be the most

sensitive among these methods.

Many related publications can be found, reporting the use of improved

algorithms, modern wireless technology and high speed data processing (Singhal

and Kiremidjian, 1996; Skjaerbaek et al., 1998; Pines and Lovell, 1998; Aktan et

al., 1998, 2000; Lynch et al., 2003a). However, in spite of rapid progress in the

hardware and the software technologies, the basic principle remains the same,

which is to identify changes in the modal and the structural parameters (or their

derivatives) resulting from damages. The main limitations of the global dynamic

techniques can be summarized as follows

(i) These techniques typically rely on the first few mode shapes and the

corresponding natural frequencies of structures, which, being global in

nature, are not sensitive enough to be altered by localized incipient damages.

For example, Pandey and Biswas (1994) reported that a 50% reduction in the

Young’s modulus of elasticity, over the central 3% length of a 2.44m long


16

beam (used by the investigators as an example), only resulted in about 3%

reduction in the observed first natural frequency. Changes of such small

order of magnitude may not be considered as reliable damage indicators in

real-life structures, in light of experimental errors of about the same order of

magnitude.

The global parameters (on which these techniques heavily rely) do

not alter significantly due to local damages. In physical terms, the reason for

this is attributed to the fact that the long wavelength stress waves associated

with the low-frequency modes may cross a local damage (such as a crack),

without sensing it. It is for this reason that Farrar and Jauregui (1998) found

that the global dynamic techniques failed to identify damage locations for

less severe damage scenarios in their experiments. It could be possible that a

damage, just large enough to be detected by global dynamic techniques, may

already be critical for the structure in question.

(ii) These techniques demand expensive hardware and sensors, such as inertial

shakers, self-conditioning accelerometers and laser velocity meters.

Typically, the cost of a single accelerometer is of the order of US$ 1000

(Lynch et al., 2003b). For a large structure, the overall cost of such sensor

systems could easily run into millions of dollars. For example, the Tsing Ma

suspension bridge in Hong Kong was instrumented with only 350 sensors in

1997 with a total cost of over US$ 8 million.

(iii) A major limitation of these techniques is the interference caused by the

ambient mechanical noise, besides the electrical and the electromagnetic

noise associated with the measurement systems. Due to low frequency, the

techniques are highly susceptible to ambient noise, which also happens to be

in the low frequency range, typically less than 100Hz.

(iv) For small miniature structural components (such as precision machinery or

computer parts), the sensors involved in these techniques prove not only

bulky, but also likely to interfere with structural dynamics due to their own

mass and stiffness. Laser vibrometers are suitable for small structures, but

are highly expensive and need to scan the entire structure for measuring

mode shapes, which proves very tedious (Giurgiutiu and Zagrai, 2002).


17

(v) The pre-requisite of a high fidelity ‘model’ of the test structure restricts the

application of the methods to relatively simple geometries and

configurations only. Because evaluation of stiffness and damping at the

supports (which are often rusted during service), is extremely difficult,

reliable identification of a ‘model’ is quite difficult in practice.

(vi) Often, the performance of these techniques deteriorates in multiple damage

scenarios (Wang et al., 1998).

Contrary to these vibration-based global methods, many researchers have

proposed methods based on global static structural response, such as the static

displacement response technique (Banan et al., 1994) and the static strain

measurement technique (Sanayei and Saletnik, 1996). These techniques, like the

dynamic techniques, essentially aim for structural system identification, but employ

static data (such as displacements or strains) instead of vibration data. Although

conceptually sound, the application of the static-response-based techniques on real

life-sized structures is not practically feasible. For example, the static displacement

technique (Banan et al., 1994) involves applying static forces at specific nodal

points and measuring the corresponding displacements. Measurement of

displacements on large structures is a mammoth task. As a first step, it warrants the

establishment of a frame of reference, which, for contact measurement, could

demand the construction of a secondary structure on an independent foundation

(Sanayei and Saletnik, 1996). Besides, the application of large loads to cause

measurable deflections (or strains) warrants huge machinery and power input. As

such, these methods are too tedious and expensive to enable a timely and cost

effective assessment of the health of real-life structures.

Many researchers have integrated the global static or dynamic methods with

neural networks (e.g. Szewczyk and Hajela, 1994; Elkordy et al., 1994; Rhim and

Lee, 1995; Jones et al., 1997; Nakamura et al., 1998; Barbosa et al., 2000; Hung and

Cao, 2002). Neural networks offer several advantages, such as ability to generalise

solutions (Flood and Kartam, 1994a, 1994b), not demanding a priori information

concerning phenomenological nature of the structure (Masri et al., 1996), and can

produce solutions within a very short time irrespective of the problem complexity.


18

Thus neural networks can reduce huge processing times involved in static and

dynamic techniques. However, they are characterised by few limitations, such as

lack of precision and limited ability to rationalise solutions. Above all, they lack

rigorous theory to assist their design and training in a well-defined manner.

In summary, the global techniques (static/ dynamic) provide only little

information about local damages unless very large numbers of sensors are

employed. They also require intensive computations to process the measurement

data. Not much information about the specifics of location/ type of damage can be

inferred without the use of high fidelity numerical models and intensive data

processing.

2.1.2 Local SHM Techniques

Another category of damage detection methods is formed by the so-called

local methods, which, as opposed to the global techniques, rely on localized

structural interrogation for detecting damages. Some of the methods in this category

are the ultrasonic techniques, acoustic emission, eddy currents, impact echo testing,

magnetic field analysis, penetrant dye testing, and X-ray analysis.

The ultrasonic methods are based on elastic wave propagation and reflection

within the material for non-destructive strength characterization and for identifying

field inhomogeneities caused by damages. In these methods, a probe (a piezo-

electric crystal) is employed to transmit high frequency waves into the material.

These waves reflect back on encountering any crack, whose location is estimated

from the time difference between the applied and the reflected waves. These

techniques exhibit higher damage sensitivity as compared to the global techniques,

due to the utilization of high frequency stress waves. Shah et al. (2000) reported a

new ultrasonic wave based method for crack detection in concrete from one surface

only. Popovics et al. (2000) similarly developed a new ultrasonic wave based

method for layer thickness estimation and defect detection in concrete. In spite of

high sensitivity, the ultrasonic methods share few limitations, such as:

(i) They typically employ large transducers and render the structure unavailable

for service throughout the length of the test.


19

(ii) The measurement data is collected in time domain that requires complex

processing.

(iii) Since ultrasonic waves cannot be induced at right angles to the surface, they

cannot detect transverse surface cracks (Giurgiutiu and Rogers, 1997).

(iv) These techniques do not lend themselves to autonomous use since

experienced technicians are required to interpret the data.

In acoustic emission method, another local method, elastic waves generated

by plastic deformations (such as at the tip of a newly developed crack), moving

dislocations and disbonds are utilized for analysis and detection of structural

defects. It requires stress or chemical activity to generate elastic waves and can be

applied on the loaded structures also (Boller, 2002), thereby facilitating continuous

surveillance. However, the main problem to damage identification by acoustic

emission is posed by the existence of multiple travel paths from the source to the

sensors. Also, contamination by electrical interference and mechanical ambient

noise degrades the quality of the emission signals (Park et al., 2000a; Kawiecki,

2001).

The eddy currents perform a steady state harmonic interrogation of structures

for detecting surface cracks. A coil is employed to induce eddy currents in the

component. The interrogated component, in-turn induces a current in the main coil

and this induction current undergoes variations on the development of damage,

which serves an indication of damage. The key advantage of the method is that it

does not warrant any expensive hardware and is simple to apply. However, a major

drawback of the technique is that its application is restricted to conductive materials

only, since it relies on electric and magnetic fields. A more sophisticated version of

the method is magneto-optic imaging, which combines eddy currents with magnetic

field and optical technology to capture an image of the defects (Ramuhalli et al.,

2002).

In impact echo testing, a stress pulse is introduced into the interrogated

component using an impact source. As the wave propagates through the structure, it

is reflected by cracks and disbonds. The reflected waves are measured and analysed

to yield the location of cracks or disbonds. Though the technique is very good for


20

detecting large voids and delaminations, it is insensitive to small sized cracks (Park

et al., 2000a).

In the magnetic field method, a liquid containing iron powder is applied on

the component to be interrogated, subjected to magnetic field, and then observed

under ultra-violet light. Cracks are detected by appearance of magnetic field lines

along their positions. The main limitation of the method is that it is applicable on

magnetic materials only. Also, the component must be dismounted and inspected

inside a special cabin. Hence, the technique not very suitable for in situ application.

In the penetrant dye test, a coloured liquid is brushed on to the surface of the

component under inspection, allowed to penetrate into the cracks, and then washed

off the surface. A quick drying suspension of chalk is thereafter applied, which acts

as a developer and causes coloured lines to appear along the cracks. The main

limitation of this method is that it can only be applied on accessible locations of

structures since it warrants active human intervention.

In X-ray method, the test structure is exposed to X-rays, which are then re-

caught on film, where the cracks are delineated as black lines. Although the method

can detect moderate sized cracks, very small surface cracks (incipient damages) are

difficult to be captured. A more recent version of the X-ray technique is computer

tomography, whereby a cross-sectional image of solid objects can be obtained.

Although originally used for medical diagnosis, the technique is recently finding its

use for structural NDE also (e.g. Kuzelev et al., 1994). By this method, defects

exhibiting different density and/ or contrast to the surroundings can be identified.

Table 2.1 summarises the typical damage sensitivities of the local NDE

methods described above. A common limitation of the local methods is that usually,

probes, fixtures and other equipment need to be physically moved around the test-

structure for recording data. Often, this not only prevents autonomous application of

the technique, but may also demand the removal of finishes or covers such as false

ceilings. Moving the probe everywhere being impractical, these techniques are often

applied at very selected probable damage locations (often based on preliminary

visual inspection or past experience), which is almost tantamount to knowing the

damage location a priori. Generally, they cannot be applied while the component is

under service, such as in the case of an aircraft during flight. Computer tomography


21

and X-ray techniques, due to their high equipment cost, are limited to very high

performance components only (Boller, 2002).

Table 2.1 Sensitivities of common local NDE techniques (Boller, 2002).Method Minimum

detectable cracklength

High probabilitydetectable cracklength (>95%)

Remarks

Ultrasonic 2mm 5-6mm Dependent upon structuregeometry and material

Eddy currents(low-frequency)

2mm 4.5-8mm Suitable for thickness<12mm only

Eddy currents(high-frequency)

2mm (surface)0.5mm (bore holes)

2.5mm (surface)1.0mm (bore holes)

X – Ray 4mm 10mm Dependent upon structureconfiguration. Better forthickness > 12mm

Magnetic particle 2mm 4mm surfaceDye penetrant 2mm 10mm surface

2.1.3 Advent of Smart Materials, Structures and Systems for SHM and NDE

The SHM/ NDE methods described so far are the conventional monitoring

techniques. They typically rely on the measurement of stresses, strains,

displacements, accelerations or other related physical responses to identify

damages. The conventional sensors, which these techniques employ, are passive

and bulky, and can only extract secondary information such as load and strain

history, which may not lead to any direct information about damages (Giurgiutiu et

al., 2000).

However, the past few years have witnessed the emergence of ‘smart’

materials, systems and structures, which have shown new possibilities for SHM and

NDE. Due to their inherent ‘smartness’, the smart materials work on fundamentally

different principles and exhibit greater sensitivities to any changes in the

environment. The next section briefly describes the principles and the recent

developments in SHM/ NDE based on smart structures and materials.


22

2.2 SMART SYSTEMS/ STRUCTURES

2.2.1 Definition of Smart Systems/ Structures

The definition of smart structures was a topic of controversy from the late

1970’s to the late 1980’s. In order to arrive at a consensus for major terminology, a

special workshop was organised by the US Army Research Office in 1988, in which

‘sensors’, ‘actuators’, ‘control mechanism’ and ‘timely response’ were recognised

as the four qualifying features of any smart system or structure (Rogers, 1988). In

this workshop, following definition of smart systems/ structures was formally

adopted (Ahmad, 1988).

“A system or material which has built-in or intrinsic sensor(s), actuator(s) and

control mechanism(s) whereby it is capable of sensing a stimulus, responding to it

in a predetermined manner and extent, in a short/ appropriate time, and reverting

to its original state as soon as the stimulus is removed”

According to Vardan and Vardan (2000), smart system refers to a device which

can sense changes in its environment and can make an optimal response by

changing its material properties, geometry, mechanical or electromagnetic response.

Both the sensor and the actuator functions with their appropriate feedback must be

properly integrated. It should also be noted that if the response is too slow or too

fast, the system could lose its application or could be dangerous (Takagi, 1990).

Previously, the words ‘intelligent’, ‘adaptive’ and ‘organic’ were also used to

characterize smart systems and materials. For example, Crawley and de Luis (1987)

defined ‘intelligent structures’ as the structures possessing highly distributed

actuators, sensors and processing networks. Similarly, Professor H. H. Robertshaw

preferred the term ‘organic’ (Rogers, 1988) which suggests similarity to biological

processes. The human arm, for example, is like a variable stiffness actuator with a

control law (intelligence). However, many participants at the US Army Research

Office Workshop (e.g. Rogers et al., 1988) sought to differentiate the terms

‘intelligent’, ‘adaptive’ and ‘organic’ from the term ‘smart’ by highlighting their

subtle differences with the term ‘smart’. The term ‘intelligence’, for example, is


23

associated with abstract thought and learning, and till date has not been

implemented in any form of adaptive and sensing material or structure. However,

still many researchers use the terms ‘smart’ and ‘intelligent’ almost interchangeably

(e.g. In the U.S.-Japan Workshop: Takagi, 1990; Rogers, 1990), though ‘adaptive’

and ‘organic’ have become less popular.

The idea of ‘smart’ or ‘intelligent’ structures has been adopted from nature,

where all the living organisms possess stimulus-response capabilities (Rogers,

1990). The aim of the ongoing research in the field of smart systems/ structures is

to enable such a structure or system mimic living organisms, which possess a

system of distributed sensory neurons running all over the body, enabling the brain

to monitor the condition of the various body parts. However, the smart systems are

much inferior to the living beings since their level of intelligence is much primitive.

In conjunction with smart or intelligent structures, Rogers (1990) defined

following additional terms, which are meant to classify the smart structures further,

based on the level of sophistication. The relationship between these structure types

is clearly explained in Fig. 2.1.

(a) Sensory Structures: These structures possess sensors that enable the

determination or monitoring of system states/ characteristics.

(b) Adaptive Structures: These structures possess actuators that enable the

alteration of system states or characteristics in a controlled manner.

(c) Controlled Structures: These result from the intersection of the sensory and

the adaptive structures. These possess both sensors and actuators integrated in

feedback architecture for the purpose of controlling the system states or

characteristics.

(d) Active Structures: These structures possess both sensors and actuators that are

highly integrated into the structure and exhibit structural functionality in

addition to control functionality.

(e) Intelligent Structures: These structures are basically active structures

possessing highly integrated control logic and electronics that provides the

cognitive element of a distributed or hierarchic control architecture.


24

It may be noted that the sensor-actuator-controller combination can be realised

either at the macroscopic (structure) level or microscopic (material) level.

Accordingly, we have smart structures and materials respectively. The concept of

smart materials is introduced in the following section.

2.2.2 Smart Materials

Smart materials are new generation materials surpassing the conventional

structural and functional materials. These materials possess adaptive capabilities to

external stimuli, such as loads or environment, with inherent intelligence. In the US

Army Research Office Workshop, Rogers et al. (1988) defined smart materials as

materials, which possess the ability to change their physical properties in a specific

manner in response to specific stimulus input. The stimuli could be pressure,

temperature, electric and magnetic fields, chemicals or nuclear radiation. The

associated changeable physical properties could be shape, stiffness, viscosity or

damping. This kind of ‘smartness’ is generally programmed by material

composition, special processing, introduction of defects or by modifying the micro-

structure, so as to adapt to the various levels of stimuli in a controlled fashion. Like

smart structures, the terms ‘smart’ and ‘intelligent’ are used interchangeably for

smart materials. Takagi (1990) defined intelligent materials as the materials which

respond to environmental changes at the most optimum conditions and manifest

ABC

D

E

A: Sensory structures; B: Adaptive structures; C: Controlled structures;D: Active structures; E: Intelligent structures.

Fig. 2.1 Classification of smart structures (Rogers, 1990).


25

their own functions according to the environment. The feedback functions within

the material are combined with properties and functions of the materials.

Optical fibres, piezo-electric polymers and ceramics, electro-rheological (ER)

fluids, magneto-strictive materials and shape memory alloys (SMAs) are some of

the smart materials. Fig. 2.2 shows the associated ‘stimulus’ and ‘response’ of

common smart materials. Because of their special ability to respond to stimuli, they

are finding numerous applications in the field of sensors and actuators. A very

detailed description of smart materials is covered by Gandhi and Thompson (1992).

2.2.3 Active and Passive Smart Materials

Smart materials can be either active or passive. Fairweather (1998) defined

active smart materials as those materials which possess the capacity to modify their

geometric or material properties under the application of electric, thermal or

magnetic fields, thereby acquiring an inherent capacity to transduce energy.

Piezoelectric materials, SMAs, ER fluids and magneto-strictive materials are active

smart materials. Being active, they can be used as force transducers and actuators.

For example, the SMA has large recovery force, of the order of 700 MPa (105 psi)

(Kumar, 1991), which can be utilized for actuation. Similarly piezoelectric

materials, which convert electric energy into mechanical force, are also ‘active’.

Fig. 2.2 Common smart materials and associated stimulus-response.

Electric field Change in viscosity(Internal damping)

Heat Original Memorized Shape

Magnetic field Mechanical Strain

Optical FibreTemperature, pressure,

mechanical strainChange in Opto-Electronic signals

(1) Stress (1) Electric Charge

(2) Electric field (2) Mechanical strainPiezoelectric

Material

Shape MemoryAlloy

Electro-rheologicalFluid

Magneto-strictivematerial


26

The smart materials, which are not active, are called passive smart materials.

Although smart, these lack the inherent capability to transduce energy. Fibre optic

material is a good example of a passive smart material. Such materials can act as

sensors but not as actuators or transducers.

2.2.4 Applications of Piezoelectric Materials

Since this thesis is primarily concerned with piezoelectric materials, some

typical applications of these materials are briefly described here. Traditionally,

piezoelectric materials have been well-known for their use in accelerometers, strain

sensors (Sirohi and Chopra, 2000b), emitters and receptors of stress waves

(Giurgiutiu et al., 2000; Boller, 2002), distributed vibration sensors (Choi and

Chang, 1996; Kawiecki, 1998), actuators (Sirohi and Chopra, 2000a) and pressure

transducers (Zhu, 2003). However, since the last decade, the piezoelectric materials,

their derivative devices and structures have been increasingly employed in turbo-

machinery actuators, vibration dampers and active vibration control of stationary/

moving structures (e.g. helicopter blades, Chopra, 2000). They have been shown to

be very promising in active structural control of lab-sized structures and machines

(e.g. Manning et al., 2000; Song et al., 2002). Structural control of large structures

has also been attempted (e.g. Kamada et al., 1997). Other new applications include

underwater acoustic absorption, robotics, precision positioning and smart skins for

submarines (Kumar, 1991). Skin-like tactile sensors utilizing piezoelectric effect for

sensing temperatures and pressures have been reported (Rogers, 1990). Very

recently, the piezoelectric materials have been employed to produce micro and nano

scale systems and wireless inter digital transducers (IDT) using advanced embedded

system technologies, which are set to find numerous applications in micro-

electronics, bio-medical and SHM (Varadan, 2002; Lynch et al., 2003b). Recent

research is also exploring the development of versatile piezo-fibres, which can be

integrated with composite structures for actuation and SHM (Boller, 2002).

The most striking application of the piezoelectric materials in SHM has been in

the form of EMI technique. This is the main focus of the present thesis and details

will be covered in the subsequent sections.


27

2.2.5 Smart Materials: Future Applications

Seasoned researchers often share visionary ideas about the future of smart

materials in conferences and seminars. According to Prof. Rogers (Rogers, 1990),

following advancements could be possible in the field of smart materials and

structures.

• Materials which can restrain the propagation of cracks by automatically

producing compressive stresses around them (Damage arrest).

• Materials, which can discriminate whether the loading is static or shock and can

generate a large force against shock stresses (Shock absorbers).

• Materials possessing self-repairing capabilities, which can heal damages in due

course of time (Self-healing materials).

• Materials which are usable up to ultra-high temperatures (such as those

encountered by space shuttles when they re-enter the earth’s atmosphere from

outer space), by suitably changing composition through transformation (thermal

mitigation).

Takagi (1990) similarly projected the development of more functional and

higher grade materials with recognition, discrimination, adjustability, self-

diagnostics and self-learning capabilities.

2.3 PIEZOELECTRICITY AND PIEZOELECTRIC MATERIALS

The word ‘piezo’ is derived from a Greek word meaning pressure. The

phenomenon of piezoelectricity was discovered in 1880 by Pierre and Paul-Jacques

Curie. It occurs in non-centro symmetric crystals, such as quartz (SiO2), Lithium

Niobate (LiNbO3), PZT [Pb(Zr1-xTix)O3)] and PLZT [(Pb1-xLax)(Zr1-yTiy)O3)], in

which electric dipoles (and hence surface charges) are generated when the crystals

are loaded with mechanical deformations. The same crystals also exhibit the

converse effect; that is, they undergo mechanical deformations when subjected to

electric fields.

In centro-symmetric crystals, the act of deformation does not induce any dipole

moment, as shown in Fig. 2.3. However, in non-centro symmetric crystals, this


28

leads to a net dipole moment, as illustrated in Fig. 2.4. Similarly, the act of applying

an electric field induces mechanical strains in the non-centro symmetric crystals.

2.3.1 Constitutive Relations

The constitutive relations for piezoelectric materials, under small field

condition are (IEEE standard, 1987)

mdimj

Tiji TdED += ε (2.1)

mEkmj

cjkk TsEdS += (2.2)

Eq. (2.1) represents the so called direct effect (that is stress induced electrical

charge) whereas Eq. (2.2) represents the converse effect (that is electric field

induced mechanical strain). Sensor applications are based on the direct effect, and

actuator applications are based on the converse effect. When the sensor is exposed

to a stress field, it generates proportional charge in response, which can be

measured. On the other hand, the actuator is bonded to the structure and an external

µ = 0 µ = 0

Fig. 2.3 Centro-symmetric crystals: the act of stretching does not cause any

dipole moment (µ = Dipole moment).

Fig. 2.4 Noncentro-symmetric crystals: the act of stretching causes dipole moment

in the crystal (µ = Dipole moment).

µ = 0 µ ≠ 0


29

field is applied to it, which results in an induced strain field. In more general terms,

Eqs. (2.1) and (2.2) can be rewritten in the tensor form as (Sirohi and Chopra,

2000b)

=

TE

sdd

SD

Ec

dTε (2.3)

where [D] (3x1) (C/m2) is the electric displacement vector, [S] (3x3) the second

order strain tensor, [E] (3x1) (V/m) the applied external electric field vector and [T]

(3x3) (N/m2) the stress tensor. Accordingly, [ Tε ] (F/m) is the second order

dielelectric permittivity tensor under constant stress, [dd] (C/N) and [dc] (m/V) the

third order piezoelectric strain coefficient tensors, and [ Es ] (m2/N) the fourth order

elastic compliance tensor under constant electric field.

Taking advantage of the symmetry of the stress and the strain tensors, these

can be reduced from a second order (3x3) tensor form to equivalent vector forms,

(6x1) in size. Thus, TSSSSSSS ],,,,,[][ 123123332211= and similarly,

TTTTTTTT ],,,,,[][ 123123332211= . Accordingly, the piezoelectric strain coefficients

can be reduced to second order tensors (from third order tensors), as [dd] (3x6) and

[dc] (6x3). The superscripts ‘d’ and ‘c’ indicate the direct and the converse effects

respectively. Similarly, the fourth order elastic compliance tensor [ Es ] can be

reduced to (6x6) second order tensor. The superscripts ‘T’ and ‘E’ indicate that the

parameter has been measured at constant stress (free mechanical boundary) and

constant electric field (short-circuited) respectively. A bar above any parameter

signifies that it is complex in nature (i.e. measured under dynamic conditions). The

piezoelectric strain coefficient cjkd defines mechanical strain per unit electric field

under constant (zero) mechanical stress and dimd defines electric displacement per

unit stress under constant (zero) electric field. In practice, the two coefficients are

numerically equal. In cjkd or d

imd , the first subscript denotes the direction of the

electric field and second the direction of the associated mechanical strain. For

example, the term d31 signifies that the electric field is applied in the direction ‘3’

and the strain is measured in direction ‘1’.


30

If static electric field is applied under the boundary condition that the crystal

is free to deform, no mechanical stresses will develop. Similarly, if the stress is

applied under the condition that the electrodes are short-circuited, no electric field

(or surface charges) will develop. For a sheet of piezoelectric material, as shown in

Fig. 2.5, the poling direction is usually along the thickness and is denoted as 3-axis.

The 1-axis and 2-axis are in the plane of the sheet.

The matrix [dc] depends on crystal structure. For example, it is different for

PZT and quartz, as given by (Zhu, 2003)

=

0000000

000000

15

24

33

32

31

dd

ddd

d c (PZT) ,

−−

−

02000000000000

11

14

14

11

11

dd

d

dd

(quartz) (2.4)

where the coefficients d31, d32 and d33 relate the normal strain in the 1, 2 and 3

directions respectively to an electric field along the poling direction 3. For PZT

crystals, the coefficient d15 relates the shear strain in the 1-3 plane to the field E1 and

d24 relates the shear strain in the 2-3 plane to the electric field E2. It is not possible

to produce shear in the 1-2 plane purely by the application of an electric field, since

all terms in the last row of the matrix [dc] are zero (see Eq. 2.4). Similarly, shear

stress in the 1-2 plane does not generate any electric response. In all poled

piezoelectric materials, d31 is negative and d33 is positive. For a good sensor, the

algebraic sum of d31 and d33 should be the maximum and at the same time, ε33 and

the mechanical loss factor should be minimum (Kumar, 1991).

1

2

3

Fig. 2.5 A piezoelectric material sheet with conventional 1, 2 and 3 axes.


31

The compliance matrix has the form

=

EEEEEE

EEEEEE

EEEEEE

EEEEEE

EEEEEE

EEEEEE

E

ssssssssssssssssssssssssssssssssssss

s

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

(2.5)

From energy considerations, the compliance matrix is symmetric, which leaves only

21 independent coefficients. Further, for isotropic materials, there are only two

independent coefficients, as expressed below (remaining terms are zero)

E

EEE

Ysss 1

332211 === (2.6)

E

EEEEEE

Yssssss ν−

====== 323123211312 (2.7)

E

EEE

Gsss 1

665544 === (2.8)

where EY is the complex Young’s modulus of elasticity (at constant electric field),

EG the complex shear modulus (at constant electric field) and ν the Poisson’s ratio.

It may be noted that the static moduli, YE and GE, are related by

)1(2 ν+=

EE YG (2.9)

The electric permittivity matrix can be written as

=TTT

TTT

TTT

T

333231

232221

131211

][

εεε

εεε

εεε

ε (2.10)

From energy arguments, the permittivity matrix can also be shown to be symmetric,

which reduces the number of independent coefficients to 6. Further, taking

advantage of crystal configurations, more simplifications can be achieved. For

example, it takes following simple forms for monoclinic, cubic and orthorhombic

crystals (Zhu, 2003)


32

][ Tε =

TT

T

TT

3313

22

3111

000

0

εεε

εε ,

T

T

T

33

22

11

000000

εε

ε ,

T

T

T

11

11

11

000000

εε

ε (2.11)

2.3.2 Second Order Effects

It should be noted that Eqs. (2.1) and (2.2) are valid under low electric fields

only. At high electric fields, the second order terms in electric fields make

significant contributions. This effect is called the electrostrictive effect. As a result

of this effect, Eq. (2.1) need to be modified as

nmmnmdimj

Tiji EEMTdED ++= ε (2.12)

where Mmn is called the electrostriction coefficient. The electro-strictive effect is

independent of the direction of the electric field (Sirohi and Chopra, 2000a). A very

common electrostrictive crystal is PMN [Pb(Mg1/3Nb2/3)O3].

The main advantage of the electrostrictive materials is that they exhibit

negligible hysteresis (which is significant in piezoelectric crystals), making them

the first choice for high voltage applications or where precision positioning of

components is warranted (Zhu, 2003). Besides, due to non-linear dependence, they

can generate larger motions, as shown in Fig. 2.6. It is for this reason that PMN is

used in actuators in the hubble space telescope.

E

S

PZT

PMN

Fig. 2.6 Strain vs electric field for PZT (piezoelectric) and PMN (electrostrictive).

monoclinic orthorhombic(e.g. PZT)

cubic


33

2.3.3 Pyroelectricity and Ferroelectricity

These phenomenon are very similar to piezoelectricity, and the three are inter-

coupled in many crystals. Pyroelectricity is the development of surface charge upon

heating. Ferroelectricity is the spontaneous presence of an electric polarization in

the absence of an applied field, as shown in Fig. 2.7. Ferroelectic materials are used

in random access memory chips. All ferroelectric crystals are simultaneously

pyroelectric and piezoelectric as well, however the converse is not necessarily true.

2.3.4 Commercial Piezoelectric Materials

Previously, piezoelectric crystals, which used to be brittle and of large

weight, were used in practice. However, now the commercial piezoelectric

materials are available as ceramics or polymers, which can be cut into a variety of

convenient shapes and sizes and can be easily bonded.

(a) Piezoceramics

Lead zirconate titanate oxide or PZT, which has a chemical composition [Pb(Zr1-

xTix)O3)], is the most widely used type piezoceramic. It is a solid solution of lead

zirconate and lead titanate, often doped with other materials to obtain specific

properties. It is manufactured by heating a mixture of lead, zirconium and titanium

oxide powders to around 800-1000oC first to obtain a perovskite PZT powder, which

is mixed with a binder and sintered into the desired shape. The resulting unit cell is

Fig. 2.7 Polarization vs electric field for ferroelctric crystals.

P

E

Spontaneouspolarization


34

elongated in one direction and exhibits a permanent dipole moment along this axis.

However, since the ceramic consists of many such randomly oriented domains, it has

no net polarization. Application of high electric field aligns the polar axes of the unit

cells along the applied electric field, thereby reorienting most of the domains. This

process is called poling and it imparts a permanent net polarization to the crystal. This

also creates a permanent mechanical distortion, since the polar axis of the unit cell is

longer than other two axes. Due to this process, the material becomes piezoelectrically

transversely isotropic in the plane normal to the poling direction i.e. d31 = d32 ≠ d33; d15

= d24, but remains mechanically isotropic (Sirohi and Chopra, 2000b).

PZT is a very versatile smart material. It is chemically inert and exhibits high

sensitivity of about 3µV/Pa, that warrants nothing more sophisticated than a charge

amplifier to buffer the extremely high source impedance of this largely capacitive

transducer. It demonstrates competitive characteristics such as light weight, low-cost,

small size and good dynamic performance. Besides, it exhibits large range of linearity

(up to electric field of 2kV/cm, Sirohi and Chopra, 2000a), fast response, long term

stability and high energy conversion efficiency. The PZT patches can be manufactured

in any shape, size and thickness (finite rectangular shapes to complicated MEMS

shapes) at relatively low-cost as compared to other smart materials and can be easily

used over a wide range of pressures without serious non-linearity. The PZT material is

characterized by a high elastic modulus (comparable to that of aluminum). However,

PZT is somewhat fragile due to brittleness and low tensile strength. Tensile strength

measured under dynamic loading is much lower (about one-third) than that measured

under static conditions. This is because under dynamic loads, cracks propagate much

faster, resulting in much lower yield stress. Typically, G1195 (Piezo Systems Inc.,

2003) has a compressive strength of 520 MPa and a tensile strength of 76 MPa (static)

and 21 MPa (dynamic) (Zhou et al., 1995). The PZT materials have negative d31,

which implies that a positive electric field (in the direction of polarization) results in

compressive strain on the PZT sheet. If heated above a critical temperature, called the

Curie temperature, the crystals lose their piezoelectric effect. The Curie temperature

typically varies from 150oC to 350oC for most commercial PZT crystals. When

exposed to high electric fields (>12 kV/cm), opposite to the poling direction, the PZT


35

loses most of its piezoelectric capability. This is called deploing and is accompanied

by a permanent change in the dimensions of the sample.

Due to high stiffness, the PZT sheets are good actuators. They also exhibit

high strain coefficients, due to which they can act as good sensors also. These

features make the PZT materials very suitable for use as collocated actuators and

sensors. They are used in deformable mirrors, mechanical micropositioners, impact

devices and ultrasonic motors (Kumar, 1991), sonic and ultrasonic sensors, filters

and resonators, signal processing devices, igniters and voltage transformers (Zhu,

2003), to name only a few. For achieving large displacements, multi layered PZT

systems can be manufactured, such as stack, moonie and bimorph actuators.

However, due to their brittleness, the PZT sheets cannot withstand bending

and also exhibit poor conformability to curved surfaces. This is the main limitation

with PZT materials. In addition, the PZT materials show considerable fluctuation of

their electric properties with temperature. Also, soldering wires to the electroded

piezoceramics requires special skill and often results in broken elements, unreliable

connections or localized thermal depoling of the elements. As a solution to these

problems, active piezoceramic composite actuators (Smart Materials Corporation),

active fibre composites (Massachusetts Institute of Technology) and macro fibre

composites, MFCs (NASA, Langley Centre) have been developed recently (Park et

al., 2003a). The MFCs have been commercially available since 2003. These new

types of PZTs are low-cost, damage tolerant, can conform to curved surface and are

embeddable. In addition, Active Control eXperts, Inc. (ACX), now owned by Mide

Technology Corporation, has developed a packaging technology in which one or

more PZT elements are laminated between sheets of polymer flexible printed

circuitry. This provides the much robustness, reliability and ease of use. The

packaged sensors are commercially called QuickPack® actuators (Mide Technology

Corporation, 2004). These are now widely used as vibration dampers in sporting

goods, buzzer alerts, drivers for flat speakers and more recently in automotive and

aerospace components (Pretorius et al., 2004). However, these are presently many

times expensive than raw PZT patches.


36

(b) Piezopolymers

The most common commercial piezopolymer is the Polyvinvylidene Fluoride

(PVDF). It is made up of long chains of the repeating monomer (-CH2-CF2-) each

of which has an inherent dipole moment. PVDF film is manufactured by

solidification from the molten phase, which is then stretched in a particular

direction and poled. The stretching process aligns the chains in one direction.

Combined with poling, this imparts a permanent dipole moment to the film.

Because of stretching, the material is rendered piezoelectrically orthotropic, that is

d31 ≠ d32, where ‘1’ is the stretching direction. However, it still remains

mechanically isotropic.

The PVDF material is characterized by low stiffness (Young’s modulus is

1/12th that of aluminum). Hence, the PVDF sensors are not likely to modify the

stiffness of the host structure due to their own stiffness. Also, PVDF films can be

shaped as desired according to the intended application. Being polymer, it can be

formed into very thin sheets and adhered to curved surfaces also due to its

flexibility. These characteristics make PVDF films more attractive for sensor

applications, in spite of their low piezoelectric coefficients (approximately 1/10th of

PZT). It has been shown by Sirohi and Chopra (2000b) that shear lag effect is

negligible in PVDF sensors.

Piezo-rubber, which consists of fine particles of PZT material embedded in

synthetic rubber (Rogers, 1990), has appeared as an alternative for PVDF. The

piezo-rubber shows much higher electrical output due to larger thickness, which is

not possible in PVDF. The piezo-rubber is used in piezoelectric coaxial cable as a

vehicle sensor. It has much longer life and is immune to rain water.

2.4 PIEZOELECTRIC MATERIALS AS MECHATRONIC IMPEDANCE

TRANSDUCERS (MITs) FOR SHM

The term mechatronic impedance transducer (MIT) was coined by Park (2000).

A mechatronic transducer is defined as a transducer which can convert electrical

energy into mechanical energy and vice versa. The piezoceramic (PZT) materials,

because of the direct (sensor) and converse (actuator) capabilities, are mechatronic

transducers. When used as MIT, their electromechanical impedance characteristics


37

are utilized for diagnosing the condition of the structures and the same patch plays

the dual roles, as an actuator as well as a sensor. The technique utilizing the PZT

based MIT for SHM/ NDE has evolved during the last nine years and is called as

the electro-mechanical impedance (EMI) technique in the literature. The following

sections describe the various aspects of this technique in detail.

2.4.1 Physical Principles

The EMI technique is very similar to the conventional global dynamic

response techniques described previously. The major difference is with respect to

the frequency range employed, which is typically 30-400kHz in EMI technique,

against less than 100Hz in the case of the global dynamic methods.

In the EMI technique, a PZT patch is bonded to the surface of the monitored

structure using a high strength epoxy adhesive, and electrically excited via an

impedance analyzer. In this configuration, the PZT patch essentially behaves as a

thin bar undergoing axial vibrations and interacting with the host structure, as

shown in Fig. 2.8 (a). The PZT patch-host structure system can be modelled as a

mechanical impedance (due the host structure) connected to an axially vibrating

thin bar (the patch), as shown in Fig. 2.8 (b). The patch in this figure expands and

Fig. 2.8 Modelling PZT-structure interaction.

(a) A PZT patch bonded to structure under electric excitation.

(b) Interaction model of PZT patch and host structure.

(a) (b)

Alternating electricfield source

l l

Point ofmechanicalfixity

PZT Patch

3 (z)1 (x)

Hoststructure

2 (y)

PZT patch

StructuralImpedancel

hw

E31

32

l

Z Z


38

contracts dynamically in direction ‘1’ when an alternating electric field E3 (which is

spatially uniform i.e. ∂E3/∂x = ∂E3/∂y = 0) is applied in the direction ‘3’. The patch

has half-length ‘l’, width ‘w’ and thickness ‘h’. The host structure is assumed to be

a skeletal structure, that is, composed of one-dimensional members with their

sectional properties (area and moment of inertia) lumped along their neutral axes.

Therefore, the vibrations of the PZT patch in direction ‘2’ can be ignored. At the

same time, the PZT loading in direction ‘3’ is neglected by assuming the

frequencies involved to be much less than the first resonant frequency for thickness

vibrations. The vibrating patch is assumed infinitesimally small and to possess

negligible mass and stiffness as compared to the host structure. The structure can

therefore be assumed to possess uniform dynamic stiffness over the entire bonded

area. The two end points of the patch can thus be assumed to encounter equal

mechanical impedance, Z, from the structure, as shown in Fig. 2.8 (b). Under this

condition, the PZT patch has zero displacement at the mid-point (x= 0), irrespective

of the location of the patch on the host structure. Under these assumptions, the

constitutive relations (Eqs. 2.1 and 2.2) can be simplified as (Ikeda, 1990)

1313333 TdED T += ε (2.13)

3311

1 EdYTS

E+= (2.14)

where S1 is the strain in direction ‘1’, D3 the electric displacement over the PZT

patch, d31 the piezoelectric strain coefficient and T1 the axial stress in direction ‘1’.

)1( jYY EE η+= is the complex Young’s modulus of elasticity of the PZT patch at

constant electric field and )1(3333 jTT δεε −= is the complex electric permittivity (in

direction ‘3’) of the PZT material at constant stress, where 1−=j . Here, η and δ

denote respectively the mechanical loss factor and the dielectric loss factor of the

PZT material.

The one-dimensional vibrations of the PZT patch are governed by the

following differential equation (Liang et al., 1994), derived based on dynamic

equilibrium of the PZT patch.

2

2

2

2

tu

xuY E

∂∂

=∂∂ ρ (2.15)


39

where ‘u’ is the displacement at any point on the patch in direction ‘1’. Solution of

the governing differential equation by the method of separation of variables yieldstjexBxAu ωκκ )cossin( += (2.16)

where κ is the wave number, related to the angular frequency of excitation ω, the

density ρ and the complex Young’s modulus of elasticity of the patch by

EYρωκ = (2.17)

Application of the mechanical boundary condition that at x = 0 (mid point of the

PZT patch), u = 0 yields B = 0.

Hence, strain in PZT patch xAexuxS tj κκω cos)(1 =∂∂

= (2.18)

and velocity xeAjtuxu tj κω ω sin)( =∂∂

=& (2.19)

Further, by definition, the mechanical impedance Z of the structure is related to the

axial force F in the PZT patch by

)()(1)( lxlxlx uZwhTF === −== & (2.20)

where the negative sign signifies the fact that a positive displacement (or velocity)

causes compressive force in the PZT patch (Liang et al., 1993, 1994). Making use

of Eq. (2.14) and substituting the expressions for strain and velocity from Eqs.

(2.18) and (2.19) respectively, we can derive

))(cos(31

a

oa

ZZlhdVZA

+=

κκ (2.21)

where Za is the short-circuited mechanical impedance of the PZT patch, given by

)tan()( ljYwh

ZE

a κωκ

= (2.22)

Za is defined as the force required to produce unit velocity in the PZT patch in short

circuited condition (i.e. ignoring the piezoelectric effect) and ignoring the host

structure.

The electric current, which is the time rate of change of charge, can be

obtained as


40

dxdyDjdxdyDIAA∫∫∫∫ == 33 ω& (2.23)

Making use of the PZT constitutive relation (Eq. 2.13), and integrating over the

entire surface of the PZT patch (-l to +l), we can obtain an expression for the

electromechanical admittance (the inverse of electro-mechanical impedance) as

+

+−=l

lYdZZ

ZYdhwljY E

a

aET

κκεω tan)(2 2

3123133 (2.24)

This equation is same as that derived by Liang et al. (1994), except that an

additional factor of 2 comes into picture. This is due to the fact that Liang et al.

(1993, 1994) considered only one-half of the patch in their derivation.

In the EMI technique, this electro-mechanical coupling between the

mechanical impedance Z of the host structure and the electro-mechanical

admittance Y is utilized in damage detection. Z is a function of the structural

parameters- the stiffness, the damping and the mass distribution. Any damage to the

structure will cause these structural parameters to change, and hence alter the drive

point mechanical impedance Z. Assuming that the PZT parameters remain

unchanged, the electromechanical admittance Y will undergo a change and this

serves as an indicator of the state of health of the structure. Measuring Z directly

may not be feasible, but Y can be easily measured using any commercial electrical

impedance analyzer. Common damage types altering local structural impedance Z

are cracks, debondings, corrosion and loose connections (Esteban, 1996), to which

the PZT admittance signatures show high sensitivity. Contrary to low-frequency

vibration techniques, damping plays much more significant role in the EMI

technique due to the involvement of ultrasonic frequencies. Most conventional

damage detection algorithms (in low-frequency dynamic techniques), on the other

hand are based on damage related changes in structural stiffness and inertia, but

rarely in damping (Kawiecki, 2001).

It is worthwhile to mention here that traditionally, in order to achieve self-

sensing, a complicated circuit was warranted (Dosch et al., 1992). This was so

because in the traditional approach, an actuating signal was first applied and the

sensing signal was then picked up and separated from the actuating signal. But due

to the high voltage, and also due to the strong dependence of the capacitance on


41

temperature, the signal was mixed with the input voltage as well as noise and was

therefore not very accurate. The EMI technique, on the other hand, offers a much

hassle free, simplified, and more accurate self-sensing approach.

At low frequencies (<1/5 th of the first resonant frequency of the PZT

patch), the term (tanκl/κl) → 1. This is called as ‘quasi-static sensor approximation’

(Giurgiutiu and Zagrai, 2002), and for this condition, Eq. (2.24) can be simplified as

+

−=a

ET

ZZZYd

hwljY 11

231332 εω (2.25)

The electromechanical admittance Y (unit Siemens or ohm-1) consists of

real and imaginary parts, the conductance (G) and susceptance (B), respectively. A

plot of G over a sufficiently wide band of frequency serves as a diagnosis signature

of the structure and is called the conductance signature or simply signature.

Fig. (2.9) shows the typical conductance and susceptance plots for a PZT patch

bonded on to the bottom flange of a steel beam (Bhalla et al., 2001). The sharp

peaks in the conductance signature correspond to structural modes of vibration. This

is how the conductance signature identifies the local structural system (in the

vicinity of the patch) and hence constitutes a unique health-signature of the

structure at the point of attachment.

Since the real part actively interacts with the structure, it is traditionally

preferred over the imaginary part in the SHM applications. It is believed that the

imaginary part (susceptance) has very weak interaction with the structure (Sun et

al., 1995). Therefore, all investigators have so far considered it redundant and have

solely utilized the real part (conductance) alone in the SHM applications.

Fig. 2.9 Conductance and susceptance plots of a PZT patch bonded

to bottom flange of a steel beam.

0.0004

0.0005

0.0006

0.0007

0.0008

140 142 144 146 148 150

Frequency (kHz)

Con

duct

ance

(S)

0.004

0.006

0.008

140 142 144 146 148 150

Frequency (kHz)

Sus

cept

ence

(S)


42

2.4.2 Method of Application

In the EMI technique, a PZT actuator/ sensor patch is bonded to the surface of

the structure (whose health is to be monitored) using high strength epoxy adhesive.

The conductance signature of the patch is acquired over a high frequency range

(30-400 kHz). This signature forms the benchmark for assessing the structural

health. At any future point of time, when it is desired to assess the health of the

structure, the signature is extracted again and compared with the benchmark

signature.

The signature of the bonded PZT patch is usually acquired by means of

commercially available impedance analyzers, such as HP 4192A impedance

analyzer (Hewlett Packard, 1996). The impedance analyzer imposes an alternating

voltage signal of 1 volts rms (root mean square) to the bonded PZT transducer over

the user specified preset frequency range (for example 140-150 kHz in Fig. 2.9).

The magnitude and the phase of the steady state current are directly recorded in the

form of conductance and susceptance signatures in the frequency domain, thereby

eliminating the requirements of domain transforms. Besides, no amplifying device

is necessary. In fact, Sun et al. (1995) reported that higher excitation voltage has no

influence on the conductance signature, but might only be helpful in amplifying

weak structural modes.

2.4.3 Major Technological Developments During Last Nine Years

Major developments and contributions made by various researchers in the field

of EMI technique during the last nine years are summarised as follows. (A very

detailed review of the various case studies and applications is covered by Park et al.,

2003b).

(1) Application of the EMI technique for SHM on a lab sized truss structure was

first reported by Sun et al. (1995). This study was then extended to a large-scale

prototype truss joint by Ayres et al. (1998).

(2) Lopes et al. (1999) trained neural networks using statistical damage quantifiers

(Area under the conductance curve, root mean square (RMS) of the curve, root

mean square deviation (RMSD) between damaged and undamaged curves and


43

correlation coefficients) using experimental data from a bolted joint structure.

The trained neural networks were found to successfully locate and quantify the

damages inflicted on the test structure in a different experiment.

(3) Park et al. (2000a) reported significant proof-of-concept applications of the EMI

technique on civil-structural components such as composite reinforced masonry

walls, steel bridge joints and pipe joints. The technique was found to be very

tolerant to mechanical noise and also to small temperature fluctuations.

(4) Park (2000) extended the EMI technique to high temperature applications

(typically > 500oC), such as steam pipes and boilers in power plants. Besides, he

also developed practical statistical cross-correlation based methodology for

temperature compensation. This paved way for application of the technique to

real situations, where the effects of damage and temperature are mixed.

(5) Soh et al. (2000) established the damage detection and localization ability of

piezo-impedance transducers on real-life RC structures by successfully

monitoring a 5m span RC bridge during its destructive load testing. Besides,

criteria were outlined for transducer positioning, damage localization and

transducer validation.

(6) Park et al. (2000b) were the first to integrate the EMI technique with wave

propagation modelling for thin beams (1D structures) under ‘free-free’

boundary conditions, by utilizing axial modes. The conventional statistical

indices of the EMI technique were used for locating the damages in the

frequency range 70-90 kHz. The damage severity was determined by spectral

finite element based wave propagation approach, in the frequency range 10-40

kHz. However, this combination necessitated the use of some additional

hardware and sensors, such as accelerometers, which are not accurate at

ultrasonic frequencies. Also, the application of the wave propagation approach

demands additional computational effort, which could restrict the application to

simple structures only. Besides, the integration of the EMI technique with wave

propagation approach was not seamless in true sense.

(7) After the year 2000, numerous papers appeared in the literature demonstrating

successful extension of the technique on sophisticated structural components


44

such as restrengthened concrete members (Saffi and Sayyah, 2001) and jet

engine components under high temperature condtions (Winston et al., 2001).

(8) Inman et al. (2001) proposed a novel technique to utilize a single PZT patch for

health monitoring as well as for vibration control

(9) Abe et al. (2002) developed a new stress monitoring technique for thin

structural elements (such as strings, bars and plates) by applying wave

propagation theory to the EMI measurement data in the moderate frequency

range (1-10kHz). This has paved way for the application of the technique for

load monitoring, besides damage detection. The major advantage is that owing

to localized wave propagation, the technique is insensitive to boundary

conditions and can make accurate stress identification. However, the suitable

frequency band for this application is very narrow, and generally difficult to

identify. Also, the method is prone to high errors, especially in 2D components,

due to imprecise modelling of the interfacial bonding layer.

(10) Giurgiutiu et al. (2002) combined the EMI technique with wave propagation

approach for crack detection in aircraft components. While the EMI technique

was employed for near field damage detection, the guided ultrasonic wave

propagation technique (pulse echo) was used for far field damage detection.

(11) Peairs et al. (2003) developed a novel low-cost and portable version of

impedance analyzer, the major hardware used in the EMI technique, paving

way for significant cost-reduction. Integration of the EMI technique with

wireless technology and development of stand-alone sensor cum processor cum

transmission units based on MEMS and inter digital transducers (IDT) is also

underway (Park et al., 2003b) which would enable large-scale instrumentation

and monitoring of civil-structures.

2.4.4 Details of PZT Patches

In the EMI technique, the same PZT patch serves the actuating as well as the

sensing functions. Fig. 2.10 shows a typical commercially available PZT patch

suitable for this particular application (PI Ceramic, 2003). The characteristic feature

of the patch is that the electrode from the bottom edge is wrapped around the

thickness, so that both the electrodes are available on one side of the PZT patch,


45

Top electrode filmBottom electrode filmwrapped to top surface

10mm

10m

m

Fig. 2.10 A typical commercially available PZT patch.

while the other side is bonded to the host structure. PZT patches of sizes ranging

from 5mm to 15mm and thickness from 0.1mm to 0.3mm are best suited for most

structural materials such as steel and RC. Such thin patches usually have thickness

resonance frequency of the order of few MHz. Hence, the frequency response

signature is relatively flat in 30-400 kHz frequency range.

2.4.5 Selection of Frequency Range

The operating frequency range must be maintained in hundreds of kHz so that

the wavelength of the resulting stress waves is smaller than the typical size of the

defects to be detected (Giurgiutiu and Rogers, 1997). Typically, for such high

frequencies, wavelengths as small as few mm are generated. Contrary to the large

wavelength stress waves in the case of low frequency techniques, these are

substantially attenuated by the occurrence of any incipient damages (such as cracks)

in the local vicinity of the PZT patch.

Sun et al. (1995) recommended that a frequency band containing major

vibrational modes of the structure (i.e. large number of peaks in the signature), such

as the one shown in Fig. 2.9, serves as a suitable frequency range. Large number of

peaks signifies greater dynamic interaction between the structure and the PZT

patch. Park et al. (2003b) recommended a frequency range from 30 kHz to 400 kHz

for PZT patches 5 to 15mm in size. According to Park and coworkers, a higher

frequency range (>200 kHz) is favourable in localizing the sensing range, while a

lower frequency range (< 70 kHz) covers a large sensing area. Further, frequency


46

ranges higher than 500Hz are found unfavourable, because the sensing region of the

PZT patch becomes too small and the PZT signature shows adverse sensitivity to its

own bonding condition rather than any damage to the monitored structure. It should

also be noted that the piezo-impedance transducers do not behave well at

frequencies less than 5kHz. Below 1kHz, the EMI technique is not at all

recommended (Giurgiutiu and Zagrai, 2002).

2.4.6 Sensing Zone of Piezo-Impedance Transducers

As MIT, the PZT patches have a localized sensing zone of influence. This is

because a PZT patch vibrating at high frequencies excites ultrasonic modes of

vibration the structure, which are essentially local in nature. Besides, damping is

much more significant at high ultrasonic frequencies, leading to localization of the

waves generated by the vibrating PZT patch. Esteban (1996) carried out extensive

numerical modelling based on wave propagation theory, as well as conducted

comprehensive parameteric studies to identify the sensing zone of the piezo-

impedance transducers. However, at such high frequencies, exact quantification of

energy dissipation proved very difficult and hence the sensing zone could not be

exactly identified. However, it was found that this zone depends on the material of

the host structure, its geometry, the frequency of excitation and the presence of

structural discontinuities. It was concluded that structural discontinuities acting as

the sources of multiple reflections cause maximum attenuation to the propagating

waves.

However, based on experimental data from a large number of case studies,

Park et al. (2000a) claimed that the sensing radius of a typical PZT patch might vary

anywhere from 0.4m on composite reinforced structures to about 2m on simple

metal beams. Tseng and Naidu (2001) reported the sensing range to be greater than

1m in their experiments on thin aluminum beams. Therefore, for effective damage

localization, in general, the structures must be instrumented with an array of PZT

patches.

Due to a localized sensing region, the technique shares a rare ability to detect

damages without being affected by far field boundary conditions, external loading


47

or normal operating conditions (Esteban, 1996). However, this advantage comes at

the cost of a limited sensing area.

2.4.7 Modes of Wave Propagation

In an unbounded 3-D elastic solid, two basic wave types exist: dilatational

and rotational. Dilational waves, (Kolsky, 1963) are described by the equation

∆∇+=∂∆∂ 222

2

)2( Gt

λρ (2.26)

where zzyyxx εεε ++=∆ (sum of principal strains) is the dilation of the medium, λ

the Lame’s constant, G the shear modulus, and ρ the mass density. In seismic

studies, the dilatational waves are called P-waves, or ‘Principal’ or ‘Pressure’

waves.

The rotational waves, on the other hand, are described by

ϖµϖρ 222

2

)2( ∇=∂∂

t (2.27)

where ϖ is the rotation vector. Rotational waves correspond to incompressible

distortion of solids, like shear, and are often referred to as S-waves or ‘Secondary’

or ‘Shear’ waves in seismic studies.

When the solid medium is not infinite, two additional aspects need to be

considered (i) wave reflection and refraction on account of boundary, and (ii)

existence of additional wave types closely related to the boundary effects. When a

pure P-wave (or S-wave) travelling at an oblique angle hits a boundary, both

pressure and shear waves are generated in the reflection process. A free boundary,

on the other hand, gives rise to two new wave types - Rayleigh waves and Lamb

waves. Rayleigh wave amplitude decreases rapidly with depth, and becomes almost

zero at a depth of approximately 1.6λ. At surface, it is the Rayleigh waves which

represents maximum proportion of wave energy. Lamb waves are only confined to

a superficial layer existing on the top of a homogeneous solid.

The wave propagation dynamics (reflection, refraction and transmission)

determines the drive-point mechanical impedance of the structure and its

modification with degradation of the material on account of damages. In the EMI

technique, typically surface waves (mainly Rayleigh waves) are generated due to


48

PZT vibrations, as shown in Fig. 2.11, and these travel radially outwards from the

patch. They play crucial role in determining the drive point impedance and in

detecting any defects which tend to obstruct their path.

2.4.8 Effects of Temperature

The conductance signatures of piezo-impedance transducers have been found

to be temperature sensitive (Sun et al, 1995; Park et al., 1999). In real situations, the

effects of damage and temperature are bound to be mixed. This necessitates a

method to decouple the two. Fortunately, over a small frequency band, the overall

effect of temperature has been observed to be a superposition of uniform horizontal

and vertical translations of the signature (Sun et al., 1995). This is absolutely

different from the signature deviation resulting from any damages, which causes an

abrupt and local variation. It was observed by Pardo De Vera and Guemes (1997)

that the horizontal shift is not uniform and depends on frequency. However, if the

frequency band is rather narrow, it can be assumed to be uniform.

Park et al. (1999) proposed statistical cross-correlation based methodologies

for temperature compensation. Bhalla (2001) studied temperature effects using

finite element simulation. It was found that the major effects of temperature on the

signatures are the horizontal shift, due to change in the host material’s Young’s

Structure under examination

PZT Patch

2D surface of structure

PZT Patch

(a) (b)

Fig.2.11 Modes of wave propagation associated with PZT patches

(Giurgiutiu and Rogers, 1997)

(a) PZT transducer patch affixed to the host structure.

(b) Surface waves generated by the vibrating PZT patch.


49

modulus, and the vertical shift, due to variations in ε33 and d31 of the PZT patch. All

the shifts were found to vary linearly with temperature over narrow frequency

bands. Out of these, the most critical was the vertical shift due to change in ε33. A

simple temperature compensation methodology was proposed which required the

acquisition of the baseline signatures at two different temperatures.

2.4.9 Effects of Noise and Other Miscellaneous Factors

Most low frequency vibration based SHM/ NDE methods on real-life

structures are likely to encounter the presence of noise. The noise could be (a)

mechanical noise, caused by sources such as vehicle movement or wind; (b)

electrical noise, generated by variations in the power supply; or (c) electromagnetic

noise, caused by communication waves, which affect the signal acquisition and

transmission through cables and other susceptible circuitry (Samman and Biswas,

1994a).

The greatest advantage of the high frequency EMI technique is that the

signal (in few hundred kHz frequency range) is not likely to be affected by

mechanical noise, since this type of noise is dominant in the low frequency ranges

only (typically less than 100Hz). Electrical noise too is not crucial in the EMI

technique since the power required by each PZT patch is in the low milliwatt range,

which does not call for the deployment of high power generating sets. Rather, it

makes possible the development of battery operated sensors (Park, 2000). The only

possible noise could be the electromagnetic noise, which can be minimized by using

coaxial cables.

Another source of error could be the parasitic electrical admittance of the

connection wires. It can be accounted for by performing zero-correction in the

impedance analyzer, prior to taking measurements. However, it could be

problematic for large arrays where each PZT patch may have a different wire

length. It is recommended that the same set of connection wire be used for

recording both the baseline signature as well as the signature at any future point of

time, so that the residual conductance (if not properly accounted for in the zero

correction) is the same in both cases. The change in signature, if any, will be due to

structural damage alone. It should also be noted that extensive experimental study


50

by Raju (1998) found that the method can still work well in-spite of variable test

wire lengths.

Park et al. (2000a) demonstrated that the technique is insensitive to distant

boundary condition changes and mass loading. The technique is also insensitive to

arbitrary ambient inputs to the structure. This is very important, especially for the

in-flight monitoring of aircraft or bridges, while under service.

However, it should be noted that care must be exercised in applying the EMI

technique on structures which are instrumented with ultrasonic transducers for

purposes of NDE. The high frequency excitations from these transducers could

generate a high frequency noise for the EMI technique. Hence, it should be made

sure that these are turned off before applying the EMI technique.

2.4.10 Thermal Stresses in Piezo-Impedance Transducers

During vibrations, thermal stresses are produced owing to the presence of

electrical and mechanical damping. In many applications, the thermal stresses could

be significant. Zhou et al. (1995) carried a detailed analysis of the problem and

found the internal thermal stresses to increase with the thickness of the PZT patch

and the rate of internal heat generation. However, they also found that for very thin

PZT patch, such as up to 0.2-0.3 mm, the thermal stress may be ignored in the

overall stress analysis, since the thickness is small enough to let the generated heat

dissipate quickly. It could be significant in the case of stacked actuators or high

voltage actuation applications, which is not the case of for the EMI technique.

2.4.11 Multiple Sensor Requirements

Since the EMI technique is essentially acousto-ultrasonic in nature, the

number of sensors necessary depends upon the geometry and material of the

monitored component. The number of sensors is small in thin beams and plates

where the acoustic waves can easily travel long distances through the material

medium. However, in complex structures with holes, notches, discontinuities and

thickness variations, a large number of sensors may be required due to greater

losses on account of energy dissipation. Also, the same would be true for materials

such as composites or concrete, which are characterized by high material damping.


51

In such scenarios, it is important that such a multi-sensor architecture to have a

built-in redundancy such that one or more sensors may be allowed to fail without

making the entire system ineffective (Boller, 2002). Also, it is important to consider

issues like sensor validation, data pre-processing, feature extraction and pattern

recognition.

Suitable locations for bonding the patches can be easily determined from the

geometry and loading conditions to which the structure is likely to be subjected

during the course of its service by preliminary structural analysis (Soh et al, 2003).

It is recommended to locate the patches at the points of maximum bending moments

and shear, which can be ascertained by the theory of structures.

It may be mentioned here that given an array of PZT patches, it can either be

excited in self-impedance fashion (The EMI technique) or transfer impedance

fashion (Esteban, 1996). In the transfer function method, one PZT patch acts as

actuator and emits acoustic signal into the structure. The signals are picked by

another patch acting as sensor. The main advantage of the transfer impedance

method (or the gain-phase) method is that it provides greater sensing range and

hence reduces the number of sensors required. Besides, this can also enable the

determination of mechanical properties of the monitored component. Impedance

analyzer can be easily utilized for the transfer impedance approach also. However,

the ‘gain’ levels encountered in the transfer impedance approach are much smaller

since the waves have to travel longer distance, besides encountering higher noise

(Park et al., 2003b). Increasing the excitation level could help overcome this

problem and this could help the two methods to supplement each other, since the

same sensor array can be utilized for both the techniques.

2.4.12 Signal Processing Techniques and Damage Quantification

The prominent effects of structural damages on the conductance signature are

the appearance of new peaks in the signature and lateral and vertical shifting of the

peaks (Sun et al., 1995), which are the main damage indicators. Samman and

Biswas (1994a, 1994b) reported many pattern recognition techniques to quantify the

variations occurring in the structural signatures (similar to conductance signatures)

due to damages; such as the waveform chain code (WCC) technique, the signature


52

assurance criteria (SAC), the equivalent level of degradation system (ELODS) and

the adaptive template matching (ATM). Similar statistical techniques have been

employed by the investigators researching on the EMI technique; such as the root

mean square deviation or RMSD (Giurgiutiu and Rogers, 1998), relative deviation

or RD (Ayres et al., 1998; Sun et al., 1995), the difference of transfer function

between damaged and undamaged conditions (Pardo de Vera and Guemes, 1997)

and the mean absolute percent deviation or MAPD (Tseng and Naidu, 2001).

The RMSD index is defined as (Giurgiutiu and Rogers, 1998; Giurgiutiu et al.,

1999)

RMSD (%)∑

−∑=

=

=N

ii

i

N

ii

G

GG

1

20

20

1

1

)(

)(x 100 (2.28)

where 1iG is the post-damage conductance at the ith measurement point and 0

iG is

the corresponding pre-damage value. Similarly, RD is based on the sum of mean

square algorithm, normalized with respect to an arbitrarily chosen maximum

amount of damage, and is defined for the ith patch (in an array) as (Sun et al., 1995)

201

1

11

20

1

1

)(

)(

k

N

kk

ik

N

kik

iGG

GGRD

∑ −

∑ −=

=

= (2.29)

where the numerator represents the mean square deviation at the ith location and the

denominator represents the deviation for the chosen reference maximum damage

location ‘1’. The MAPD index is defined as (Tseng and Naidu, 2001)

∑−

==

N

i i

ii

GGG

NMAPD

10

01100 (2.30)

The covariance (Cov) and correlation coefficient (CC) are respectively defined as

(Tseng and Naidu, 2001)

∑=

−−=N

iii

o GGGGN

GGCov1

11001 ))((1),( (2.31)

10

10 ),(σσ

GGCovCC = (2.32)


53

where σ0 and σ1 are the standard deviations of the baseline signature and the

signature after damage respectively. 0G and 1G respectively are the mean values of

the baseline signature and the signature after damage.

Following observations by different investigators regarding statistical indices

are worth being taken note of.

(1) The author performed a comparative study of the RMSD, the SAC, the WCC

and the ATM techniques, as a part of M. Eng. Research (Bhalla, 2001) and

found the RMSD algorithm as the most robust and most representative of

damage progression among these indices.

(2) Tseng and Naidu (2001) demonstrated the use of MAPD, covariance (Cov) and

correlation coefficient (CC) to quantify damages in thin aluminium beams. They

found Cov and CC to be very good indicators when quantifying increase in

damage size at one particular location. When the peaks of one signature match

with the peaks of the other signature, the covariance value obtained is positive.

When valleys of one signature match with peaks of the other, and vice versa,

covariance is negative. When values in both signatures are unrelated, covariance

is nearly zero. Thus, the damages can be characterized by the fact that when the

deviation between the signatures is large, the covariance is closer to zero or is

negative.

(3) Giurgiutiu et al. (2002) reported comprehensive investigations of CC as damage

index in their experiments on thin circular aluminium plates. It was

experimentally found by these researchers that (1-CC)3 decreased linearly as the

distance between the sensor and the damage (a simulated crack) increased.

Although the statistical methods are easy to implement and share the advantage

of being non-parametric (Soh et al., 2000), their main drawback is that they do not

provide any clear picture of the associated damage mechanism or any change in

mechanical parameters of the structure under question. For example, in many

situations, incipient damage and the high order damage can be found to lead to an

RMSD index of the same order of magnitude. As such, the particular “threshold

value” demanding an alarm could vary from structure to structure (Soh et al., 2000).

In such situations, one needs to rely on the slope of the RMSD curve rather than its


54

absolute magnitude. However, this may also prove unreliable. It is probably for this

reason that Giurgiutiu et al. (2002) have remarked “…Further work is needed to

systematically investigate the most appropriate damage metric that can be used for

processing the frequency spectra successfully…”.

2.5 ADVANTAGES OF EMI TECHNIQUE

The major advantages of the EMI techniques over the prevalent global and local

SHM techniques are summarized below

(i) The EMI technique shows far greater damage sensitivity than the

conventional global methods. Typically, the sensitivity is of the order of the

local ultrasonic techniques (Park et el., 2003b). Yet the technique is very

straightforward to implement on large structures as compared to the local

methods, whose application is quite cumbersome. It does not warrant very

expensive hardware like the ultrasonic techniques and also does not warrant

any probe to be physically moved from one location to other. The data

acquisition is much more simplified as compared to the traditional

accelerometer-shaker combination in the global vibration techniques since

the data is directly obtained in the frequency domain. Thus, the EMI

technique provides a very nice interface between global vibration based

techniques and local ultrasonic techniques.

(ii) The PZT patches are bonded non-intrusively on the structure, possess

negligible weight and demand low power consumption. Small and non-

intrusive sensors can monitor inaccessible locations of structures and

components. Hence, this could save the expensive time and effort involved

in dismantling machines and structural components for inspection purposes.

Easy installation (no sub-surface installation) makes the piezo-impedance

transducers equally suitable for existing as well as to-be-built structures.

(iii) The use of the same transducer for actuating as well as sensing saves the

number of transducers and the associated wiring.

(iv) The limited sensing area of the PZT patches helps in isolating changes due to

far field variations such as boundary conditions and normal operational

vibrations. Also, multiple damages in different areas can be picked easily.


55

(v) The technique is practically immune to mechanical, electrical and electro-

magnetic noise. This makes the technique very suitable for implementation

during operating conditions, such as in aircraft during flight.

(vi) The PZT patches can be produced at very low costs, typically US$1 (Peairs

et al., 2003) to US$10 (Giurgiutiu and Zgrai, 2002), in contrast to

conventional force balance accelerometers, which may be as expensive as

US$1000 (Lynch et al., 2003b) and at the same time bulky and narrow-

banded.

(vii) The technique is very favourable for autonomous and online implementation

since the requirements for data processing are minimal. The data is directly

recorded in the frequency domain thereby saving expensive domain

transform efforts.

(viii) The method can be implemented at any time in the life of a structure. For

example, the PZT patches can be installed on structures after an earthquake

to monitor the growing cracks or loosening connections. Many other

methods warrant installation of the sensors at the time of construction and

hence not suitable for existing structures. However, it should be noted that

the PZT patches would be able to detect any structural damages appearing in

the post-installation period only. Hence, they cannot detect “existing”

damages in the structures.

(ix) Being non-model based, the technique can be easily applied to complex

structures.

(x) The PZT patches are orders of magnitude below the stiffness and mass of the

monitored structures. Hence the dynamics of the host structure are not

modified and accurate structural identification is possible.

(xi) PZT sensors are non-resonant devices with wide bad capabilities and exhibit

large range of linearity, fast response, light weight, high conversion

efficiency and long-term stability.

(xii) Commercial availability of portable and low-cost impedance analyzers will

further enhance the applicability of the technique.


56

It is therefore needless to say that the EMI technique has evolved as a universal

NDE method, applicable to almost all engineering materials and structures. If the

damage location could be predicted in advance (i.e. ‘where to expect damage’), the

EMI technique would be most powerful technique in such applications (Park et al.,

2003b).

2.6 LIMITATIONS OF EMI TECHNIQUE

In spite of many advantages over other techniques, the EMI technique shares

several limitations as outlined below

(i) A PZT patch is sensitive to structural damages over a relatively small

sensing zone, ranging from 0.4m to 2m only, depending upon the material

and geometrical configuration. Though sufficient for monitoring miniature

components and mechanical/ aerospace systems, the small sensing zone

warrants the deployment of several thousands of PZT patches for real-time

monitoring of large civil-structures, such as bridges or high rise buildings.

The large number of PZT patches would warrant significant cost and effort

for laying out the wiring system, data collection and data processing. Hence,

critical locations must be judiciously decided based on the theory of

structures.

(ii) Since all civil and mechanical structures are statically indeterminate,

cracking of a few joints might not necessarily affect the overall safety and

stability of the monitored structure. Thus, a drawback of the EMI technique

as compared to the global SHM techniques is its inability to assess the

overall structural stability. Rather, in this respect, global SHM techniques

and the EMI techniques could easily complement each other.

(iii) PZT materials and the related technologies are only supplementary steps in

addition to good designs of structures and machines. Many academicians

argue that more research should be focused on improving material strength

and design rather than on sensors. But even the best-designed structures

could have problems, therefore it is justified to explore the application smart

materials to sense or detect damages in advance (Reddy, 2001).


57

2.7 NEEDS FOR FURTHER RESEARCH IN EMI TECHNIQUE

2.7.1 Theoretical and Data Processing Considerations

(i) In spite of key advantages over other NDE technologies, the difficulties in

developing a theoretical model at high frequencies renders the EMI

technique unable to correlate the changes in signature with specific changes

in structural properties (Lopes et al., 1999). Hence, no comparison can be

found in the literature between theoretical and experimental electrical

admittance spectra, especially in 100-200 kHz frequency range. Giurgiutiu

et al. (2000) acknowledged that the main barrier to the widespread industrial

application of the EMI technique is the meager understanding of the multi-

domain interaction between the PZT patch and the host structure. The wave

propagation dynamics associated with vibrating PZT patches has also not

been thoroughly investigated so far.

(ii) Till date, all the existing damage quantification approaches are non-

parametric and statistical in nature and are able to utilize the real part of

signature only. The information about damage possessed by the imaginary

part is therefore lost. Giurgiutiu and Zagrai (2002) employed the imaginary

part to check the sensor bonding conditions, but not for any damage related

information.

(iii) No attempt has been made to extract the mechanical impedance of the

interrogated structure from the electro-mechanical signatures, partly due to

the non-existence of suitable impedance models.

(iv) No ready calibration is currently available so as to realistically predict

damage level based on the measured signatures.

(v) The influence of shear lag caused by finite thickness adhesive layer used for

bonding the PZT patches to the surface of host structures has not been

thoroughly investigated so far.

(vi) For practical application of the technique, it is very important to address the

issues of sensor calibration, validation and self-diagnostics.

(vii) Uncertainties in signature deviation due to damage have also been noted by

various investigators. These need to be taken into consideration more

scientifically, using the tools of probability and statistics.


58

(viii) At the present moment, the sensing region of the patch can only be

estimated very crudely.

2.7.2 Hardware/ Technology Considerations

(i) Presently, the commercial impedance analyzers used in the EMI technique

are very expensive (> US$16000) and bulky (typically, HP 4192A

impedance analyzer measures 425x235x615mm in size and weighs 19kg).

Besides, the requirements of wiring could seriously limit the practical

application of the technique on real-life structures. Although Peairs et al.

(2003) developed a novel low-cost and portable impedance analyzer, the

data acquisition, processing and signal transmission are still elementary.

This calls for the integration of the EMI technique with wireless

technologies and development of stand-alone sensor cum processor cum

transmission units based on MEMS and IDT. Efforts for developing stamp

sized chips capable of replacing impedance analyzers are also underway

Park et al. (2003a).

(ii) Park et al. (2003b) suggested the integration of local computing units with

sensor systems so as to save energy consumption in data transmission to any

central processing unit. Utilizing ambient vibrations for deriving necessary

operational power can also be of great practical advantage since this would

eliminate the requirement of replacing batteries periodically in wireless

applications.

(iii) Many practical aspects such as protection of PZT patches against harsh

environmental conditions for long serviceability and the reliability of the

adhesive bonding under extreme conditions need to be investigated.

(iv) Signal multiplexing can significantly reduce sensor interrogation times,

especially for critical large-sized structures. Suitable algorithms and

technological solutions for multiplexing and de-multiplexing the signatures

need to be developed.


59

2.8 CONCLUDING REMARKS

This chapter has presented a detailed review of the state of the art in SHM, with

a special emphasis on the EMI technique. The chapter also introduced the concept

of smart materials and structures. The physical principles underlying the EMI

technique and the details of the previous work undertaken by prominent research

groups of the world (Liang, Rogers and coworkers; Inman, Park and co-workers;

Giurgiutiu and coworkers) have been presented. The needs for further research to

improve this technique were also highlighted.

This research has primarily focused on understanding the structure-PZT

interaction mechanism to develop analytical tools for realistically calibrating the

piezo-impedance transducers for damage prediction. The next chapter will deal with

the structure-PZT interaction mechanism inherent in the piezo-impedance

transducers.

Chapter 3: PZT-Structure Electro-Mechanical Interaction

60

Chapter 3

PZT-STRUCTURE ELECTRO-MECHANICAL

INTERACTION

3.1 INTRODUCTION

The electro-mechanical interaction between the piezo-impedance transducer

and the host structure is key to damage detection in the EMI technique. On the

application of an alternating voltage across a bonded PZT patch, deformations are

produced in the patch as well as in the local area of the host structure surrounding it.

The response of this area to the imposed mechanical vibrations is transferred back

to the PZT wafer in the form of electrical response, as conductance and susceptance

signatures. As a result of this interaction, the structural characteristics are reflected

in the signatures. An understanding of the PZT-structure electro-mechanical

interaction is therefore very vital for an effective implementation of the EMI

technique for NDE. Important aspects of PZT-structure interaction are addressed in

this chapter.

3.2 MECHANICAL IMPEDANCE OF STRUCTURES

A harmonic force, acting upon a structure, can be represented by a rotating

phasor on a complex plane (to differentiate it from a vector), as shown in Fig. 3.1.

Let Fo be the magnitude of the phasor and let it be rotating anti-clockwise at an

angular frequency ω (same as the angular frequency of the harmonic force). At any

instant of time ‘t’, the angle between the phasor and the real axis is ‘ωt’. The

instantaneous force (acting upon the structure) is equal to the projection of the

phasor on the real axis i.e. Focosωt. The projection on the ‘y’ axis can be deemed as

the ‘imaginary’ component. Hence, the phasor can be expressed, using complex

notation as


61

tjooo eFtjFtFtF ωωω =+= sincos)( (3.1)

The resulting velocity response, u& , at the point of application of the force, is also

harmonic in nature. However, it lags behind the applied force by a phase angle φ,

due to the ‘mechanical impedance’ of the structure. Hence, velocity can also be

represented as a phasor, as shown in Fig. 3.1, and expressed as)()sin()cos( φωφωφω −=−+−= tj

ooo eutujtuu &&&& (3.2)

The mechanical impedance of a structure, at any point, is defined as the

ratio of the driving harmonic force to the resulting harmonic velocity, at that point,

in the direction of the applied force. Mathematically, the mechanical impedance, Z,

can be expressed as

φφω

ωj

o

otj

o

tjo e

uF

eueF

uFZ

&&&=== − )( (3.3)

Based on this definition, the mechanical impedance of a pure mass ‘m’ can

be derived as ‘mωj’ (Hixon, 1988). Similarly, the mechanical impedance of an

ideal spring possessing a spring constant ‘k’ can be derived as ‘–jk/ω’, and that of a

damper can be obtained as ‘c’ (the damping constant). For a parallel combination of

‘n’ mechanical systems, the equivalent mechanical impedance is given by (Hixon,

1988)

Fig. 3.1 Representation of harmonic force and velocity by rotating phasors.

X (Real Axis)

Y (Imaginary Axis)

ωt

Fo

φ

Force phasor

Velocity phasor

uo


62

∑=

=n

iieq ZZ

1 (3.4)

Similarly, for a series combination,

∑=

=n

i ieq ZZ 1

11 (3.5)

The main advantage of the impedance approach is that the differential

equations of Newtonian mechanics are reduced to simple algebraic equations and a

black-box concept is introduced. Critical forces and velocities only at one or two

points of interest alone need to be considered, thereby eliminating the need of a

complex analysis of the system.

3.3 MECHANICAL IMPEDANCE OF PZT PATCHES

As a general practice, the mechanical impedance of the PZT patches is

determined in short circuited condition, as shown in Fig. 3.2, so as to eliminate the

piezoelectric effect and to invoke pure mechanical response alone. If F is the force

applied on the PZT patch, then from Eq. (3.3), the short-circuited mechanical

impedance of the patch, Za, can be determined as

)(

)(1

)(

)(1

)(

)(

lx

lxE

lx

lx

lx

lxa u

SYwhu

whTuF

Z=

=

=

=

=

= ===&&&

(3.6)

where T1 is the axial stress in the patch, S1 the corresponding strain, EY the

complex Young’s modulus of elasticity of the patch, u& the velocity response and l,

w and h the patch dimensions as shown in Fig. 3.2. It should be noted that we are

considering one symmetrical half of the PZT patch (l = half length) in accordance

with the developments in section 2.4.1). As derived in Chapter 2, the displacement

response of a vibrating PZT patch is given by

Stress = T1

lh

w

Fig. 3.2 Determination of mechanical impedance of a PZT patch.

1

32

Resultant force = F

PZT patch


63

tjexAu ωκ )sin(= (3.7)

Calculating S1(x=l) and u& (x=l) with the aid of Eq. (3.7) (by differentiation with respect

to ‘x’ and ‘t’ respectively), and substituting in Eq. (3.6), we can derive the

mechanical impedance of the PZT patch as

)(tan ωκκ

jY

ll

lwhZ

E

a

= (3.8)

Za, which is a function of frequency, is a complex quantity, and can therefore be

expressed as

jyxZ aaa += (3.9)

On substituting

ll

κκtan by (r + tj) and EY by )1( jY E η+ in Eq. (3.8), and

simplifying, we can obtain following expressions for xa and ya

)()(

22 trltrwhY

xE

a +

−=

ωη

and )(

)(22 trl

trwhYy

E

a +

+−=

ωη

(3.10)

If the operating frequency is very low as compared to the first resonant frequency,

(typically, resωω51

<< ), the term

klkltan can be approximated as unity (quasi-

static sensor approximation) and Eq. (3.8) will be reduced to

=

ljY

whZE

a ω (3.11)

Hence, under low frequencies, PZT patches act like linear output devices,

independent of frequency (Liang et al., 1993). However, this is normally not the

case in the EMI technique since the operating frequency is typically in the range of

30-400 kHz, often containing first few resonant frequencies (corresponding to PZT

vibrations along length) for the finite sized (5-15mm long) patches. The advantage

is that this high frequency ensures greater sensitivity to structural damages.

However, this might also lead to the appearance of ‘false’ peaks under rare

circumstances, as will be shown in the latter part of this chapter. In this work, the

limitations of quasi-static sensor approximation adopted by previous investigators

(Liang et al., 1994) have been lifted and Eq. (3.8) is used in all deductions. The

resonant frequency of the PZT patch can be determined from the condition


64

2)12(

)1(π

ηρωκ −

=+

=n

jYll Eres (3.12)

where n is any positive integer. At these frequencies, the term tan(κl) assumes

infinitely large value, thereby reducing Za close to zero. Denoting, EY/ρ (which

is a complex number) by (Cr + Cij), and replacing ωres by 2πfres , we can obtain

following expression for the resonant frequency

lCCjCCnf

ir

irres )(4

))(12(22 +−−

= (3.13)

Fig. 3.3 shows a plot of the real part (xa), the imaginary part (ya) and the absolute

value |Za| (= 22aa yx + ) against frequency for a PZT patch possessing PZT

parameters (except η) shown in Table 3.1. Two different values of mechanical loss

factor, η= 0 and η= 3% have been considered. The points of resonance are

apparent as sharp valleys in the plot of |Za|. The first resonance occurs at 14.123kHz

for η= 0 and at 14.126kHz for η= 3%. Similarly, at frequencies where

πκ nl = (3.14)

the term tan(κl) approaches zero, thereby rendering the magnitude of Za infinitely

large (see Eq. 3.8). This phenomenon is called as ‘anti-resonance’, and such

frequencies appear as sharp peaks in the plot of |Za|. The anti-resonance frequencies

are related to the corresponding resonant frequencies by

resar fn

nf

−=

122 (3.15)

Using Eq. (3.15), the first anti-resonance frequency can be found as 28.246 kHz for

η= 0 and at 28.252kHz for η= 3%.

It should be noted from Fig. 3.3(a) that in the absence of material damping

(η= 0), the real part of mechanical impedance is zero throughout the frequency

range. This is because in this case,κl is real which means that t is also zero. Hence,

from Eq. (3.10), xa = 0. Also, as can be observed from Fig. 3.3(c) the mechanical

impedance approaches near zero value at the points of resonance and near infinite

values at the points of anti-resonance. However, in real scenarios, the presence of

finite damping (η = 3% in the present analysis) introduces finite value to ‘x’ (Fig.


65

3.3a) and reduces the peaks of ‘y’ (Fig. 3.3b). Accordingly, it tends to flattens the

peaks of |Za| (Fig. 3.3c).

3.4 ELECTRO-MECHANICAL INTERACTION IN SINGLE DEGREE OF

FREEDOM (SDOF) SYSTEMS

Liang et al. (1993, 1994) proposed ‘impedance method’ to accurately model

and predict the behaviour of active material based smart systems. Consider one such

system, represented by a parallel spring-mass-damper combination, coupled to a

PZT patch, as shown in Fig. 3.4. With regard to the definition of smart system in

Chapter 2, this is a smart system since the PZT patch, acting as piezo-impedance

Fig. 3.3 Variation of actuator impedance with frequency.

(a) Real part vs Frequency.

(b) Imaginary part vs Frequency.

(c) Absolute value of impedance vs Frequency.

0.1

10

1000

100000

0

1000

0

2000

0

3000

0

4000

0

5000

0

6000

0

7000

0

8000

0

9000

0

1000

00

Frequency (Hz)

Za (N

s/m

)

(c)

Resonance

Anti-resonanceWithoutdamping

With 3%damping

-4000

-2000

0

2000

4000

0

2000

0

4000

0

6000

0

8000

0

1000

00

Frequency (Hz)

y a (N

s/m

)

(b)

Withoutdamping

With 3%damping

(a)

0.1

10

1000

1000000

2000

0

4000

0

6000

0

8000

0

1000

00

Frequency (Hz)

xa (N

s/m

)

Withoutdamping = 0

With 3%damping


66

transducer (hence as a sensor and an actuator), can detect any variation in the

structural parameters (stimulus), by displaying variations in the electro-mechanical

admittance signatures (response).

The PZT patch in this system is assumed to possess the parameters listed in

Table 3.1. Liang et al. (1994) and Fairweather (1998) considered same parameters

in their numerical examples. However, the numerical studies reported by these

investigators were restricted to one special case out of the many possible PZT-

structure interaction scenarios, which may arise depending upon the mechanical

impedance of the PZT patch relative to that of the host structure. In the present

study, all possible interaction scenarios are considered in depth. Although both

Liang et al. (1994) and Fairweather (1998) reportedly considered

Table 3.1 Key parameters of PZT patch.

S. NO. PHYSICAL PARAMETER VALUE

1 Young’s modulus at constant electric field, EY 6.3x1010 N/m2

2 Piezoelectric strain coefficient, d31 -166x10-12 m/V

3 Electric permittivity at constant stress, T33ε 1.5x10-8 Farad/m

4 Density, ρ 7650 kg/m3

5 Dielectric loss factor, δ 0.012

6 Mechanical loss factor, η 0.001

7 Length of PZT patch, l 0.0508 m

8 Width of PZT patch, w 0.0254 m

9 Thickness of PZT patch, h 2.54x10-4 m

Fig. 3.4 A PZT patch coupled to a spring-mass-damper system.

Structural Mechanical Impedance Z

PZT patch

l

hw

E31

32

Alternating electric field source

k

c

m


67

different thickness of the PZT patch (Liang: 0.2 cm; Fairweather: 0.0245 cm), they

reported identical interaction plots in their respective works. After careful

computations, the author found the thickness reported by Liang et al. (1994) to be

incorrect, probably a typographical error.

Case Study I:

Let the driven SDOF system (shown in Fig. 3.4) has mass ‘m’ = 2 kg,

damping constant ‘c’ = 125.7 Ns/m (damping ratio ξd = 0.01) and stiffness, ‘k’ =

1.974x107 N/m. This SDOF system has a natural frequency (undamped) equal to

500 Hz. Using Eq. (3.4), its complex mechanical impedance can be determined as

ωω

kjjmcZ −+= (3.16)

In other words, yjxZ += (3.17)

where cx = and

−=

ωω kmy

2

(3.18)

The conductance and the susceptance plots for this smart system can be obtained by

the use of Liang’s impedance methodology (Liang et al., 1994) and can be

expressed as

+

+−=l

lYdZZ

ZYdhwljY E

a

aET

κκεω tan)( 2

3123133 (3.19)

which is different from Eq. (2.24), by a factor of 2, due to different PZT boundary

conditions.

Let the structural parameters, c, k and m be now altered one by one; ‘c’ be

increased by 20%, ‘k’ be reduced by 20%, and ‘m’ be increased by 20%, so as to

simulate different types of ‘damages’ in the system. Figs. 3.5(a) and 3.5(b)

respectively show the plots of the conductance (G) and the susceptance (B), for the

pristine state as well as for the various damage states. Figs. 3.5(c) and 3.5(d)

respectively show the real part (x, xa) and the imaginary part (y, ya) of the

mechanical impedances of the structure and the PZT patch, whereas Fig. 3.5(e)

shows the absolute mechanical impedance, |Z| of the structure and |Za| of the PZT

patch.


68

10

100

1000

10000

400

450

500

550

600

650

Frequency (Hz)

|Z|,

|Za|

(N

s/m

)

-5000

-3000

-1000

1000

3000

5000

400

450

500

550

600

650

Frequency (Hz)

y, y

a (N

s/m

)

1

10

100

1000

400

450

500

550

600

650

Frequency (Hz)

x, x

a (N

s/m

)

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

400

450

500

550

600

650

Frequency (Hz)

B (S

)

Fig. 3.5 Signatures for SDOF- Case I, m = 2.0 kg, k = 1.974 x 107 N/m, c = 125.7 Ns/m.

(a) Conductance vs Frequency. (b) Susceptance vs Frequency.

(c) Real impedance vs Frequency (pristine). (d) Imaginary impedance vs Frequency (pristine).

(e) Absolute impedance vs Frequency (pristine).

(e)

(a) (b)

(c) (d)

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

400

450

500

550

600

650

Frequency (Hz)

G (S

)

Pristine StatePristine State

20% increase in ‘c’

20% reduction in ‘k’20% increase in ‘m’

20% reduction in ‘k’

20% increase in ‘m’

Structure, x

PZT patch, xa

y

ya

Structure, y

PZT patch, ya

Pristine State



PZT patch, |Za|

20% increase in ‘m’ Structure, |Z|



69

It is observed from Fig. 3.5 (a) that the G-plot for the pristine state exhibits a

peak at a frequency of 593 Hz for the pristine state. At this point, it is observed

form Fig. 3.5(d) that a special condition ‘y = -ya’ occurs, that is, the imaginary

components of the mechanical impedance of the host structure and the patch

counteract each other. It was claimed by Liang et al. (1994) that the structure and

the PZT impedances are ‘complex conjugates’ of each other at the point where peak

occurs (like the present case), that is ‘x = xa’, in addition to ‘y = -ya’. However, on

examining Fig. 3.4(c) this is found incorrect, since there is no match at all for ‘x’

and ‘xa’ in the frequency range considered in this study (At f = 593 Hz, x = 125.7

Ns/m, xa = 2.148 Ns/m). This matches well with the observation of Fairweather

(1998), that impedances are rarely complex conjugate (that is impedance matching)

in real practice. At 593 Hz, because of the condition ‘y = -ya’, the imaginary part of

(Z+Za) in Eq. (3.19) vanishes and therefore ‘G’ plot exhibits a maxima.

Any variation in the structural parameters viz. k, c or m (or in other words

any ‘damage’ inflicted on the host structure) causes detectable changes in the G-plot

as well as the B-plot. Whereas any reduction in ‘k’ or an increase of ‘m’ manifests

itself as leftward shift of the peaks of the G-plot and the B-plot, an increase in ‘c’ is

reflected as a suppression of the peak response. Increase in ‘c’ also leads to a

marginal increase in the natural frequency, though hardly discernible from the

figures. It should be noted that the absolute value of Z (Fig. 3.5e) is of comparable

magnitude to that of Za in the frequency range under consideration. At 593 Hz, the

magnitudes of |Z| and |Za| are very close but this may not be true in general. This

could not be the governing criteria for the occurrence of peak since there are two

possible values of y (i.e. ±y) leading to |Z|= |Za|. In fact the second condition arises

very close to 400 Hz (Fairweather, 1998), however, no peak is observed to occur

around 400Hz.

It should also be noted that the peak of the G-plot occurs at a frequency

higher than the structural resonant frequency (which is 500 Hz). This shifting of the

‘system natural frequency’ from 500Hz to 593Hz is due to the additional stiffness

and mass contributed by the PZT wafer transducer, since the dynamic stiffness of

the PZT patch (or the mechanical impedance) is comparable to that of the structure.


70

Case Study II:

Let us consider another SDOF system driven by the same PZT patch,

however with parameters: ‘m’ = 200 kg, ‘c’ = 12566.4 Ns/m (damping ratio, ξd =

0.01) and ‘k’ = 1.974x109 N/m. This system also exhibits a resonant frequency of

500 Hz. The interaction plots for this case are shown in Fig. 3.6. As can be seen

from Fig. 3.6(e), the magnitude of the structural impedance is much higher than that

of the PZT patch, and nowhere in the frequency range of interest do their

magnitudes match each other. However, as can be seen from Fig. 3.6(d), the

condition ‘y = -ya’ does occur at a frequency almost equal to 500 Hz. Since the

magnitude of ‘y’ exhibits a very large fluctuation as compared to that of ‘ya’,

(changing from negative to positive) the condition ‘y = -ya’ occurs at a frequency

only slightly higher than the resonant frequency of the system (500Hz). It is at this

frequency that the G-plot exhibits a sharp peak (Fig. 3.6a). Thus, in this situation

also, Z and Za are not complex conjugates of each other at the point of occurrence

of the peak (as claimed by Liang, 1994), since it is evident from Fig. 3.6(c) that ‘x’

and ‘xa’ exhibit a large difference of magnitude.

The effect of variation in the structural parameters (due to damage) on G-

plot and B-plot is also shown in the figure. The variations in the G-plot caused by

the damages (Fig. 3.6a) are of similar nature as for the Case Study I. On the other

hand, the imaginary part B, as can be seen in Fig. 3.6 (b), is largely insensitive to

damages. This is because the excessive capacitive contribution of the PZT patch

camouflages the structural impact on the signatures.

For all practical purposes, the ‘system natural frequency’ is equal to that of

the host structure, due to very negligible additional stiffening effect caused by the

PZT patch. This is very much desirable in the real world applications, so that the

peak of the G-plot can signify the resonant frequency of the structure accurately.

Case Study III:

Consider another SDOF system, with ‘m’= 0.2 kg, ‘k’= 1.974x106 N/m, and

‘c’= 12.57 Ns/m (damping ratio, ξd = 0.01), again implying a resonant frequency of

500 Hz. The plots for this particular case are shown in Fig. 3.7. As can be clearly

seen from Fig. 3.7(e), the absolute magnitude of Z (pristine state) is much less than


71

100

1000

10000

100000

1000000

400

450

500

550

600

650

Frequency (Hz)

|Z|,

|Za|

(N

s/m

)

-300000

-200000

-100000

0

100000

200000

300000

400

450

500

550

600

650

Frequency (Hz)

y, y

a (N

s/m

)

1

10

100

1000

10000

100000

400

450

500

550

600

650

Frequency (Hz)

x, x

a (N

s/m

)

0

0.0001

0.0002

0.0003

0.0004

400

450

500

550

600

650

Frequency (Hz)

B (S

)

0

0.000002

0.000004

0.000006

0.000008

0.00001

400

450

500

550

600

650

Frequency (Hz)

G (S

)

Fig. 3.6 Signatures for SDOF- Case II, m = 200 kg, k = 1.974 x 109 N/m, c = 12566.4 Ns/m.




(a)

(c) (d)

(e)

Pristine State






Pristine State20% increase in ‘c’


PZT patch



Pristine State

(b)

Structure, x Structure, y

PZT patch, xa PZT patch, ya

|Za|


72

that of Za, that is, the stiffness of the PZT patch is much higher than that of the

‘driven’ host structure. From Fig. 3.7(d), it is observed that whereas ‘x’ and ‘xa’

have comparable magnitudes (though still they do not match in the frequency range

of interest), ‘y’ and ‘ya’, on the other hand, as seen from Fig. 3.7(e), differ

significantly and that the condition ‘y = -ya’ does not occur anywhere in the

frequency range of interest. Therefore, the PZT patch is not able to capture the

structural resonant frequency as no peak is observed to occur in the G-plot (Fig.

3.7a). This result clearly verifies the fact that the necessary condition for the

occurrence of peak in the G-plot is ‘y = -ya’.

However, in spite of the absence of any peak, the PZT patch is able to detect

any changes in the structural parameters (damage), as is evident from Figs. 3.7(a)

and 3.7(b), however, with much less sensitivity as compared to case I and II (a

much more severe damage is needed, e.g. k: 80%; c: 500%; and m: 1000% to cause

detectable changes in the G-plot). It may also be noted that in this particular case,

an increase in the value of ‘c’ has shifted the G-plot upwards, in contrast to case I

and II, where the G-plot was shifted down by similar damage.

This is a typical case of ‘over stiffening’ caused by the PZT patch, hence the

admittance response is governed by the PZT patch rather than the structure itself.

This condition is undesirable in the real world applications of piezo-impedance

transducers for structural identification and SHM/ NDE.

Case Study IV:

Consider case IV, with ‘m’= 2500 kg, ‘k’= 2.46x1010 N/m, and ‘c’= 3927

Ns/m (a damping ratio of ξd = 2.5x10-4), again implying a resonant frequency of

500 Hz. The plots for this particular case are shown in Fig. 3.8. This case is very

much similar to case II; the only difference being that the structural impedance is

still much higher in magnitude as compared to the PZT patch, as can be observed

from Fig. 3.8(e). Here, the magnitude of ‘y’ is also much larger than that for case II

(almost 20 times). In other words, the PZT encounters a much more stiff structure

as compared to case II. As in Case II, the condition ‘y = - ya’ was found to occur at

a frequency almost equal to the resonant frequency of the structural system

(Fig.3.8d). It is at this point that a peak (though less prominent as compared to


73

1

10

100

1000

10000

100000

1000000

10000000

400

450

500

550

600

650

Frequency (Hz)

|Z|,

|Za|

(Ns/

m)

-5000

-4000

-3000

-2000

-1000

0

1000

400

450

500

550

600

650

Frequency (Hz)

y, y

a (N

s/m

)

1

4

7

10

13

400

450

500

550

600

650

Frequency (Hz)

x, x

a (N

s/m

)

0

0.0001

0.0002

0.0003

0.0004

400

450

500

550

600

650

Frequency (Hz)

B (S

)

0

0.000001

0.000002

0.000003

0.000004

0.000005

0.00000640

0

450

500

550

600

650

Frequency (Hz)

G (S

)

Fig. 3.7 Signatures for SDOF-Case III, m = 0.2 kg, k = 1.974 x 106 N/m, c = 12.57 Ns/m.




(c) (d)

(b)

(e)

Pristine State





Pristine State



PZT patch, xa

PZT patch, ya

PZT patch1000% increase in ‘m’

Pristine State


(a)

Structure, x

Structure, y

|Za|


74

1

10

100

1000

10000

100000

1000000

10000000

400

450

500

550

600

650

Frequency (Hz)

|Z|,

|Za|

(Ns/

m)

-6000000

-4000000

-2000000

0

2000000

4000000

6000000

400

450

500

550

600

650

Frequency (Hz)

y, y

a (N

s/m

)

0.1

1

10

100

1000

10000

100000

400

450

500

550

600

650

Frequency (Hz)

x, x

a (N

s/m

)

0.00015

0.0002

0.00025

0.0003

400

450

500

550

600

650

Frequency (Hz)

B (S

)

0.000002

0.0000025

0.000003

0.0000035

0.000004

400

450

500

550

600

650

Frequency (Hz)

G (S

)

Fig. 3.8 Signatures for SDOF-Case IV, m = 2500 kg, k = 2.46 X 1010 N/m, c = 3927 Ns/m.




(c) (d)

(a) (b)

(e)

Pristine State


Pristine State20% increase in ‘c’


PZT patch, xa

PZT patch, ya

PZT patch20% increase in ‘m’


Pristine State

Structure, x Structure, y

|Za|


75

Case II) occurs in the G-plot. It is clearly evident from Fig. 3.8 (a) and (b) that the

sensitivity of the patch to detect any variation in the structural parameters has

reduced considerably. Except for ∆c = 20%, all other curves virtually coincide with

the curves of the pristine sate. Also, the variation in ‘c’ causes an upward shift of

the G-plot, which is in contrast to case II.

It was also found that any further increase in the magnitude of ‘y’ caused the

sensitivities of both the G-plot as well as the B-plot to go down significantly.

Hence, sensitivity of the signatures goes down appreciably beyond a limiting

impedance ratio.

Case Study V:

Consider case V, in which the imaginary component of the structural

mechanical impedance is assumed to have a constant value, ‘y’ = 30 Ns/m, whereas

the real part, ‘x’, is assumed to be frequency dependent. Strictly speaking, this is not

a SDOF system. However, it will be helpful in understanding PZT-structure

interaction mechanism. The various plots for this case are shown in Fig. 3.9. This

case is quite similar to Case III (|Z a | > |Z|), with the exception that the real part of

the structural impedance is frequency dependent and the imaginary part is constant

(which is possible in real world over small frequency intervals). The relative order

of magnitude of the impedance of the host structure and the PZT patch are similar

to case III, i.e. the patch is much stiffer than the host structure.

In this case, the magnitude of ‘y’ is very small as compared to ‘ya’ (Fig. 3.9d).

It can be clearly observed that the G-plot exhibits similar variation as the damping

constant ‘c’ (or ‘x’). In this case, two types of damages were induced: (i) increasing

‘x’ increased by 20%, (b) Reducing ‘y’ reduced by 50%. It is found that the G-plot

is only sensitive to variation in ‘x’ (which is increased only by 20%) rather than ‘y’

(which is reduced by 50%). At the same time, the B-plot exhibits an extremely

feeble sensitivity to damages, and it can be deemed useless from NDE point of view


76

10

100

1000

10000

400

450

500

550

600

650

Frequency (Hz)

|Z|,

|Za|

(Ns/

m)

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

400

450

500

550

600

650

Frequency (Hz)

y, y

a (N

s/m

)

0

4

8

12

16

20

400

450

500

550

600

650

Frequency (Hz)

x, x

a (N

s/m

)

0.00015

0.0002

0.00025

0.0003

0.00035

0.0004

400

450

500

550

600

650

Frequency (Hz)

B (S

)

0.000002

0.0000025

0.000003

0.0000035

0.00000440

0

450

500

550

600

650

Frequency (Hz)

G (S

)

Fig. 3.9 Signatures for Case V.



(e) Absolute Impedance vs Frequency (pristine).

(b)

(d)

(e)

Pristine State


Pristine State


50% reduction in the imaginary part

PZT patch, xa

PZT patch, ya

PZT patch


Pristine State

(a)

(c)

Structure, x

Structure, y


77

Case Study VI:

All the case studies described so far were characterized by low operating

frequencies as compared to the first resonant frequency of the PZT patch. Consider

case VI, where the structural parameters are ‘k’= 197.4 N/m, ‘m’= 0.0002 kg, and

‘c’= 0.01257 Ns/m (damping ratio ξd = 0.03), thus implying a system resonant

frequency of 158 Hz. The frequency range chosen for this case is from 5 kHz to

40 kHz, which includes the first resonant frequency of the patch (14.123 kHz) and

the second resonant frequency is also quite close (42.369kHz). The plots for Case

VI are shown in Fig. 3.10. It is observed that the G-plot and the B-plot exhibit two

very sharp peaks, although the structure does not have any resonant frequency in

this particular range (structural resonant frequency = 158 Hz!). The peaks occur at

the points where the condition ‘y = -ya’ (see Fig. 3.10d) is satisfied, however not at

structural resonant frequency. Thus the structural modes are identified falsely.

Both the real part and the imaginary part of admittance are very feebly

affected by any changes in the structural parameters. From Figs. 3.10 (a) and (b), it

is observed that only the change in ‘m’ is detectable whereas for all other simulated

damages, the plots virtually coincide with the plot for the pristine state.

This particular case study is characterized by two features:

(i) The order of magnitude of the structural impedance and the PZT mechanical

impedance are similar over certain frequency ranges, such as around 15 000

Hz and 40 000 Hz (see Fig. 3.10e).

(ii) The frequency range under consideration includes the resonant frequencies

of the PZT patch.

These two situations occurring concurrently must be avoided under all

circumstances. This case shows that for highly stiff PZT patches, the peaks of the

signatures could actually be near the natural frequencies of the patch, rather than the

structure.

In order to illustrate the extent to which false peaks can dominate the electro-

mechanical admittance spectrum, consider a hypothetical case of a PZT patch,

30x30m wide and 0.25mm thick, actuating a SDOF system with parameters: ‘m’=

200 kg, ‘k’=1.97x109 N/m, and ‘c’=12566.4 Ns/m (ξd = 0.01). This system also has

a natural frequency of 500 Hz. The plots for this case are shown in Fig. 3.11. It is


78

-3.00E+04

-1.00E+04

1.00E+04

3.00E+04

5000

1000

0

1500

0

2000

0

2500

0

3000

0

3500

0

4000

0

Frequency (Hz)

y, y

a (N

s/m

)

0.001

0.1

10

1000

100000

5000

1000

0

1500

0

2000

0

2500

0

3000

0

3500

0

4000

0

Frequency (Hz)

x, x

a (N

s/m

)

0.01

1

100

10000

5000

1000

0

1500

0

2000

0

2500

0

3000

0

3500

0

4000

0

Frequency (Hz)

|Z|,

|Za|

(Ns/

m)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

5000

1000

0

1500

0

2000

0

2500

0

3000

0

3500

0

4000

0

Frequency (Hz)

B (S

)

(b)

(c) (d)

(e)

Fig. 3.10 Signatures for SDOF-Case VI, m= 0.0002 kg, k= 197.4 N/m, c= 0.01257 Ns/m.



(e) Absolute impedance vs Frequency (pristne).

0.00001

0.0001

0.001

0.01

0.1

150

00

1000

0

1500

0

2000

0

2500

0

3000

0

3500

0

4000

0

Frequency (Hz)

G (S

)

(a)

Pristine State


20% increase in ‘m’Pristine State

PZT patch, xa

PZT patch, ya

Pristine State

PZT patch


Structure, x

Structure, y

y =-yay =-ya


79

100

1000

10000

100000

1000000

10000000

100000000

0

400

800

1200

1600

2000

Frequency (Hz)

|Z|,

|Za|

(Ns/

m)

100

1000

10000

100000

1000000

10000000

0

400

800

1200

1600

2000

Frequency (Hz)

x, x

a (N

s/m

)

(c) (d)

Fig. 3.11 Appearance of large number of ‘false’ peaks.


(c) Real impedance vs Frequency. (d) Imaginary impedance vs Frequency.

(e) Absolute impedance vs Frequency.

0

1

2

3

4

5

6

720

0

300

400

500

600

700

800

900

1000

1100

1200

Frequency (Hz)

G (S

)

0

100

200

300

400

500

600

700

800

0

400

800

1200

1600

2000

Frequency (Hz)

B (S

)

(a)

Structure

PZT patch

Structure, x

PZT patch, xa

-6000000

-4000000

-2000000

0

2000000

4000000

6000000

0

400

800

1200

1600

2000

Frequency (Hz)

y, y

a (N

s/m

)

Structure, y

PZT patch, ya

(e)

(b)


80

observed that instead of one peak (at 500 Hz, the natural frequency of the structure),

there are a total of 9 peaks in the G-plot (Fig. 3.11a). These peaks appear at all those

frequencies which are characterised by the condition ‘y = -ya’, typically near the

natural frequencies of the PZT patch.

3.5 STRUCTURE-PZT INTERACTION IN COMPLEX SYSTEMS

In the previous section, various cases were investigated for structure-PZT patch

interaction for a simple SDOF system. This section considers a multiple degree of

freedom (MDOF) system for understanding the structure-PZT interaction. Consider

a 2D-MDOF structure, driven by a PZT patch, as shown in Fig. 3.12(a). The PZT

patch is assumed to be 10 mm long and 0.2 mm thick, and extending along the

width of the host structure, thereby ensuring plain strain conditions. The patch is

assumed to possess the properties listed in Table 3.1.

The mechanical impedance for this system was determined using Eq. (3.3), by

computing the drive point harmonic velocity corresponding to a finite harmonic

actuating force, using dynamic harmonic finite element method (FEM). Taking

advantage of symmetry, the finite element model of one half of the structure, shown

in Fig. 3.12(b), is sufficient for analysis. Fine meshing was carried out in the region

surrounding the PZT patch in order to realistically simulate the transfer of the PZT

forces (Liang et al., 1993). Material properties of unalloyed and low-alloyed steels

at 25oC (source: Richter, 1983) were considered for the host structure (see Table

3.2). The real and the imaginary parts of the electrical admittance were then

determined by using Eq. (2.24) in the frequency range 115-165 kHz, at an interval

of 200Hz. A Visual Basic program listed in appendix A was used to perform

computations.

In order to ensure adequacy of finite element meshing, modal analysis was

additionally performed. According to Makkonen et al. (2001), while carrying out

dynamic harmonic analysis by FEM, the element size should be sufficiently small

(typically 3 to 5 nodal points per half-wavelength) to ensure accuracy of solution.

All the modes of vibration in the frequency range of interest were analysed, from

which the wavelengths of the excited modes were found to be quite large as

compared to the element size considered. Fig. 3.13 typically shows mode 48

(highest excited mode), characterised by a natural frequency of 162.46 kHz. From


8

the figure, the wavelength of the excitation can be seen to be quite large as

compared to the element size. Hence, the criteria of Makkonen et al. (2001) is

clearly satisfied.

100

B (Point of attachmenof PZT patch.)

200 mm = =

StructurePZT patch

A B

10 mm = =

50m

m

Fig. 3.12 A MDOF system consid

(a) 2D host structure.

(b) Finite element model of the r

(a)

mm

50m

m

t

(b)

1

ered for PZT-structure interaction.

ight half of structure.


82

Table 3.2 Key material properties of structure.

Physical Parameter Value

Density (kg/m3) 7850

Young’s Modulus (N/m2) 2.1267 x 1011

Shear Modulus (N/m2) 8.2815 x 1010

Poisson’s Ratio 0.2840

Fig. 3.14 shows the interaction plots for this structure. As can be seen from Fig.

3.14(e), the mechanical impedance of the structure varies with frequency, attaining

minimum values at the points of resonance and maximum at the points of anti-

resonance. From Fig. 3.14(f), it is seen that the structural impedance is much higher

as compared to the PZT patch. Both the real as well as the imaginary parts of the

structural mechanical impedance are of very high order of magnitude as compared

to their PZT counterparts (Fig.3.14c and 3.14d). As such, like in Case II, the B-plot

is a straight line and the G-plot captures the variation in the real part of the

structural impedance. The magnitude of ‘x’ (the real component of structural

impedance) shows many peaks (e.g. points x1 and x2 in Fig.3.14c). The G-plot

exhibits peaks at almost same frequencies (e.g. points G1 and G2 in Fig.3.14a,

corresponding to x1 and x2). Thus the G-plot reflects the dynamic characteristics of

the structure. Although the frequency range considered includes the resonant

Fig. 3.13 Graphical representation of Mode 48 (f = 162.46 kHz).

Wavelength of excited mode


83

Fig. 3.12 Mode shape 48 (f = 162.459 kHz). (a) 2 –D structure used in the study. (b) Finite element model of the right half of the structure.

60000

80000

100000

120000

140000

160000

180000

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

|Z| (

Ns/

m)

0.007

0.008

0.009

0.01

0.011

0.012

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

G (S

)

1

10

100

1000

10000

100000

1000000

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

|Z|,

|Za|

(Ns/

m)

0

20000

40000

60000

80000

100000

120000

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

x, x

a (N

s/m

)

-140000

-120000

-100000

-80000

-60000

-40000

-20000

0

20000

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

y, y

a (Ns/

m)

(a) (b)

(e)

Fig. 3.14 Signatures for MDOF system considered in Fig. 3.12.


(c) Real impedance vs Frequency. (d) Imaginary impedance Vs Frequency.

(e) Structural absolute impedance vs Frequency.

(f) Relative absolute impedance of structure and PZT patch.

x1 x2

y1 y2

Structure, x

PZT patch, xa

Structure, y

PZT patch, ya

PZT patchStructure

G1

G2

(c) (d)

(f)

0

0.2

0.4

0.6

0.8

1

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

B (S

)


84

frequency of the PZT patch (143.6 kHz, see Fig. 3.14f), no false peak is visible in

the G and B plots. This is because the PZT mechanical impedance is sufficiently

low as compared to its structural counterpart. This is the desirable criteria for any

real-life structural system.

3.6 IMPLICATIONS OF STRUCTURE-PZT INTERACTION

It is apparent from the case studies discussed in Sections 3.4 and 3.5 that the

nature of the interaction between the PZT patch and the host structure, and the

resulting electrical admittance spectra, are both governed entirely by the relative

magnitudes of x, xa, and y, ya. As a guideline, the PZT patch should typically

possess negligible mass and stiffness as compared to the structure (to ensure |Za| <<

|Z|), so that the signature response captures the essence of the structure, rather than

influenced by the PZT patch itself. Even if the frequency range includes resonant

frequencies of the PZT patch, the structure is expected to be identified reasonably

accurately if the impedance of the PZT patch is much lower in magnitude than its

structural counterpart. If this condition is satisfied, the nature of the conductance

plot will be essentially the same as that of the real mechanical impedance of the

structure, i.e. ‘x’.

However, from these case studies, it may be apparent that the B-plot is useless

from structural identification point of view (for e.g. Figs. 3.6b, 3.14b). However, the

next section introduces a new concept which could change this belief and render the

susceptance signature equally meaningful for structural identification as well as

SHM/ NDE.

3.7 DECOMPOSITION OF COUPLED ELECTRO-MECHANICAL

ADMITTANCE

For a PZT patch bonded to any structure, such as the one shown in

Fig. 3.12(a), the coupling between the structural parameters and the complex

electro-mechanical admittance represented by Eq. (2.24) is valid for skeletal

structures. By rearranging the various terms, the equation can be rewritten as

[ ] ( )

++−=

llYd

ZZZ

hwljYd

hwljY E

a

aET

κκωεω tan22 2

3123133 (3.20)

Part I Part II


85

From this equation, it is observed that whereas the first part depends solely on the

parameters of the PZT patch, the second part depends partly on the structural

parameters and partly on the parameters of the PZT patch. In fact, part II represents

the electro-mechanical coupling between the structure and the PZT patch (since

both Z and Za appear in the expression of part II). Hence, Eq. (3.20) can be written

as

AP YYY += (3.21)

where PY denotes the PZT contribution and AY represents the contribution arising

from the structure-PZT interaction. AY can be termed as the ‘active’ component,

since it represents the coupling between the structure and the patch. Also, it is

sensitive (or responsive) by any damage to the structure (any change in Z) in the

vicinity of the patch. On the contrary, PY can be regarded as the ‘passive’

component, since it not affected by any damage in the vicinity of the patch. PY can

be decomposed into real and imaginary parts by expanding )1(3333 jTT δεε −= and

)1( jYY EE η+= and substituting in part I of Eq.(3.20), which results in

{ } { }

−+

+= ETET

P YdhwljYd

hwlY 2

313323133 22 εωηδεω (3.22)

or PPP jBGY += (3.23)

where GP and BP are the real and the imaginary components of PY . BP has large

magnitude (comparable to B) whereas GP has a small magnitude, due to the

presence of δ and η, which are of very small order of magnitude (see Table 3.1). In

the measured susceptance signature, BP camouflages the active component, which is

why the raw-susceptance signature is traditionally not considered ideal for SHM.

Thus, the electro-mechanical conductance and susceptance signatures, as

obtained from direct measurement, contain the contribution of the PZT patch. So far

all the investigators have employed the raw conductance signatures directly for

SHM/ NDE. The susceptance signature has been deemed as redundant, considering

the high contribution arising out of the patch (Sun et al., 1995). However, since the


86

PZT parameters are known, the PZT contribution can be filtered off. From Eq.

(3.21),

PA YYY −= (3.24)

)()( PP jBGjBG +−+=

or jBBGGY PPA )()( −+−= (3.25)

Thus, the active conductance GA and the active susceptance BA can be determined

as

GA = G - GP (3.26)

and BA = B - BP (3.27)

Fig. 3.15 shows the plots of GA and BA for the MDOF system discussed in

the previous section. On comparison with Figs. 3.14 (a) and (b), it can be observed

that the plots have changed drastically after the removal of the PZT contribution.

The plot of GA is almost exactly similar to the variation of the real mechanical

impedance (Fig. 3.14c) and the plot of BA is similar to the variation of the

imaginary mechanical impedance (Fig. 3.14d). Raw-susceptance (Fig. 3.14b), as

observed in the previous section, hardly reflected any information regarding the

structure. But after filtering the passive component, it is reflecting structural

characteristics as prominently as the real component, as seen from Fig. 3.15(b).

Similar pattern can be observed with respect to the susceptance B for Case I

to Case VI (SDOF systems) on filtering off the passive component. For example, in

Case II, it was earlier noticed that the B-plot was unable to capture any damage

(Fig. 3.6b). However, the plot of BA, shown in Fig. 3.16, on the contrary, shows

identifiable response to damages.

Hence, the active components are more realistic representations of the

structural behaviour. Also, signature decomposition can facilitate the utilization of

the imaginary part. It is possible to derive useful information from the susceptance

signature, which was so far lacking, and could be utilized for better structural

identification as well as SHM/ NDE.


(a) (b)

0

0.0004

0.0008

0.0012

0.0016

0.002

0.0024

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

GA

(S)

0.0004

0.0008

0.0012

0.0016

0.002

0.0024

0.0028

0.0032

0.0036

1150

00

1200

00

1250

00

1300

00

1350

00

1400

00

1450

00

1500

00

1550

00

1600

00

1650

00

Frequency (Hz)

BA(

S)

Fig. 3.15 Active conductance and active susceptance (modified signatures after

filtering out the PZT contribution) for a MDOF system.

(a) Active conductance. (b) Active susceptance.

87

Fig. 3.16 Active-susceptance plot for Case II. (should be seen in

comparison to Fig. 3.6(b)

-0.000003

-0.000002

-0.000001

0

0.000001

0.000002

0.000003

400

450

500

550

600

650

Frequency (Hz)

Xs (S

)



Pristine State

BA (S

)


88

From Eq. (3.20) it can be noted that the passive component can be

completely eliminated if the PZT parameters are adjusted such that

E

T

Yd

11

3331

ε= (3.28)

However, this is difficult to achieve in practice, since PZT parameters are

temperature sensitive.


This chapter has focused on understanding PZT-structure interaction

mechanism in PZT driven smart systems. It is found that the relative magnitudes of

impedances of the host structure and the PZT influence the nature of the resulting

signatures as well as their sensitivity to damages. For an accurate structural

identification, it is necessary to ensure that |Z| >> |Za|. Otherwise, resonance peaks

could be shifted in the conductance plot and in worst case, false peaks might also

occur. It is shown that the raw-conductance and the raw-susceptance signatures

contain passive contribution from the PZT patch, which is insensitive to damage.

Especially, the imaginary part is drowned by the passive PZT contribution. Filtering

out the PZT contribution using signature decomposition could significantly improve

the quality of signature, particularly the susceptance signature, which has so far not

been utilized by researchers. Subsequent chapters will show how both the

conductance as well as the susceptance could be employed to derive more

meaningful information about the structural parameters and for improved SHM/

NDE.

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

89

Chapter 4

DAMAGE ASSESSMENT OF SKELETAL STRUCTURESVIA EXTRACTED MECHANICAL IMPEDANCE

4.1 INTRODUCTION

This chapter presents a new method of damage diagnosis, based on changes

in structural mechanical impedance at high frequencies. The mechanical impedance

is extracted from the electro-mechanical admittance signatures of piezo-impedance

transducers, by means of signature decomposition, which was introduced in the

preceding chapter. The main feature of this approach is that both the real and the

imaginary components of the admittance signature are utilized in damage

quantification. A complex damage metric is proposed to quantify damage

parametrically, based on the extracted structural parameters. As proof of concept,

the chapter reports a damage diagnosis study conducted on a model RC frame

subjected to base vibrations on a shaking table. The proposed methodology was

found to perform better than the existing damage quantification approaches i.e. the

low frequency vibration methods as well as the traditional raw-signature based

damage quantification using EMI technique.

4.2 ANALOGY BETWEEN ELECTRICAL AND MECHANICAL SYSTEMS

The concept of mechanical impedance, introduced in the previous chapter, is

analogous to the concept of electrical impedance in electrical circuits (Halliday et

al., 2001). The impedance approach allows a simplified analysis of complex

mechanical systems by reducing the differential equations of Newtonian mechanics

into simple algebraic equations, as in the electrical circuits.


90

Consider a SDOF spring-mass-damper system, subjected to a dynamic

excitation force Fo at an angular frequency ω, as shown in Fig. 4.1(a). Let the

instantaneous velocity response (which is same for each component of the system

due to parallel connection) be

)cos( φω −= txx o&& (4.1)

where ox& is the velocity amplitude and φ the phase lag of the velocity with respect

to the applied force. Displacement and acceleration can be determined from Eq.

(4.1) by integration and differentiation respectively. Hence, the force associated

with each structural element i.e. the spring (the elastic force), the damper (the

damping force) and the mass (the inertial force) can be determined. Thus,

Damping force, ( )φω −== txcxcF od cos&& (4.2)

Inertial force,

+−==

2cos πφωω txmxmF oi &&& (4.3)

Spring force,

−−

==

2cos πφω

ωt

xkxkF os

s

& (4.4)

From Eqs. (4.2) to (4.4), it is observed that this system is mathematically

analogous to a series LCR circuit in the classical electricity. The term x& is

analogous to the current (which is same for each element of the LCR circuit) and

the mechanical force is analogous to the electro-motive force (voltage). The damper

is analogous to the resistor, since Fd is in phase with x& (Eq. 4.2). The mass is

analogous to the inductor, since Fi leads x& by 90o (Eq. 4.3). Similarly, the spring is

Fig. 4.1 (a) A single degree of freedom (SDOF) system under dynamic excitation.

(b) Phasor representation of spring force (Fs), damping force (Fd) and inertial

force (Fi).

(a) (b)

FResultant

Fd

Fi

(Fi - Fs)

Fs

φ xx(t) = xocos(ωt - φ)

F(t) = Focosωt

k

c

m


91

analogous to the capacitor, since Fs lags behind x& by 90o (Eq. 4.4). These terms can

be analogously represented by a phasor diagram, as shown in Fig. 4.1(b). Hence the

amplitude of the resultant force (analogous to voltage across the entire LCR circuit)

is given by

22 )( soiodoo FFFF −+= (4.5)

where the subscript ‘o’ denotes amplitude of the concerned force. Substituting

expressions for the amplitudes from Eqs. (4.2) to (4.4) and solving, we can obtain

the amplitude of the mechanical impedance of the structure as22

2||

−+==

ωω kmc

xFZ

o

o

& (4.6)

The quantity Z is analogous to the electrical impedance (ratio of voltage to current)

of an LCR circuit. Using complex number notation, analogous to that used in

classical electricity, it may be expressed in Cartesian and polar coordinates as

φ

ωω jeZjkmcyjxZ =

−+=+=

2

(4.7)

The phase lag ‘φ’ of the velocity ‘ x& ’with respect to the resultant driving force ‘F’ is

given by (Fig. 1b)

ωωφc

kmF

FF

d

si −=

−=

2

tan (4.8)

Here, ‘x’ is the dissipative or real part and ‘y’ is the reactive or imaginary part of

the mechanical impedance. It should be noted that damping can be included in the

stiffness itself, by adopting complex stiffness, as given by

)1( jkk η+= (4.9)

where the term η, commonly known as mechanical loss factor, can be expressed as

kcωη = (4.10)

4.3 MEASUREMENT OF MECHANICAL IMPEDANCE

The concept of mechanical impedance, introduced above for SDOF systems,

can be easily extended to any complicated real-life MDOF system. Although Eqs.

(4.7) and (4.8) have been derived here for the SDOF system, complex structural


92

systems too essentially possess a mechanical impedance consisting of the real

(dissipative) and imaginary (reactive) components. These two terms can be

considered to represent a purely resistive element (such as damper) connected in

parallel to a purely reactive element (such as spring or mass or a combination of the

two). The two terms can be considered to be the “equivalent SDOF” representation

of the actual system.

However, analytical determination of mechanical impedance for complex

MDOF systems is very tedious. It can be measured experimentally by applying a

sinusoidal force at a point on the structure and measuring the resulting velocity at

that point in the direction of the force. Conventionally, this is done by using

impedance head, which consists of a force transducer and an accelerometer (Hixon,

1988). The force transducer is an electro-magnetic shaker, which produces a

sinusoidal force proportional to the input sinusoidal voltage. The accelerometer

measures acceleration at the point of interest, again in the form of a proportional

sinusoidal voltage signal. Being harmonic, velocity can be deduced from

acceleration by integration. The magnitude of the mechanical impedance is thus

determined from the ratio of the measured force and the velocity amplitudes (Eq.

4.6), and the phase difference between the two is given by the phase difference

between the corresponding measured voltage signals. However, the conventional

impedance heads possess very small operational bandwidth, which prohibits their

application for high frequencies. The same holds equally true for conventional

accelerometers. Even the high-tech miniaturized accelerometers share the

disadvantages of high cost and small operational bandwidth (Giurgiutiu and Zagrai,

2002; Lynch et al., 2003b). The next sections demonstrate how this difficulty can

be overcome with the aid of the EMI technique.

4.4 DECOMPOSITION OF ADMITTANCE SIGNATURES

In the previous chapter, the coupled electro-mechanical admittance signature,

acquired by using EMI technique, was decomposed into active and passive

components. At this point, the author would like to introduce the following

definitions.


93

(a) Raw Complex Admittance

The complex electro-mechanical admittance of a piezo-impedance transducer

bonded to a structure, obtained from direct measurement through the EMI

technique, is called the raw complex admittance. It is denoted by Y and can be

expressed as

BjGY += (4.11)

The real part, G, is called the raw conductance and the imaginary part, B, is called

the raw susceptance. A frequency plot of the raw conductance is called the raw

conductance signature (RCS) and that of the raw susceptance is called the raw

susceptance signature (RSS).

(b) Passive Complex Admittance

The contribution of the PZT patch in the complex electro-mechanical admittance, or

in other words the passive component, is called the passive complex admittance or

the PZT admittance. It is denoted by PY and can be expressed as

jBGY PPP += (4.12)

The real part, GP, is called the passive conductance and the imaginary part, BP, is

called the passive susceptance. Expressions for GP and BP are given by (From Eq.

3.22)

{ }ηδεω ETP Yd

hwlG 2

31332 += (4.13)

{ }ETP Yd

hwlB 2

31332 −= εω (4.14)

(c) Active Complex Admittance

Active complex admittance is that part of the raw complex admittance, which arises

from the electro-mechanical interaction between the PZT patch and the host

structure. It is denoted by AY and can be expressed as

jBGY AAA += (4.15)

The real part, GA, is called the active conductance and the imaginary part, BA, is

called the active susceptance. The active conductance and susceptance can be


94

obtained from Eqs. (3.26) and (3.27) respectively. A frequency plot of active

conductance is called the active conductance signature (ACS) and that of active

susceptance is called the active susceptance signature (ASS).

Traditionally, the researchers working in the field of EMI technique have

utilized the deviation in the RCS alone for damage assessment, using statistical

indices such as RMSD, RD, MAPD etc., which have been described in detail in

Chapter 2, with their shortcomings highlighted. As seen in the analysis presented in

Chapter 3, the raw-conductance is mixed with the non-interactive passive-

conductance of the PZT patch, which masks its damage detection ability.

Most of the published work related to the EMI technique has been focused

on relatively light structures. In majority of the reported investigations, the damage

was typically simulated non-destructively such as by loosening bolts or similar

components (Sun et al., 1995; Ayres et al., 1998; Park et al., 2001). Only few

destructive tests on the structures instrumented with PZT patches have been

reported (Soh et al., 2000; Park et al., 2000a). In many structures, simply the

‘detection’ of damage might be more than sufficient, which can be done

conveniently by means of the conventional statistical indices. However, in civil-

structures, we often need to find out whether the damage is ‘incipient’ or ‘severe’.

We might even tolerate an incipient damage without endangering the lives or

properties. This fact has motivated us to extract the drive point structural impedance

from measured raw signatures for damage quantification.

4.5 EXTRACTION OF STRUCTURAL MECHANICAL IMPEDANCE OF

SKELETAL STRUCTURES

4.5.1 Computational Procedure

Electro-mechanical admittance relations were derived in Chapter 1 for piezo-

impedance transducers bonded to ‘skeletal’ structures. Using the principle of

signature decomposition introduced in Chapter 3, for a skeletal structure, the active

complex admittance, AY , can be expressed as a function of structural impedance,

PZT impedance and frequency as (using Eq. 3.20)


95

( )

+

=+=l

lYd

ZZZ

hwljjBGY E

a

aAAA κ

κω

tan2 2

31 (4.16)

Substituting Za = xa + yaj (actuator impedance), Z = x + yj (structure impedance),

tjrl

l+=

κκtan , )1( jYY EE η+= , and after eliminating the imaginary term from the

denominator, we obtain

])()[(])()[(

])())[((5.0 23122 jrttrYd

yyxxjyyxxjyx

hwljY E

aa

aaaaA ηηω ++−

++++−++

= (4.17)

Denoting (x + xa) by xT and (y + ya) by yT, this equation can be rewritten in a

simplified form as

)())((5.0 22

TTA yx

TjRQjPjKY+

++= ω (4.18)

where the terms K, P, Q, R and T are defined as

EYdhwlK 2

31= (4.19)

TaTa yyxxP += TaTa yxxyQ −= (4.20)

trR η−= rtT η+= (4.21)

AY can now be decomposed into the real and imaginary parts, GA and BA

respectively as

)()(5.0 22

TTA yx

PTQRKG++−

=ω

(4.22)

)()(5.0 22

TTA yx

QTPRKB+−

=ω

(4.23)

Dividing Eq. (4.22) by Eq. (4.23) and solving, we can obtain the ratio c = Q / P as

RTBGTRBG

PQc

AA

AA

−+

==)/()/( (4.24)

Further, by using Eq. (4.20), we can obtain the ratio Ct = yT/xT as

aa

aa

T

Tt xcy

cxyxyC

+−

== (4.25)

Hence, we can solve Eqs. (4.20) and (4.22) using the constants Ct (determined from

PZT parameters) and c (determined using GA and BA) to obtain following

expression for xT and yT


96

)1(5.0))((

2tA

ataT CG

xCyTcRKx+

++−=

ω , and TtT xCy = (4.26)

It should be noted that xa and ya can be determined from the PZT parameters, as

given by Eq. (3.10). Hence, xT and yT , can be determined from ‘x’ and ‘y’ as

aT xxx −= and aT yyy −= (4.27)

Following this computational procedure, ‘x’ and ‘y’ can be extracted from

the measured conductance and susceptance signatures alone. Only the PZT

parameters are assumed known. No a-priori information about the structure is

warranted. It is important to predict the PZT mechanical impedances xa and ya

accurately. In this chapter, this is done using Eq. (3.10), on the basis of the data

provided by the manufacturer. Methods for more accurate predictions will be

covered in the subsequent chapters.

Further, in order to ensure smooth computations, |x| > |xa| and |y| > |ya|.

Otherwise, false peaks could appear in the impedance spectra. This is consistent

with the findings reported in Chapter 2.

4.5.2 Determination of (tan κl/κl)

In these computations, the quantity tjrll +=κκ /tan must be determined

precisely using the theory of complex algebra (Kreyszig, 1993). This term was

approximated by Liang et al. (1994) as unity under the assumption that the

operational frequency is much lower than the first resonant frequency of the PZT

patch (or in other words “quasi-static sensor approximation”). However, this is not

the case in SHM applications where the frequency range is typically in few hundred

kHz. Denoting κl by z, we can write

zzz

ll

cossintan

=κκ (4.28)

Noting from the theory of complex numbers (Kreyszig, 1993) that

2cos

jzjz eez−+

= andjeez

jzjz

2sin

−−= , (4.29)

and substituting z = rl + (im)j, we can derive, after algebraic manipulations,


97

jvubuav

vubvau

ll

++

−

+−

= 2222

tanκκ (4.30)

where )sin(][ rleea jmjm += − (4.31)

)cos(][ rleeb jmjm −= − (4.32)

)cos(][ rleec jmjm += − (4.33)

)sin(][ rleed jmjm −= − (4.34)

)()( imdrlcu −= (4.35)

)()( imcrldv += (4.36)

4.5.3 Physical Interpretation of Drive Point Impedance

As mentioned before, all the previous reported works so far utilized only the

real part of the raw complex admittance to quantify structural damages. However, in

the newly developed methodology, both the real and imaginary parts are utilized.

They are first filtered, using signature decomposition, to remove the PZT

contribution and to yield active signatures. The active signatures are further

processed to extract the real and the imaginary parts of the drive point mechanical

impedance, which are direct functions of the structural parameters. The drive point

impedance is typically a function of frequency. The absolute mechanical

impedance, 22|| yxZ += , attains minimum values at the points of structural

resonance and maximum values at the points of anti-resonance.

The real part, ‘x’, is the equivalent SDOF damping of the structure and the

imaginary part, ‘y’, is the equivalent SDOF stiffness- mass factor (see Eq. 4.7) of

the host structure, at the ‘drive point’ of the PZT patch. In other words they are the

structural parameters ‘apparent’ to the PZT patch at its ends. Being direct structural

parameters, these should be more sensitive to structural damages than stresses or

strains, which are secondary effects. In this manner, a piezo-impedance transducer

identifies the ‘equivalent’ parameters of the ‘black-box’ host structure in the form of

SDOF damping and stiffness-mass factors.


98

4.6 DEFINITION OF DAMAGE METRIC BASED ON EXTRACTED

STRUCTURAL IMPEDANCE

As pointed out in the preceding section, a damage index based on drive point

structural impedance is expected to be more realistic than those based on RCS, as in

the conventional approaches. This has motivated the author to define a complex

damage metric as

jDDD yx += (4.37)

where Dx denotes the damage metric of the real part of the structural impedance

(equivalent SDOF damping) and Dy the corresponding value for the imaginary part

(equivalent SDOF stiffness-mass factor) .

Dx is defined as the average of Dxi, which is the value of the metric at the ith

frequency point, defined as follows. If 1)/( <ioi xx , then, )/(1 ioixi xxD −= else,

)/(1 iioxi xxD −= . Here, xio is the baseline value at the ith frequency point and xi is

the value at the current state. The other component, Dy, is similarly defined using

‘y’ in place of ‘x’. This definition of the damage metric quantifies the damage on a

uniform 0-1 (fractional) or 0-100 (percentage) scale.

Hence, Dx measures the changes in equivalent SDOF damping associated with

the drive point of the PZT patch and Dy similarly measures the variation in the

equivalent mass-stiffness factor. Being based on the extracted impedance rather

than the raw-signatures, this method quantifies the damage parametrically.

4.7 PROOF OF CONCEPT APPLICATION: DIAGNOSIS OF VIBRATION

INDUCED DAMAGES

The proposed mechanical impedance based methodology was tested for

damage detection on a model RC frame subjected to base vibrations. The test

structure was a two-storeyed portal frame, made of reinforced concrete, as shown in

Fig. 4.2. This model represented a prototype frame with storey height of 2.9m and

span length of 3.3m, at a scale of 1:10. The shaker was an electromagnetic shaking

table, rated to a maximum acceleration of 120g and a maximum frequency of

3000Hz.


The t

patch #1 a

adhesive

beam, ver

view of s

face of th

point of v

thick and

mechanic

was a typ

impedanc

applied.

The t

frequenci

motions d

was perfo

(a) (b)

130

Patch #2

Patch #1

20

1233.5

25

30

COLUMN

BEAM

Fig. 4.2 (a) Details of test frame (All dimensions are in mm).

(b) Test frame just before applying loads.

99

est frame was instrumented with two PZT patches, shown in Fig. 4.2 as

nd patch #2, which were bonded to the structure using RS 850-940 epoxy

(RS Components, 2003). Patch #1 was instrumented on the first floor

y close to the beam-column joint, a location very critical from the point of

hear cracks. Patch #2, on the other hand, was instrumented at the bottom

e second floor beam, near the mid point, a location very critical from the

iew of flexural cracks. Both the patches were 10mm square and 0.2mm

conformed to grade PIC 151 (PI Ceramic, 2003). The electrical and

al parameters of the PZT patches are as listed in Table 4.1. The test frame

ical skeletal structure and hence signature decomposition and mechanical

e extraction outlined in the preceding sections can be conveniently

est loads were applied in the form of vertical base motions of varying

es and amplitudes. The buildings are normally subjected to such base

uring earthquakes and underground explosions (Lu et al., 2001). The test

rmed in eight phases according to the range of the imposed base motion


100

frequencies and the velocity and acceleration amplitudes. The induced base motions

are graphically shown in Table 4.2. After each excitation, the patches were scanned

to acquire the raw-signatures in the frequency range of 100-150 kHz, at an interval

of 100Hz, using HP 4192A impedance analyzer (Hewlett Packard, 1996). The

signatures were decomposed to obtain active components first, which were then

processed to extract the drive point mechanical impedance. Damage metric was

determined by the procedure outlined in the previous sections. A program written in

Visual Basic and listed in Appendix B was used for computations.

The test structure was also instrumented with conventional sensors such as

accelerometers, LVDTs and strain gauges. This part of the instrumentation was

carried out by another research group (Lu et al., 2000), which was interested in

monitoring the condition of the frame by low frequency vibration techniques.

4.7.1 Flexural Damage Prediction by PZT Patch #2

The raw-signatures of PZT patch #2 are shown in Fig. 4.3. Fig. 4.4 shows the

components Dx and Dy of the complex damage metric at various states. It also shows

the RMSD index (conventional index in the EMI technique) for comparison. From

State 1 to State 3, only minor deviations could be noticed in the raw-signatures. This

observation was consistent with previous prediction (Lu et al., 2000) that flexural

cracks will start from State 4 onwards. At State 4, a prominent shift was observed in

the conductance signature (Fig. 4.3a). The inherent cause of the shift can be

correlated with damage indices shown in Fig. 4.4(a). This shift in the signature is

accompanied by a prominent rise in the value of Dx. This signifies a change in the

Table 4.1 Key properties of PZT patches (PI Ceramic, 2003).



Electric Permittivity, T33ε (farad/m) 2.124 x 10-8

Piezoelectric Strain Coefficient, d31 (m/V) -2.10 x 10-10

Young’s Modulus, EY (N/m2) 6.667 x 1010

Dielectric loss factor, δ 0.015


101

Table 4.2 Base motions and time-histories to which test frame was subjected.PHASE LOAD

DESCRIPTIONTYPICAL BASE MOTION TIME HISTORIES

BASELINE

Phase1Freq.(850~200)Hz

Acceleration12.48g /Velocity0.027m/s

-10.0

0.0

10.0

0.00 0.05 0.10 0.15 0.20Time(s)

Base Acceleration

STATE 1

Phase 2(150-15)Hz

3.016g / 0.057m/s-2.0

0.0

2.0

0.00 0.20 0.40 0.60 0.80Time (s)

Base Acceleration

STATE 2

Phase 3700Hz

20.36g / 0.131m/s-50

0

50

0.00 0.10 0.20 0.30Time (s)

Base Acceleration

STATE 3

Phase 4700Hz

25.62g / 0.203m/s -50

0

50

0.00 0.10 0.20 0.30Time (s)

Base Acceleration

STATE 4

Phase 5200Hz

23.67g / 0.443m/s -50

0

50

0.00 0.20 0.40 0.60Time (s)

Base Acceleration

STATE 5

Phase 6200Hz

13.46g / 0.376m/s -50

0

50

0.00 0.20 0.40 0.60Time (s)

Base Acceleration

STATE 6

Phase 7200Hz

25.12g / 0.744m/s

-50

0

50

0.00 0.20 0.40 0.60Time (s)

Base Acceleration

STATE 7

Phase 8200Hz

25.12g / 0.744m/s

-50

0

50

0.00 0.20 0.40 0.60Time (s)

Base Acceleration

STATE 8


102

equivalent SDOF damping associated with the drive point impedance of the PZT

patch. An increase in damping is an expected phenomenon associated with crack

development. At State 5, further upward shift of the conductance signature (Fig.

4.3a) as well as increase of Dx (Fig. 4.4a) can be observed. No major change in Dy is

observed from State 1 to 5. This is because damping is much more sensitive to

Fig. 4.3 Raw-signatures of PZT patch #2 at various damage states (1, 2, 3, .., 6).

(a) Raw-conductance. (b) Raw-susceptance.

(a) (b)

0.0020.003

0.0040.005

0.0060.007

0.008

100 110 120 130 140 150Frequency (kHz)

Susc

epta

nce

(S) 4, 5

6

1,2,3

0.00025

0.00045

0.00065

0.00085

0.00105

0.00125

100 110 120 130 140 150

Frequency (kHz)

Con

duct

ance

(S) 4

5 6

21

Baseline

3

Fig. 4.4 Damage prediction by patch #2.

(a) Real and imaginary components of complex damage metric.

(b) RMSD (%) in raw-conductance.

(a)

020406080

100120140

0 1 2 3 4 5 6 7 8

Damage States

RM

SD (

%)

Visible cracksPatch founddamaged

(b)

0

20

40

60

80

1 2 3 4 5 6

Damage States

Dx,

Dy Dx

Dy


103

damage at high frequencies as compared to the stiffness or inertia related effects

(Esteban, 1996).

The area around the patch was continuously monitored and observable

flexural cracks could only be detected at State 6. The patch however provided the

necessary warning much earlier, at State 4 itself. State 6 was accompanied by a

reduction of Dx and a rise in Dy. A reduction in the ‘apparent damping’ could be due

to the development of disbonding between the patch and the host structure and

possible damage to the patch itself. This is also reflected in the equivalent stiffness-

mass factor since the associated index Dy shows an abrupt rise in magnitude,

signalling a reduction in the equivalent spring stiffness. This is further supported by

the fact that the patch was found to be damaged at State 7 (a crack was detected

running through the patch). However, the patch provided the necessary warning

much earlier, at State 4. As can be observed from Fig. 4.4(a), the conventional

processing approach, the RMSD, failed to respond to damage to the patch itself at

State 6 and continued to show a rising trend.

4.7.2 Shear Damage Prediction by PZT Patch #1

Fig. 4.5 shows the raw-signatures of PZT patch #1 and Fig. 4.6 shows the

components of the complex damage metric and the RMSD index at various states.

From the Baseline State to State 6, the raw-conductance signature of patch #1 did

not undergo any substantial change. The indices Dx and Dy also did not display any

prominent rise (Fig. 4.6a). Again, higher sensitivity of the associated equivalent

damping to incipient damage was confirmed by the relatively large magnitude of Dx

as compared to Dy (State 1 to State 6). At State 7, an observable shift was observed

in the conductance signature (Fig. 4.5a). This can be seen to be accompanied by rise

of Dx (Fig. 4.6a). At State 8, a sudden and more prominent vertical shift of the

signature was observed. From Fig. 4.6(a), it is observed that at this stage, both Dx

and Dy attained relatively large values, suggesting development of an abrupt

damage. Close examination of the region surrounding the patch in fact showed the

development of a hairline shear crack near the beam-column joint. The patch

however provided the information of the imminent damage at State 7 itself, in the

form of an abrupt and significant variation in signature.


104

4.7.3 Damage Sensitivity of the Proposed Methodology

It is worthwhile to compare the sensitivity of the proposed damage diagnosis

methodology with the low frequency vibration based methods as well as conventional

approach based on raw-conductance signatures utilizing statistical quantifiers. Shown in

Fig. 4.7(a) are the reductions in the natural frequency associated with the local

0.003

0.004

0.005

0.006

0.007

100 110 120 130 140 150

Frequency (kHz)

Susc

epta

nce

(S) 7

8

Baseline, 1,2,3,4,5,6

0.0002

0.0004

0.0006

0.0008

0.001

100 110 120 130 140 150

Frequency (kHz)

Cond

ucta

nce

(S) 8

76

Baseline,1,2,3,4,5

Fig. 4.5 Raw-signatures of PZT patch #1 at various damage states (1, 2, 3, .., 8).

(a) Raw-conductance. (b) Raw-susceptance.

(a) (b)

Fig. 4.6 Damage prediction by PZT patch #1.

(a) Real and imaginary components of complex damage metric.

(b) RMSD (%) in raw-conductance.

0

20

40

60

80

1 2 3 4 5 6 7 8

Damage States

Dx,

Dy

Dx

Dy

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8

Damage States

RM

SD (

%)

(a) (b)


105

vibrations of the second floor beam (on which PZT patch #2 was instrumented). These

were obtained using conventional accelerometers, which have a relatively small

frequency bandwidth, generally the upper limit is of the order of 100-200Hz. From

overall structural point of view, these frequencies correspond to a ‘higher’ mode. These

are compared with the RMSD of the raw-conductance (traditional approach in EMI

technique) as well as with the RMSD of the extracted real part of structural impedance,

‘x’ in Fig. 4.7(b). The higher sensitivity of ‘x’ to damage as compared to the low

frequency vibration techniques as well as the conventional damage quantification

approach (based on G) in EMI technique is clearly evident from Fig. 4.7(b).

Thus the new methodology enables us to derive greater information about the

nature of damages occurring in the vicinity of the PZT patches, viz. the equivalent SDOF

stiffness, the damping and the mass associated with the drive point of the PZT patch. It

predicts the damage on a uniform 0-1 (fractional) or 0-100 (percent) scale. It is therefore

more pragmatic than the previously reported non-parametric statistical approaches. It is

recommended that (Dx+Dy) ≤ 20% indicates incipient nature of damage and (Dx+Dy) ≥

50% indicates severe nature of damage. Tests will be reported in the Chapter 6 for

calibrating damage with specific changes in ‘x’ and ‘y’.

0 1 2 3 4 5 6 7 80

50

100

150

200

Damage States

Freq

uenc

y (H

z)

Fig.4.7 (a) Natural frequency of vibration of floor #2 beam at various damage states.

(b) Evaluation of damage based on natural frequency, raw-conductance and

extracted mechanical impedance.

��

��

��

��

��0

50

100

150

200

250

1 2 3 4 5

Damage States

RM

SD (%

)

%Reduction in natural frequency�� RMSD Based on G

RMSD based on x

(a) (b)


106

4.8 DISCUSSIONS

To author’s best knowledge, this is the first attempt to extract structural

parameters from the measured electrical admittance signatures of piezo-impedance

transducers. In the proposed derivation, only 1D vibrations have been considered.

The electro-mechanical coupling in the other direction has been neglected. This is

justifiable in the present case due to the skeletal nature of the test structure. In other

structures, significant coupling could be present in the other direction. Analysis for

structures in which two-dimensional interaction is dominant will be presented in

next chapter. Nonetheless, the present method can still be applied to structures

where 2D coupling is significant. In this case, the extracted parameters will

represent the ‘equivalent 1D parameters’. The 1D analysis, as presented in this

chapter, offers a simple and convenient approach to make meaningful

interpretations about damage.


In this chapter, a new method of analyzing the electro-mechanical

admittance signatures obtained from the PZT patches bonded to structures has been

presented. The proposed method extracts the ‘apparent’ drive point structural

impedance associated with the bonded PZT patch. A complex damage metric has

been proposed to quantify structural damages, based on changes in the drive point

mechanical impedance of the host structure. The real part of the damage metric

measures changes in the equivalent SDOF damping caused by damages, and the

imaginary part similarly represents the changes in the equivalent SDOF stiffness-

mass factor associated with the drive point of the PZT patch.

The proposed method was tested on a model frame structure that was

subjected to base vibrations on a shaking table. The instrumented PZT patches were

found to provide a meaningful insight into the changes taking place in the structural

parameters as a result of damages. The patches were successful in identifying

flexural and shear cracks, two prominent types of incipient damages in RC frames.

The proposed method was found to have a higher sensitivity to damages as

compared to the existing approaches.

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

107

Chapter 5 GENERALIZED ELECTRO-MECHANICAL IMPEDANCE FORMULATIONS: THEORETICAL DEVELOPMENT AND SHM APPLICATIONS

5.1 INTRODUCTION

It was demonstrated in Chapter 4 that structural mechanical impedance is far

more reliable for SHM as compared to raw admittance signatures. However, the

methodology based on signature decomposition covered in Chapter 4 is in principle

valid for skeletal structures only. This chapter introduces new generalized PZT-

structure electro-mechanical formulations valid for the more general class of

structures where significant 2D coupling exists between the PZT patch and the host

structure. The proposed formulations can be easily employed to extract the

mechanical impedance of any ‘unknown’ structural system from the admittance

signatures of a surface bonded PZT patch. The chapter also outlines a new

methodology to quantify structural damages using the extracted impedance spectra,

suitable for diagnosing damages in structures ranging from miniature precision

machine components to large civil-structures.

5.2 EXISTING PZT-STRUCTURE INTERACTION MODELS

Two well-known approaches for modelling the behaviour of PZT-based

electro-mechanical smart systems are the static approach and the impedance

approach.

The static approach, proposed by Crawley and de Luis (1987), assumes

frequency independent actuator force, determined from static equilibrium and strain

compatibility between the PZT patch and the host structure. In this approach, the

patch is assumed to be a thin bar (length ‘l’, width ‘w’ and thickness ‘h’), under


108

static equilibrium with the structure, which is represented by its static stiffness Ks,

as shown in Fig. 5.1. In this configuration, owing to static condition, the imaginary

component of the complex terms in the PZT constitutive relations (Eqs. 2.13 and

2.14) can be dropped. Hence, from Eq. (2.14), the axial force in the PZT patch can

be expressed as

EP YEdSwhwhTF )( 33111 −== (5.1)

Similarly, the axial force in the structure can be determined as

1lSKxKF SSS −=−= (5.2)

The negative sign signifies that a positive displacement ‘x’ causes compressive

force in the spring (the host structure). Force equilibrium in the system implies that

FP and FS should be equal, which leads to the equilibrium strain, Seq, given by

+

=

whYlK

EdS

ES

eq

1

331 (5.3)

Hence, from Eq. (5.2), the magnitude of the force in the PZT (or the structure) can

be worked out as eqSeq lSKF = .

In order to determine the response of the system under an alternating electric

field, the static approach recommends that a dynamic force with amplitude

eqSeq lSKF = be applied to the host structure, irrespective of the frequency of

actuation. Since the static approach employs only static PZT properties, the effects

of damping and inertia, which significantly affect PZT output characteristics, are

completely ignored. Because of these reasons, the static approach often leads to

Fig. 5.1 Modelling of PZT-structure interaction by static approach.

Static electric field

PZT patch

Structure KS

lh

w

E3 1(x)

3(z)2(y)

x


109

significant errors, especially near the resonant frequency of the structure or the

patch. (Liang et al., 1993; Fairweather 1998).

In order to alleviate this inaccuracy, impedance approach was proposed by

Liang et al. (1993, 1994), based on dynamic equilibrium rather than static

equilibrium and by rigorously including dynamic PZT properties and structural

stiffness. In the impedance approach, the host structure is represented by

mechanical impedance Z, rather than a pure spring, as shown in Fig. 2.8(b). The

force-displacement relationship for the structure (Eq. 5.2) is replaced by impedance

based force-velocity relationship (Eq. 2.20). Further, instead of actuator’s static

stiffness, the impedance approach considers actuator impedance Za, similar in

principle to structural impedance. Impedance model based electromechanical

formulations have already been derived for 1D structures in Chapter 2.

Zhou et al. (1995, 1996) extended 1D impedance approach to model the

interactions of a generic PZT element coupled to a 2D host structure. The analytical

model of Zhou et al. is schematically shown in Fig. 5.2. In this approach, the

structural impedance is represented by direct impedances Zxx and Zyy, and the cross

impedances Zxy and Zyx, which are related to the planar forces F1 and F2 (in

directions 1 and 2 respectively) and the corresponding planar velocities 1u& and 2u&

by

−=

2

1

2

1

uu

ZZZZ

FF

yyyx

xyxx

&

& (5.4)

Applying D’Alembert’s principle along the two principal axes and after imposing

boundary conditions, Zhou et al. (1995) derived following expression for the

electro-mechanical admittance across PZT terminals

Fig. 5.2 Modelling PZT-structure 2D physical coupling by impedance

approach (Zhou et al., 1995).

x, 1 y, 2

z, 3

l

wZxx

Zxy

Zyy Zyx

E3


110

−+

−−=+= −

11sinsin

)1()1(2 1

231

231

33 Nw

wl

lYdYdhwljBjGY

EET κκ

ννεω (5.5)

where κ, the 2D wave number, is given by

EY

)1( 2νρωκ −= (5.6)

and N is a 2x2 matrix, given by

+−

−

−

+−=

ayy

yy

ayy

yx

axx

xx

axx

xy

ayy

yy

ayy

yx

axx

xx

axx

xy

ZZ

ZZ

wlw

ZZ

ZZ

lwl

ZZ

ZZ

wlw

ZZ

ZZ

lwl

Nνκκνκκ

νκκνκκ

1)cos()cos(

)cos(1)cos( (5.7)

where Zaxx and Zayy are the two components of the mechanical impedance of the

PZT patch in the two principal directions, derived in the same manner as in the 1D

impedance approach.

5.3 LIMITATIONS OF EXISTING MODELLING APPROACHES

The inability of the static approach in accurately modelling system

behaviour has already been pointed out in the previous section. Although Liang et

al. (1993, 1994) proposed more accurate formulations using impedance approach,

they however ignored the two-dimensional effects associated with PZT vibrations.

Their formulations are strictly valid for skeletal structures only, such as the test

frame described in Chapter 4. In other structures, where 2D coupling is significant,

Liang’s model might introduce serious errors. Zhou et al. (1995, 1996) addressed

this problem by extending Liang’s approach to planar vibrations, assuming a four-

parameter impedance model for the host structure (Eq. 5.4). Although the

analytical derivations (Eqs. 5.4-5.7) of Zhou et al. (1995, 1996) are accurate in

themselves, the experimental difficulties prohibit their direct application for

extraction of host structure’s mechanical impedance. For example, using the EMI

technique, we can only obtain two quantities- G and B (Eq. 5.5). If we need to

acquire complete information about the structure, we need to solve Eq. (5.5) for 4

complex unknowns- Zxx, Zyy, Zxy, Zyx (or 8 real unknowns). Thus, the system of

equations is highly indeterminate (8 unknowns with 2 equations only). As such,


111

the method could not be employed for experimental determination of the drive point

mechanical impedance.

To alleviate the shortcomings inherent in the existing models, a new concept

of ‘effective impedance’ is introduced in the next section, followed a step-by-step

derivation of electro-mechanical admittance across the PZT terminals. The new

formulations aim to bridge the gap between 1D model of Liang et al. (1993, 1994)

and the 2D model of Zhou et al. (1995, 1996).

5.4 DEFINITION OF EFFECTIVE MECHANICAL IMPEDANCE

Conventionally, the mechanical impedance at a point on the structure is

defined as the ratio of the driving harmonic force (acting on the structure at the

point in question) to the resulting harmonic velocity at that point. The existing

models are based on this definition, the point considered being the PZT end point.

The corresponding impedance is called the ‘drive point mechanical impedance’.

However, the true fact is that the mechanical interaction between the patch and the

host structure is not restricted at the PZT end points alone, rather it extends all over

the finite sized PZT patch.

This section introduces a new definition of mechanical impedance based on

‘effective velocity’ rather than ‘drive point velocity’. In the derivations that follow,

we assume that the force transmission between the PZT patch and the structure

occurs along entire boundary of the patch, and that plane stress conditions exist

within the patch. Besides, the patch is assumed square shaped and infinitesimally

small as compared to the host structure, so as to possess negligible mass and

stiffness. Opposite edges of the patch therefore encounter equal dynamic stiffness

from the structure, irrespective of the location of the patch on the host structure.

Hence, the nodal lines invariably coincide with the two axes of symmetry in the

PZT patch. At the same time, we ignore the effects of the PZT vibrations in the

thickness direction, assuming the frequency range of interest to be much lower than

the dominant modes of thickness vibration.

Consider a finite sized square PZT patch, surface bonded to an unknown

host structure, as shown in Fig. 5.3, subjected to a spatially uniform electric field

( )0// =∂∂=∂∂ yExE , undergoing harmonic variations with time. The patch has


112

half-length equal to ‘l’. Its interaction with the structure is represented in the form

of boundary traction ‘f’ per unit length, varying harmonically with time. This planar

force causes planar deformations in the PZT patch, leading to variations in its

overall area. The ‘effecive mechanical impedance’ of the patch is hereby defined as

effeff

Seffa u

Fu

dsnfZ

&&

r

==∫ ˆ.

, (5.8)

where n̂ is a unit vector normal to the boundary and ‘F’ represents the overall

planar force (or effective force) causing area deformation in the PZT patch.

ueff = δA/po is defined as ‘effective displacement’, where δA is the change in the

surface area of the patch and po its perimeter in the undeformed condition. More

precisely, po is equal to the summation of the lengths of ‘active boundaries’, i.e. the

boundaries undergoing mechanical interaction with the host structure.

Differentiation with respect to time of the effective displacement yields the

effective velocity, effu& . It should be noted that in order to ensure overall force

equilibrium,

∫ =S

dsf 0r

(5. 9)

The effective drive point (EDP) impedance of the host structure can also be defined

on similar lines. However, for determining structural impedance, force needs to be

Fig. 5.3 A PZT patch bonded to an ‘unknown’ host structure.

‘Unknown’ host structure

f (Interaction force at boundary)

PZT patch

E3

Boundary S

l l

l

l


113

applied on the surface of the host structure along the boundary of the proposed

location of the PZT patch.

5.5 ELECTRO-MECHANICAL FORMULATIONS BASED ON EFFECTIVE

IMPEDANCE

Consider a square PZT patch, as shown in Fig. 5.4, under in-plane

excitation, caused by a spatially uniform and harmonic electric field, with an

angular frequency ω. Since the nodal lines coincide with the axes of symmetry, it

suffices to consider the interaction of one quarter of the patch with the

corresponding one-quarter of host structure, since only the ratio of the two

mechanical impedances that will govern the electrical admittance across the

terminals of the PZT patch.

Let the patch be mechanically and piezoelectrically isotropic in the x-y

plane. Hence, EEE YYY == 2211 and 3231 dd = . Therefore, the PZT constitutive

relations (Eqs. 2.1 and 2.2) can be reduced to

)( 21313333 TTdED T ++= ε (5.10)

33121

1 EdY

TTSE

+−= ν (5.11)

Fig. 5.4 A square PZT patch under 2D interaction with host structure.

x

y

Nodal line

Nodal line

u1o

u2o

T1

T2

l

l

Area ‘A’


114

33112

2 EdY

TTSE

+−= ν (5.12)

where ν is the Poisson’s ratio of the PZT patch. By algebraic manipulation, we can

obtain

ν−

−+=+1

)2( 3312121

EYEdSSTT (5.13)

If the PZT patch is in short-circuited condition (i.e. zero electric field), Eq. (5.13)

can be reduced to

ν−

+=+ − 1)()( 21

21

E

circuitedshortYSSTT (5.14)

As derived by Zhou et al. (1996), the displacements of the PZT patch in the two

principal directions are given by

tjexAu ωκ )sin( 11 = and tjeyAu ωκ )sin( 22 = (5.15)

where the wave number κ is given by Eq. (5.6) and A1 and A2 are constants to be

determined from boundary conditions. The corresponding velocities can be

obtained by differentiating these equations with respect to time. Hence,

tjexjAtuu ωκω )sin( 1

11 =

∂∂=& and tjeyjA

tuu ωκω )sin( 2

22 =

∂∂=& (5.16)

Similarly, corresponding strains can be obtained by differentiation with respect to

the two coordinate axes. Hence,

tjexAxuS ωκκ )cos( 1

11 =

∂∂= and tjeyA

yuS ωκκ )cos( 2

22 =

∂∂= (5.17)

From Fig. 5.4, the effective displacement of the PZT patch, considering

displacements at the active boundaries of one-quarter of the patch (the boundaries

along the nodal axes are ‘inactive’ boundaries) can be deduced as

l

uululupAu oooo

oeff 2

2121 ++== δ ≈2

21 oo uu + (5.18)

where u1o and u2o are edge displacements, as shown in Fig. 5.4.

Differentiating with respect to time, we obtain the effective velocity as

22)(2)(121 lylxoo

eff

uuuuu == +=+=

&&&&& (5.19)


115

From Eqs. (5.8) and (5.19), we can obtain the short-circuited effective mechanical

impedance of the quarter PZT patch as

++

===

−==

2

)(

)(2)(1

)(2)(1,

lylx

circuitedshortlylxeffa uu

lhTlhTZ

&& (5.20)

Making use of Eq. (5.14), we obtain

or

+−

+=

==

==

2)1(

)(

)(2)(1

)(2)(1,

lxlx

Elylx

effa uuYlhSlhS

Z&&

ν (5.21)

Substituting the values of the velocities and strains (Eqs. 5.16 and 5.17 respectively)

at the two active edges of the PZT patch, and upon solving, we obtain

)1)((tan

2, νω

κ−

=klj

YlhZE

effa (5.22)

The overall planar force (or the effective force), F, is related to the EDP impedance

of the host structure by

effeffsS

uZdsnfF &,ˆ. −== ∫ (5.23)

As in the 1D case, negative sign signifies that a positive effective displacement

causes compressive force on the patch (due to reaction from the host structure).

Since we are considering a square patch, Eq. (5.23) can be simplified as

+−=+ ==

== 2)(2)(1

,)(2)(1lylx

effslylx

uuZhlThlT

&& (5.24)

Making use of Eq.(5.13), we get

+−=

−−+ ====

2)1()2( )(2)(1

,331)(2)(1 lylx

effs

Elylx uu

ZhlYEdSS &&

ν (5.25)

Substituting the expressions for ( 1u& + 2u& )x=l and (S1+S2) x=l from Eqs. (5.16) and

(5.17) respectively, and with E3 = (Vo/h)ejωt, we can derive

)()(cos2

,,

,3121

effaeffs

effao

ZZkhklZVd

AA+

=+ (5.26)


116

The electric displacement (or the charge density) over the surface of the PZT patch

can then be determined from Eq. (5.10). Substituting Eq. (5.13) into Eq. (5.10) and

with E3=(Vo/h)ejωt, we get

−+

−+= tjo

EtjoT e

hVdSSYde

hVD ωω

νε 3121

31333 2

)1( (5.27)

The instantaneous electric current, which is the time rate of change of charge, can

be derived as

dxdyDjdxdyDIAA∫∫∫∫ == 33 ω& (5.28)

Substituting D3 from Eq. (5.27) and S1 and S2 from Eq.(5.17), and integrating from

‘–l’ to ‘+l’ with respect to both ‘x’ and ‘y’, we obtain

+−+

−−=

ll

ZZZYdYd

hljVI

effaeffs

effaEE

T

κκ

ννεω

tan)1(

2)1(

24

,,

,231

231

33

2

(5.29)

where tjoeVV ω= is the instantaneous voltage across the PZT patch. Hence, the

complex electro-mechanical admittance of the PZT patch is given by

+−+

−−=+==

ll

ZZZYdYd

hljBjG

VIY

effaeffs

effaEE

T

κκ

ννεω tan

)1(2

)1(2

4,,

,231

231

33

2

(5.30)

which is the desired coupling equation for a square PZT patch. It should be noted

that a factor of 4 is introduced in the final expression, since ‘l’ represents half-

length of the patch. In the previous models (1D- Liang et al., 1994 and 2D- Zhou et

al., 1996), only one half and one-quarter of the PZT patch (from the nodal point to

the end of the patch) respectively were considered as the generic elements (See Fig.

5.2). The governing equations in those models (such as Eq. 5.5) correspond to one-

half and one-quarter of the patch only.

The main advantage of the present approach is that a single complex term

for Zs,eff accounts for the two dimensional interaction of the PZT patch with the host

structure. This makes the equation simple enough to be utilized for extracting the

mechanical impedance of the structure from Y , which can be measured at any

desired frequency using commercially available impedance analyzers. The related

computational procedure is presented in the sections to follow.


117

5.6 EXPERIMENTAL VERIFICATION

5.6.1 Details of Experimental Set-up

Fig. 5.5 shows the experimental test set-up to verify the new effective impedance

based electro-mechanical formulations. The test structure was an aluminum block,

48x48x10mm in size, conforming to grade Al 6061-T6. Table 5.1 lists major physical

properties of Al 6061-T6. The test block was bonded to a much larger and stiffer base

plate to simulate base support. The test block was instrumented with a PZT patch,

10x10x0.3mm in size, conforming to grade PIC 151 (PI Ceramic, 2003). Table 4.1 (page

100) lists the key properties of PIC 151. The patch was bonded to the host structure using

RS 850-940 epoxy adhesive (RS Components, 2003), and was wired to a HP 4192A

impedance analyzer (Hewlett Packard, 1996) via a 3499B multiplexer module (Agilent

Technologies, 2003). In this manner, the electro-mechanical admittance signature,

consisting of the real part (conductance- G) and the imaginary part (susceptance- B), was

acquired in the frequency range 0-200 kHz.

Table 5.1 Physical Properties of Al 6061-T6.



Young’s Modulus, EY11 (N/m2) 68.95 x 109

Poisson ratio 0.33

Fig. 5.5 Experimental set-up to verify effective impedance based new electro-

mechanical formulations.

10mm 48mm

N2260 multiplexer and 3499A/B switching box

Personal Computer

HP 4192Aimpedance analyzer

48mm

PZT patch 10x10x0.3mm

Host structure

Base Plate


118

5.6.2 Determination of Structural EDP Impedance by FEM

Before using Eq. (5.30) to derive theoretical signatures for comparison with

experimental signatures, we need to evaluate the effective mechanical impedance of

the PZT patch (Za,eff) as well as the EDP impedance of the structure (Zs,eff). Though

a closed form expression has been derived for Za,eff (Eq. 5.22), it is not possible to

derive such closed-form expression for Zs,eff, especially for complex structural

systems characterized by non-trivial 3D geometries. This holds true for most real-

life structures and systems where NDE is of prime importance. Hence, in this

research, a method based on 3D dynamic finite element analysis has been

developed to determine the EDP impedance of the host-structure. The main strength

of the FEM lies in its ability to accurately model real-life complex shapes and

boundaries. It should be noted that FEM is solely employed for verifying the new

impedance formulations derived above. In actual application of the formulations for

SHM, no finite element analysis is required, as will be illustrated in the later part of

this chapter.

The excitation of this smart system by a harmonic electric field is a typical

case of linear steady state forced vibrations. Investigations by Makkonen et al.

(2001) showed that fairly accurate results can be obtained for dynamic harmonic

problems by FEM, even for frequencies in the GHz range. In FEM, the physical

domain (such as the aluminum block) is discretized into elementary volumes called

elements. Fig. 5.6 shows the finitely discretized volume of the aluminum block.

Because of symmetry about the x and y axes, it suffices to perform computations

using only one quadrant of the actual structure. Appropriate boundary conditions

were imposed on the planes of symmetry, that is, the x and y components of

displacements were set to zero on the yz and the zx planes of symmetry

respectively. In addition, displacements of the bottom of the block were set to zero

to simulate bonding with the base plate. The finite element meshing was carried out

using the preprocessor tool of ANSYS 5.6 (ANSYS, 2000), with 1.0 mm sized

linear 3D brick elements (solid 45), possessing three degrees of freedom at each

node. Since the stiffness and the damping of the PZT patch are separately lumped in

the term Za,eff (Eqs. 5.22 and 5.30), we need not mesh the PZT element (Liang et al.,

1993).


119

In general, for a forced harmonic structural excitation, as in the present case,

Galerkin finite element discretization of the 3D domain leads to the following

differential equation (Zienkiewicz, 1977)

][]][[]][[]][[ FuKuCuM =++ &&& (5.31)

where [K] is the stiffness matrix, [M] the mass matrix, [C] the damping matrix, [F]

the force vector and [u] the displacement vector. The continuous field quantities i.e.

the mechanical displacements are approximated in each element through linear

sums of the interpolation functions or the shape functions (linear in the present

case). The natural boundary conditions are included in the load vector, and the

essential boundary conditions are imposed by adjusting the load vector and the

stiffness matrix (Bathe, 1996).

The simplest approach to determine the EDP impedance of the host structure

is to apply an arbitrary harmonic force (at the desired frequency) on the surface of

the structure (along the boundary of the PZT patch), perform dynamic harmonic

analysis by FEM, and obtain the complex displacement response at those points.

The applied mechanical load can be expressed as

Fig. 5.6 Finite element model of one-quarter of test structure.

24 mm

Boundary of PZT patch

Displacements in y-direction = 0

x

y z

Displacement in x-direction = 0

10 mm

24 mm

B A

C D

Origin of coordinate system

O


120

tjejFFF ω][][ 21 += (5.32)

The resulting displacements, which are also harmonic functions of time (at same

frequency as the loads) can be similarly expressed as

tjejuuu ω][][ 21 += (5.33)

Complex notation is employed here to account for the phase lag caused by the

‘impedance’ of the system (Zienkiewicz, 1977). Substituting Eqs. (5.32) and (5.33)

into Eq. (5.31) and noting that ][][ uju ω=& , ][][ 2 uu ω−=&& , we obtain

{ ][][][ 2 MCjK ωω −+ } ][][ 2121 jFFjuu +=+ (5.34)

which can be written in a form similar to the static analysis as

][]*][[ FuA = (5.35)

The only difference from the static case being that all the terms are complex. Eq.

(5.35) can be decomposed into two coupled equations involving real numbers only,

and can be written as

=

+−−+−

2

1

2

12

2

][][][][][][

FF

uu

KMCCKM

ωωωω

(5.36)

This set of equations can be solved to obtain the displacement components u1 and

u2. This solution method is called the full solution method. Reduced solution

method (Makkonen et al., 2001) is another approach but it is not as accurate as the

full solution method employed presently. It should be noted that computing the

frequency response requires the solution of the FEM equations at each desired

frequency throughout the range of interest.

If the boundary of the PZT patch consists of N equal divisions on each

adjacent edge (N = 5 in the present case, as shown in Fig. 5.6) , we can obtain

effective displacement as

oeff p

Au δ= (5.37)

Substituting expression for δA and po, we get

lNluu

Nluu

Nluu

Nluu

Nluu

uyNNyyyxNNxxxxx

eff2

)(21...)(

21)(

21...)(

21)(

21

)1(21)1(3221

+++++

++++++

=++ (5.38)

Solving, )(21

,, yeffxeffeff uuu += (5.39)


121

where

Nuuuuu

u NxxxxNxxeff

)..()(5.0 32)1(1,

+++++= + (5.40)

Nuuuuu

u NyyyyNyyeff

)..()(5.0 32)1(1,

+++++= + (5.41)

Further, by splitting the real and the imaginary terms we can alternatively write,

)(21

,, juuu ieffreffeff += (5.42)

We can then obtain the EDP structural impedance from Eq. (5.8), noting that

effeff uju ω=& . If a uniformly distributed planar force, with an effective magnitude

jFFF ir += is applied, from Eqs. (5.8) and (5.42), the EDP impedance of the host

structure can be derived as

( ) ( )j

uuuFuF

uuuFuF

juujjFFZ

ieffreff

ieffireffr

ieffreff

ieffrreffi

ieffreff

ireffs

++

−

+−

=+

+=)()(

2)(

22

,2

,

,,2

,2

,

,,

,,, ωωω

(5.43)

We can simplify the computations by applying a purely real force (Fi = 0), in which

case, the effective impedance will be given by

juu

uFuu

uFZ

ieffreff

reffr

ieffreff

ieffreffs

+−

+−=

)(2

)(2

2,

2,

,2

,2

,

,, ωω

(5.44)

This procedure enables the determination of the EDP structural impedance using

any commercial FEM software, without any adjustment or warranting the inclusion

of electric degrees of freedom in the finite element model.

5.6.3 Modelling of Structural Damping

In most commercial FEM software, the damping matrix is determined from

the stiffness and the mass matrices as

][][][ KMC βα += (5.45)

where α is the mass damping factor and β is the stiffness-damping factor. This type

of damping is called Rayleigh damping. Further simplification can be achieved by

defining damping as a function of the stiffness alone

=ωη

][C [K] (5.46)


122

Then, after substituting in Eq. (5.34), this simplification renders the stiffness matrix

complex, as defined by

])[1(][ KjK η+= (5.47)

where η is called the mechanical loss factor of the material. Its equivalent Rayleigh

damping coefficients are α = 0 and ωηβ /= . This type of damping is frequency

independent. The present analysis considered α = 0 and β = 3 x 10-9, resulting in η

≈ 0.002 on an average for the frequency range considered.

5.6.4 Wavelength Analysis and Convergence Test

In dynamic harmonic problems, in order to obtain accurate results, a

sufficient number of nodal points (3 to 5) per half wavelength should be present in

the finite element mesh (Makkonen et al., 2001). In order to ensure this

requirement, modal analysis was additionally performed. The frequency range of

0-200 kHz was found to contain a total of 24 modes. The modal frequencies are

listed in Table 5.2, computed for four different element sizes- 2mm, 1.5mm, 1mm

and 0.8mm. It can be observed from the table that good convergence of the modal

frequencies is achieved at an element size of 1mm (which is the element size used

in the present analysis). Thus, fairly accurate results are expected from the present

analysis using FEM. In addition, Figs. 5.7(a), 5.7(b) and 5.7(c) respectively show

the plots of the displacements ux, uy and uz, corresponding to the 24th mode (the

highest excited mode), over the top surface of the block (z = 10mm). Also, the

displacements in the three principal directions are plotted for the edge AB (see Fig.

5.6) to illustrate that there are sufficient number of nodes per half wavelength so as

to ensure adequate accuracy of the analysis.

5.6.5 Comparison Between Theoretical and Experimental Signatures

Using the EDP structural impedance obtained by FEM, as described in the

preceding sub-sections, the admittance functions were derived using Eq. (5.30). The

values of T33ε and δ for the PZT patch were determined experimentally. The

Poisson’s ratio of the patch was assumed as 0.3. A MATLAB program listed in

Appendix C was used to perform computations. Fig. 5.8 shows a comparison


123

between the experimental and the theoretical signatures, based on the proposed

approach as well as that based on the model of Zhou et al. (1995, 1996). The

prediction by the present method is quite close to that by Zhou and coworkers’

model. However, the present formulations are much easier to apply than the

approach of Zhou et al. (1995, 1996), as evident from the very complex nature of

the governing equation (Eqs. 5.5-5.7) in the approach of Zhou et al. (1995, 1996).

It is observed that reasonably good agreement exists between the

experimental and the theoretical plots of the real part- the conductance, predicted by

the proposed model (Fig. 5.8a). Major peaks are accurately predicted, though the

experimental spectrum contains few unpredicted peaks (mainly due to edge

roughness and due to the inability of FEM to accurately model solid-air interactions

at the boundaries). However, in the plots of the susceptance (Fig. 5.8b), large

discrepancy is clearly evident, especially the difference in slopes of the curves. This

discrepancy is attributed to the deviation of the PZT behavior from the ideal

Table 5.2 Details of modes of vibration of test structure.

MODAL FREQUENCY (kHz) MODE

2mm 1.5mm 1mm 0.8mm

DESCRIPTION OF MODE

1 81.710 81.480 81.320 81.256 Thickness shear (diagonal) 2 89.354 89.105 88.944 88.884 Face shear 3 90.991 90.765 90.610 90.547 Thickness shear (diagonal) 4 106.667 106.335 106.101 106.016 Face Shear + Flexure 5 125.847 125.125 124.623 124.464 Thickness flexure 6 139.579 138.916 138.521 138.367 Bending about diagonal 7 139.910 139.227 138.845 138.691 Bending about diagonal + Rotation 8 142.425 141.406 140.745 140.525 Thickness Flexure 9 146.653 145.852 145.420 145.249 Flexure 10 148.645 148.017 147.624 147.484 Flexure 11 150.387 149.511 149.000 148.801 Flexure 12 156.807 155.576 154.882 154.623 Flexure 13 157.744 156.706 156.119 155.905 Flexure+Thickness extension 14 165.482 164.333 163.660 163.417 Flexure 15 168.217 166.960 166.207 165.941 Flexure 16 176.823 174.370 172.701 172.186 Thickness flexure 17 181.411 180.035 179.145 178.841 Flexure 18 183.001 181.943 181.222 180.984 Flexure 19 185.590 183.573 182.242 181.808 Flexure 20 191.910 189.760 188.364 187.902 Flexure 21 192.133 190.116 188.776 188.345 Flexure 22 195.335 193.208 191.869 191.424 Flexure 23 196.805 194.432 192.986 192.519 Flexure 24 200.887 199.026 197.845 197.457 Flexure


124

-15-10-505

10152025

0 4 8 12 16 20 24

Distance along edge (mm)

Nor

mal

ized

dis

plac

emen

t Displacement in x direction Displacement in y direction Displacement in z direction

(a) (b)

Fig. 5.7 Examination of mode 24 to check adequacy of mesh size of 1mm (a) Displacements in x direction on surface z = 10mm. (b) Displacements in y direction on surface z = 10mm. (c) Displacements in z direction on surface z = 10mm. (d) Displacements in principal directions along the line defined by the

intersection of surfaces y = 24mm and z = 10mm (see Fig. 5.6).

(d)

x (mm)

ux

y (mm) x (mm)

uy

y (mm)

x (mm)

uz

y (mm)

(c)


125

behavior predicted by Eq. (5.22). Besides, many parameters of the PZT patch could

deviate from the values provided by the manufacturer. Fortunately, we had obtained

the admittance signatures of the PZT patch in ‘free-free’ condition prior to bonding

it on the structure. Hence, it was possible to investigate the behavior of free PZT

Fig. 5.8 Comparison between experimental and theoretical signatures.

(a) Conductance plot. (b) Susceptance plot.

(a)

(b)

Theoretical (Model of Zhou et al)

Experimental

Theoretical (Proposed model)

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

0 40 80 120 160 200

Frequency (kHz)

G (S

)

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

0 40 80 120 160 200

Frequency (kHz)

B (S

)


126

patch and use this information to obtain more accurate plots. The next section

describes the investigations in detail.

5.7 REFINING THE MODEL OF PZT SENSOR-ACTUATOR PATCH

The properties of piezoceramics are strongly dependent upon the process

route and exhibit statistical fluctuations within a given batch (Giurgiutiu and Zagrai,

2000). The fluctuations are caused by inhomogeneous chemical composition,

mechanical differences in the forming process, chemical modification during

sintering and the polarization method (Sensor Technology Ltd., 1995). A variance

of the order of 5-20% in properties is not uncommon. In the EMI technique, we

solely depend upon PZT patches to predict the mechanical impedance spectra of the

structures. Hence, it is very important to accurately model the behavior of the PZT

patches when using the formulations derived in the previous sections. For this

purpose, it is recommended that the signatures of the PZT patches be recorded in

the ‘free-free’ condition prior to their bonding on to the host structure.

Looking back at Eq. (5.30), for a free (unbonded) PZT patch, the complex

electro-mechanical admittance can be derived (by substituting Zs,eff = 0 and and

simplifying) as

−

−+= 1tan

)1(2

4231

33

2

llYd

hljY

ET

freeκ

κν

εω (5.48)

Substituting )1( jYY EE η+= , )1(3333 jTT δεε −= , tjrl

l +=κ

κtan and fπω 2= (‘f’

being the frequency of vibrations in Hz), and simplifying we get

jBGY fffree += (5.49)

where

{ }

+−

−−= tr

YdhflG

ET

f )1()1(

28 231

33

2

ην

δεπ (5.50)

{ }

−−

−+= tr

YdhflB

ET

f ην

επ )1()1(

28 231

33

2

(5.51)


127

Further, under very low frequencies (typically < one-fifth of the first resonance

frequency of the PZT patch), 1tan →l

lκ

κ(i.e. r→1, t→0) (Liang et al., 1993, 1994),

thereby leading to quasi-static sensor approximation (Giurgiutiu and Zagrai, 2002)

hflG

T

qsfδεπ 33

2

,8= (5.52)

hflB

T

qsf33

2

,8 επ= (5.53)

Rearranging the various terms, Eqs. (5.52) and (5.53) can be rewritten as

fl

hGG T

qsfqsf δ

επ==

332,*

, 8 (5.54)

fl

hBB Tqsf

qsf 332,*

, 8ε

π== (5.55)

From Eqs. (5.54) and (5.55), we can determine the electrical constants T33ε and δ as

the slopes of the frequency plots of *,qsfB (unit S/m) and *

,qsfG (unit S/F) for

sufficiently low frequencies (typically < 10 kHz for 10mm long PZT patches). Figs.

5.9 (a) and (b) respectively show the typical plots of these functions in the

frequency range 0-10 kHz for two PZT patches labelled as S2002-5 and S2002-6.

Fig. 5.9 Plots of quasi-static admittance functions of free PZT patches to

obtain electric permittivity and dielectric loss factor.

(a) *,qsfB vs frequency. (b) *

,qsfG vs frequency.

0

100

200

300

400

0 2000 4000 6000 8000 10000

f (Hz)

S2002-5

S2002-6

Gf,q

s ( S

/F )

*

0

0.00005

0.0001

0.00015

0.0002

0 2000 4000 6000 8000 10000f (Hz)

S2002-5

S2002-6

Bf,q

s ( S

/m )

*

(a) (b)


128

Patch S2002-5 was used as piezo-impedance transducer in the experiment described

in the previous section. From these plots, T33ε was worked out to be 1.7919x10-8 F/m

and 1.7328x10-8 F/m respectively for S2002-5 and S2002-6, against a value of

2.124x10-8 F/m supplied by the manufacturer). Similarly, δ was worked out to be

0.0238 and 0.0225 respectively, against a value of 0.015 supplied by the

manufacturer.

Using the values of the PZT parameters obtained above, free conductance

and susceptance signatures of the PZT patches s2002-5 and s2002-6 were obtained

in the ‘free-free’ condition in the frequency range 1-1000 kHz, using Eqs. (5.50)

and (5.51) respectively. These are shown in Fig. 5.10 and compared with the

experimental free PZT signatures. Although a quick look at the figures suggests

reasonable agreement between the analytical and the experimental signatures, there

are some underlying discrepancies, which need closer examination. A close look in

frequency range 0-300kHz (Figs. 5.10a and 5.10c) shows an unpredicted mode at

around 240kHz. In the case of S2002-5 (Fig. 5.10a), twin peaks are observed in the

experimental spectra around each of the prominent resonance frequencies.

Besides, a general observation is that the experimental resonance frequency is

slightly higher than the theoretical frequency.

The twin peaks are due to the deviation in the shape of the PZT patch from

perfect square shape during manufacturing. This leads to somewhat partly

independent resonance peaks corresponding to the two slightly unequal edge

lengths. The unpredicted modes in the admittance spectra are due to edge roughness

induced secondary vibrations. Somewhat higher experimental natural frequency

suggests additional 2D stiffening, which is unaccounted for in the present model. A

similar comparison was reported by Giurgiutiu and Zagrai (2000, 2002), but

considering 1D vibrations only. They assumed the patch to possess widely

separated values for length, width and thickness so that length, width and thickness

vibrations are practically uncoupled. Their analytical predictions matched the

experimental results only for aspect ratios higher than 2.0 only.

Presently, the frequency range of interest is 0-200 kHz. The unpredicted

mode does not come into picture in this frequency range. In order to further

‘update’ the model of the PZT patch with respect to peaks, a correction factor is


129

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

0 200 400 600 800 1000

Frequency (kHz)

G (S

)

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

0 200 400 600 800 1000

Frequency (kHz)

G (S

) 120

140

160

180

200

(a) (b)

(c) (d)

Fig. 5.10 Experimental and analytical plots of free PZT signatures.

(a) S2002-5: Conductance (G) vs Frequency.

(b) S2002-5: Susceptance (B) vs Frequency.

(c) S2002-6: Conductance (G) vs Frequency.

(d) S2002-6: Susceptance (B) vs Frequency.

-4.00E-02

-2.00E-020.00E+00

2.00E-02

4.00E-026.00E-02

8.00E-02

0 200 400 600 800 1000

Frequency (kHz)

B (S

)

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

0 200 400 600 800 1000

Frequency (kHz)

B (S

)120

140

160

180

200

Analytical Experimental

Twin peaks

Unpredicted mode

Unpredicted mode


130

introduced in the term ( ll κκ /tan ). In the case of PZT patch S2002-5, where twin

peaks are observed, this term may be replaced by

+

lClC

lClC

κκ

κκ

2

2

1

1 )tan()tan(21

By trial and error, values of C1 = 0.94 and C2 = 0.883 were found to update the

model of the PZT patch. Further, following values of the PZT parameters were

determined from the experimental plot using the techniques of curve fitting.

9231 1016.5

)1(2 −=

−= xYdK

E

ν NV-2 and η = 0.03

The value of K based on data supplied by the manufacturer is determined as

8.4x10-9 NV-2. Using these values and the correction factors C1 and C2, the free PZT

signatures were again worked out in the frequency range 0-200 kHz. Figs. 5.11(a)

and 5.11(b) compare the updated signatures with the experimental signatures. A

very good agreement is observed between the two.

Similarly, for the PZT patch S2002-6, a coefficient C =0.885 was found,

such that the term )/(tan ll κκ , when replaced by [ lClC κκ /)tan( ] yielded a good

agreement between the experimental and the analytical plots of free PZT signatures.

Further, K was computed to be 4.63x10-9 NV-2 and η again worked out to be 0.03.

Figs. 5.11(c) and 5.11(d) compare the analytical and the experimental plots. Again,

a good agreement is observed between the experimental signatures and the

signatures using the updated PZT model.

Hence, considering the necessity of updating the model of the PZT patch,

Eq. (5.30) can be modified as

+−+

−−=+= T

ZZZYdYd

hljBjGY

effaeffs

effaEE

T

,,

,231

231

33

2

)1(2

)1(2

4νν

εω (5.56)

where the term T is the complex tangent ratio (ideally tanκl/κl), which can be

expressed as


131

lC

lCκ

κ )tan( for single-peak behaviour.

=T (5.57)

+

lClC

lClC

κκ

κκ

2

2

1

1 tantan21

for twin-peak behaviour.

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

0 25 50 75 100 125 150 175 200 225

Frequency (kHz)

G (S

)

(a)

Fig. 5.11 Plots of free-PZT admittance signatures using an updated PZT model.

(a) S2002-5: Conductance (G) vs Frequency.

(b) S2002-5: Susceptance (B) vs Frequency.

(c) S2002-6: Conductance (G) vs Frequency.

(d) S2002-6: Susceptance (B) vs Frequency.

(b)

(c) (d)

0.000001

0.00001

0.0001

0.001

0.01

0.1

0 25 50 75 100 125 150 175 200 225

Frequency (kHz)

G (S

)

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

6.00E-02

100 125 150 175 200 225

Frequency (kHz)

B (S

)

Experimental Analytical

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

0 25 50 75 100 125 150 175 200 225Frequency (kHz)

B (S

)


132

Further, the corrected actuator effective impedance (earlier expressed by Eq. 5.22)

can be written as

Tj

YhZE

effa )1(2

, νω −= (5.58)

As mentioned before, PZT patch S2002-5 was the one bonded to the host

structure shown in Fig. 5.5. The theoretical signatures for this test structure (with

the PZT patch bonded on the surface) were again worked out using the updated PZT

model (Eqs. 5.56, 5.57 and 5.58). A MATLAB program listed in Appendix D was

used to perform computations. Fig. 5.12 compares the theoretical signatures based

on the proposed effective impedance based model (after updating PZT model) and

the experimental signatures. This time, a much better agreement is found between

the two.

Fig. 5.13(a) compares the idealized and the corrected effective impedance

for the PZT patch S2002-5. The influence of twin peaks is clearly reflected in the

plot of the updated impedance. If we were to solely depend upon the idealized

model of PZT patch to identify the structure, significant errors could have been

introduced, as can be clearly observed in Fig. 5.13(b), which shows the plot of

|Zs,eff|-1. Further, Fig. 5.13 shows the plots of |Zs,eff| and |Za,eff| derived

experimentally to illustrate that the present system satisfies the criteria |Zs,eff| >

|Za,eff|. Hence, this case falls in the category of Case II described in Chapter 3.

It should be noted that Giurgiutiu and Zagrai (2002) also evaluated the

electro-mechanical admittance across PZT terminals using analytical and numerical

methods. However, they could only model very simple structures, such as thin

beams, under extremely simple boundary conditions (such as ‘free-free’). There

were orders of magnitude of error between the experimental and the analytical

impedance spectra. The present work, on the other hand, is more general in nature

and is valid for all types of structures, whether 2D or 3D. The agreement between

the analytical and experimental results is also much better as compared to the

results of the previous researchers. The present work, which is a semi-analytical

approach (numerical + analytical) is the first attempt to compare theoretical

modelling for 3D structures with experimental data for such high frequencies.


133

Fig. 5.12 Comparison between experimental and analytical signatures based

on updated PZT model.

(a) Conductance (G) vs frequency. (b) Susceptance (B) vs Frequency.

(b)

(a)

Experimental Theoretical

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

0 40 80 120 160 200

Frequency (kHz)

G (S

)

0

0.002

0.004

0.006

0.008

0 40 80 120 160 200

Frequency (kHz)

B (S

)

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

134

5.8 DECOMPOSITION OF COUPLED ELECTRO-MECHANICAL

ADMITTANCE

As in the case of 1D impedance model (covered in Chapters 2 to 4), the electro-

mechanical admittance (given by Eq. 5.56) can be decomposed into two

components as

−−=

)1(2

4231

33

2

νεω

ET Yd

hljY + jT

ZZZ

hlYd

effaeffs

effaE

+− ,,

,22

31

)1(8

νω

(5.59)

Part I Part II

1

10

100

1000

10000

0 50 100 150 200 250Frequency (kHz)

Effe

ctive

Impe

danc

e (N

s/m)

0

0.002

0.004

0.006

0.008

0.01

0 25 50 75 100 125 150 175 200Frequency (kHz)

|Z|-1

(mN-1

s-1)

Fig. 5.13 (a) PZT effective impedance, based on idealized and updated models.

(b) Error in extracted structural impedance in the absence of updated

PZT model.

(c) Relative magnitudes of structure and PZT impedances.

Based onupdated PZTmodel

Based on idealized PZT model

Based on idealized PZT model

Based on updatedPZT model

1

10

100

1000

10000

100000

0 50 100 150 200

Frequency (kHz)

|Zs,

eff|,

|Za,

eff|

(Ns/

m)

Structure

PZT patch

(a) (b)

(c)


135

It can be observed that the first part solely depends on the parameters of the PZT

patch and is independent of the host structure. The structural parameters make their

presence felt in part II only, in the form of the EDP structural impedance, Zs,eff.

Therefore, Eq. (5.59) can be written as

AP YYY += (5.60)

where AY is the ‘active’ component and PY the ‘passive’ component. PY can be

broken down into real and imaginary parts by expanding )1(3333 jTT δεε −= and

)1( jYY EE η+= and can be expressed as

jBGY PPP += (5.61)

where { }ηδεω KhlG T

P += 33

24 (5.62)

{ }KhlB T

P −= 33

24 εω (5.63)

and)1(

2 231

ν−=

EYdK (5.64)

We can predict GP and BP with reasonable accuracy if we record the conductance

and the susceptance signatures of PZT patch in ‘free-free’ condition, prior to its

bonding to the host structures, as demonstrated in section 5.7. Hence, the PZT

contribution can be filtered off from the raw signatures and the active component

deduced as

PA YYY −= (5.65)

or )()( jBGBjGY PPA +−+= (5.66)

Thus, the active components (GA and BA) can be derived from the measured raw

admittance signatures (G and B) as

PA GGG −= (5.67)

and PA BBB −= (5.68)

In the complex form, we can express the active component as

jTZZ

Zh

lYdjBGY

effaeffs

effaE

AAA

+−=+=

,,

,22

31

)1(8

νω

(5.69)


136

It was demonstrated in Chapter 4, using 1D interaction model, that the

elimination of the passive component renders the admittance signatures more

sensitive to structural damages. The same holds true for the 2D PZT-structure

interaction considered in this chapter. Therefore, it is more pragmatic to employ

active components rather than raw signatures for SHM and NDE.

5.9 EXTRACTION OF STRUCTURAL MECHANICAL IMPEDANCE

Chapter 4 outlined a computational procedure for extracting 1D drive point

mechanical impedance of skeletal structures using the active admittance signatures

of surface-bonded piezo-impedance transducers. This section outlines the

corresponding procedure for the more general class of structures, based on the new

electro-mechanical admittance formulations.

Substituting )1( jYY EE η+= and tjrT += into Eq. (5.69), and rearranging

the various terms, we obtain

)(,,

, SjRZZ

ZNjM

effaeffS

effa +

+=+ (5.70)

where 24 KlhBM A

ω= and 24 Kl

hGN A

ω−= (5.71)

trR η−= and rtS η+= (5.72)

Further, expanding yjxZ effS +=, and jyxZ aaeffa +=, , and upon solving, we can

obtain the real and imaginary components of the EDP structural impedance as

aaaaa x

NMRySxNSyRxMx −

+++−

= 22

)()( (5.73)

aaaaa y

NMSyRxNRySxMy −

+−−+

= 22

)()( (5.74)

In all these computations, the term T (which plays a significant role), depends

upon (tanκl/κl) (see Eqs. 5.56-5.58), where κl is a complex number. It is essential to

determine this quantity precisely, by the procedure outlined in Chapter 4. Further,

it should again be noted that |x| > |xa| and |y| > |ya| in order to ensure smooth

computations. Else, the extracted impedance spectra might exhibit false peaks.

The simple computational procedure outlined above results in the

determination of the drive point mechanical impedance of the structure,


137

Zs,eff = x + yj, at a particular frequency ω, from the active admittance signatures.

Following this procedure, ‘x’ and ‘y’ can be determined for the entire frequency

range of interest. This procedure was employed to extract the structural EDP

impedance of the test aluminium block (used for validating the new impedance

model in sections 5.6 and 5.7). A MATLAB program listed in Appendix E was used

to perform the computations. In the present case, |Z| > |Za|, as apparent from Fig.

5.13(c). Fig. 5.14 shows a plot of |Zeff|-1, worked out by this procedure, comparing it

with the plot determined using FEM, as discussed in the preceding sections.

Reasonable agreement can be observed between the two. The main reason for

plotting |Zs,eff|-1 (instead of Zs,eff) is that the resonant frequencies can be easily

identified as peaks of the plot.

As will be demonstrated in the forthcoming sections, this procedure enables

us to ‘identify’ any unknown structure without demanding any a-priori information

governing the phenomenological nature of the structure. The only requirement is an

‘updated’ model of the PZT patch, which can be derived from preliminary

specifications of the PZT patch and by recording its admittance signatures in the

‘free-free’ condition, prior to bonding to the host structure. It was demonstrated in

Chapter 4 that the utilization of ‘x’ and ‘y’ (rather than raw signatures) leads not

only to higher damage sensitivity but also facilitates greater insight into the

mechanism associated with structural damage. The next section will present a

simple procedure to derive system parameters from the structural EDP impedance.

0.0001

0.001

0.01

0.1

0 40 80 120 160 200

Frequency (kHz)

|Zef

f|-1(m

/Ns)

Experimental

Numerical

Fig. 5.14 Comparison between |Zeff|-1 obtained experimentally and numerically.


138

5.10 SYSTEM PARAMETER IDENTIFICATION FROM EXTRACTED

IMPEDANCE SPECTRA

The structural EDP impedance, extracted by means of the procedure

outlined in the previous section, carries information about the dynamic

characteristics of the host structure. In Chapter 4, the host structure (1D skeletal

structure) was idealized as a parallel combination of a resistive element (damper)

and a reactive element (stiffness-mass factor). The extracted structural parameters

were employed in evaluating structural damages.

This section presents a more general approach to ‘identify’ the equivalent

structural system. Before considering any real-life structural system for this

purpose, it would be a worthwhile exercise to observe the impedance pattern of few

simple systems. Fig. 5.15 shows plots of the real and the imaginary components of

the mechanical impedance of basic structural elements- the mass, the spring and the

damper. These basic elements can be combined in a number of different ways

(series, parallel or a mixture) to evolve complex mechanical systems. Table 5.3

shows the impedance plots (x, y vs frequency) for some possible combinations of

the basic elements (Hixon, 1988). In general, for any real-life structure, the two

components (real and imaginary) of the extracted EDP impedance may not display

an ideal behavior, such as pure mass or pure stiffness or pure damper. Both the

‘resistive’ and the ‘reactive’ terms might vary with frequency, similar to a

combination of the basic elements. The ‘unknown’ structure can thus be idealized

as an ‘equivalent’ structure (series or parallel combination of basic elements), and

the equivalent system parameters can thereby be determined.

Fig. 5.15 Impedance plots of basic structural elements- spring, damper and mass.

(a) Real part (x) vs frequency. (b) Imaginary part (y) vs frequency.

(b)(a)

y

Mass

Spring

Damper

Frequency

0

0

x

Spring, mass

Damper

Frequency


139

Table 5.3 Mechanical impedance of combinations of spring, mass and damper.No. COMBIN—

ATIONx y x vs Freq. Y vs Freq.

1 cωk

−

2 c ωm

3 0ω

ω km −

4 cω

ω km −

522

1

)( −−

−

+ mccω

22

1

)()(

−−

−

+ mcmω

ω

6 01)()/(

1−−

−mk ωω

722

1

)/1/( mkcc

ωω −+−

−

22 )/1/()/1/(mkcmkωωωω

−+−−

−

8 cmk

mk2ω

ω−

922

1

)( −−

−

+ mccω [ ]22

2221

)()(

−−

−−−−

++−mcmckm

ωωω

1022

1

)/( kccω+−

− [ ]22

1222

)/()(kc

kkcmωωω

+−+

−

−−−

1122

22

)/( ωωωkmc

cm−+

22

2

)/(

)/(

ωω

ωωω

ω

kmc

kmkcm

−+

−−

12[ ]222

1

)/( ωω mkcc−+−

−

[ ]222

2

)/()/(

ωωωωmkcmk−+

−−−

1322

22

)/(/

ωωωkmc

ck−+

22

2

)/(

)/(

ωωω

ωω

kmcmkckmkm

−+

+−−

0

0

0 0

0 0

0

0

0 0

0 0

00

0

0

0

0

0

0

0

0

0

0

0

0


140

To demonstrate this approach, let us consider another aluminum (grade Al

6061-T6) block, 50x48x10mm in size, representing an unknown structural system.

The PZT patch S2002-6 (10x10x0.3mm in size), whose updated model was derived

in section 5.7, was bonded to the surface of this specimen. Experimental set-up

similar to that shown in Fig. 5.5 was employed to acquire the raw admittance

signatures (conductance and susceptance) of this PZT patch. The passive

components were filtered off from the raw signatures and the structural EDP

impedance was extracted out, using the MATLAB program listed in Appendix E

(considering the parameters of patch S 2002-6 derived experimentally).

A close examination of the extracted impedance components in the

frequency range 25-40 kHz suggested that the system behavior was similar to a

parallel spring-damper (k-c) combination (system 1 in Table 5.3). For this system,

cx = and ωky −= (5.75)

Using Eq. (5.75) and the actual impedance plots, the average “equivalent” system

parameters were worked as: c = 36.54 Ns/m and k = 5.18x107 N/m. The analytical

plots of ‘x’ and ‘y’ obtained by these equivalent parameters match well with their

experimental counterparts, as shown in Fig. 5.16.

0

50

100

150

200

25 30 35 40

Frequency (kHz)

x (N

s/m

)

Fig. 5.16 Mechanical impedance of aluminium block in 25-40 kHz frequency range.

The equivalent system plots are obtained for a parallel spring-damper combination.


(b)(a)

Experimental Equivalent system

-350

-300

-250

-200

-150

25 30 35 40

Frequency (kHz)

y (N

s/m

)


141

Similarly, in the frequency range 180-200 kHz, the system behavior was

found to be similar to a parallel spring-damper (k-c) combination, in series with

mass ‘m’ (system 11 in Table 5.3). For this combination,

22

22

−+

=

ωω

ωkmc

cmx and 22

2

−+

−−

=

ωω

ωω

ωω

kmc

kmkcmy (5.76)

and the peak frequency of the x-plot is given by

kcm

ko 2

−=ω (5.77)

If x = xo (the peak magnitude) at ω = ωo and x = x1 (somewhat less than the peak

magnitude) at ω = ω1 (<ωo), using Eqs. (5.76) and (5.77), the system parameters can

be determined, by algebraic manipulations, as2/1

2

22

−±−=

AACBBm (5.78)

222

22

oo

oo

mxxmcω

ω+

= (5.79)

mcxk o= (5.80)

where

)( 141

4oo xxA −= ωω (5.81)

341

21

221

4221

221

21 )(2 ooooooo xxxxxB ωωωωωωωω −+−= (5.82)

14222

1 )( xxC ooωω −= (5.83)

A set of system parameters c = 1.1x10-3 Ns/m, k = 4.33 x 105 N/m and m =

3.05 x 10-7 kg produced similar impedance pattern, as shown in Fig. 5.17. Further

refinement was achieved by adding a spring K* = 7.45x107 N/m and a damper C* =

12.4 Ns/m in parallel, to make the equivalent system appear as shown in Fig. 5.18.

Hence, Eq. (5.76) may be refined as


142

22

22*

−+

+=

ωω

ωkmc

cmCx and ω

ωω

ωω

ωω *

22

2

Kkmc

kmkcmy −

−+

−−

= (5.84)


Fig. 5.17 Mechanical impedance of aluminium block in 180-200 kHz frequency

range. The equivalent system plots are obtained for system 11(Table 5.3).

(a) Real part. (b) Imaginary part.

0

50

100

150

180 185 190 195 200

Frequency (kHz)

x (N

s/m

)

-150

-100

-50

0

50

100

180 185 190 195 200

Frequency (kHz)

y (N

s/m

)

(b)(a)

Fig. 5.18 Refinement of equivalent system by introduction of additional

spring K* and additional damper C*.

k

cm

K*

C*


143

Fig. 5.19 shows the comparison between the experimental plots with the analytical

plots for this equivalent system. Extremely good agreement can be observed

between the plots obtained experimentally and those pertaining to the equivalent

system. Hence, the structural system is identified with a reasonably good accuracy.

The next section explains how this methodology can be used to evaluate damages

aerospace and mechanical structures.

5.11 DAMAGE DIAGNOSIS IN AEROSPACE AND MECHANICAL

SYSTEMS

This section describes a damage diagnosis study, carried out on the

aluminum block (50x48x10mm in size), identified using a piezo-impedance

transducer, as described in the previous section. This is a typical small-sized rigid

structure, characterized by high natural frequencies in the kHz range. Many critical

aircraft components such as turbo engine blades are small and rigid, and are

(b)(a)

0

50

100

150

180 185 190 195 200

Frequency (kHz)

x (N

s/m

)

-150

-100

-50

0

50

180 185 190 195 200

Frequency (kHz)

y (N

s/m

)


Fig. 5.19 Mechanical impedance of aluminium block in 180-200 kHz frequency

range for refined equivalent system (shown in Fig. 5.18)

(a) Real part. (b) Imaginary part.


144

characterized by typically high natural frequencies in the kHz range (Giurgiutiu and

Zagrai, 2002), and hence exhibit similar dynamic behavior.

Damage was induced in this test structure by drilling holes, 5mm in

diameter, through the thickness of the specimen. Three different levels of damage

were induced- incipient, moderate and severe, as shown in Figs. 5.20(b), 5.20(c),

and 5.20(d) respectively. The number of holes was increased from two to eight in

three stages, so as to simulate a gradual growth of damage from the incipient level

to the severe level. After each damage, the admittance signatures of the PZT patch

were recorded and the equivalent structural parameters were worked out in

25-40 kHz and 180-200 kHz range.

Fig. 5.20 Levels of damage induced on test specimen (aluminium block).

(a) Pristine state. (b) Incipient damage.

(c) Moderate damage. (d) Severe damage.

(b)(a)

(d)(c)

50 mm

= =

48 m

m

=

=

7 mm

7 mm

7 mm 7 mm

5mm φ holesHost structure

PZTpatch


145

Fig. 5.21 shows the effect of these damages on the real and the imaginary

components of the extracted mechanical impedance in the frequency range

25-40 kHz. Fig. 5.22 shows the effect of the damages on the identified structural

parameters. As expected, with damage progression, the stiffness can be observed to

reduce, and the damping can be observed to increase. The stiffness was found to

reduce by about 12% and the damping was found to increase by about 7% after the

incipient damage. Thereafter, with further damage propagation, very small further

drop/ increase was observed in these parameters. However, it should be noted that

the incipient damage was captured reasonably well.

(a)

Fig. 5.21 Effect of damage on extracted mechanical impedance in 25-40 kHz range.


20

30

40

50

60

70

80

25 30 35 40Frequency (kHz)

x (N

s/m

)

Pristine state

Incipient, Moderate,Severe damage

-400

-300

-200

-100

25 30 35 40

Frequency (kHz)

y (N

s/m

)

Pristine state

Incipient, severedamage

Moderatedamage

(b)

Fig. 5.22 Effect of damage on equivalent system parameters in 25-40kHz range.

(a) Equivalent damping constant. (b) Equivalent spring constant.

36.54221

39.14065 39.1132739.58451

36

37

38

39

40

41

Pristinestate

Incipientdamage

Moderatedamage

Severedamage

c (N

s/m

)

5.18E+07

4.55E+074.37E+07

4.53E+07

4.00E+07

4.50E+07

5.00E+07

5.50E+07

Pristinestate

Incipientdamage

Moderatedamage

Severedamage

k (N

/m)

(a) (b)


146

Fig. 5.23 shows the effect of these damages on the impedance spectra in the

frequency range of 180-200 kHz. Equivalent lumped system parameters were

determined for each damage state using the procedure outlined in the preceding

section. Fig. 5.24 shows a comparison between the experimental impedance plots

and the plots based on the equivalent system parameters for each damage state. A

very good agreement between the two demonstrates reasonably accurate structural

identification for the damaged structure also.

The effect of damages on the equivalent parameters for 180-200 kHz range

is shown in Fig. 5.25. Again, the trend is very consistent with expected behavior,

and much more prominent than for the frequency range 25-40 kHz. With damage

progression, mass and stiffness can be seen to reduce, and the damping can be

observed to increase. The stiffness was found to reduce gradually- 17% for the

incipient damage, 31% for the moderate damage and 47% for the severe damage.

Mass was also found to similarly reduce with damage severity- 16% for the

incipient damage, 28% for the moderate damage and 42% for the severe damage.

The damping values (c and C*) were also found to increase with damage (Figs. 13c

and 13e), though ‘c’ displayed a slight decrease after the incipient damage. The

only exception is found in the parallel stiffness K*, which remains largely

insensitive to all the levels of damage. Contrary to the 25-40 kHz range, the

180-200 kHz range was found to diagnose the damages much better, as

demonstrated by the significant variation in the parameters for moderate and severe

damages in addition to incipient damages.

Fig. 5.26 shows a plot between the area of the specimen, ‘A’ (a measure of

the residual capacity of the specimen) and the equivalent spring stiffness ‘k’

identified by the PZT patch. Following empirical relation was found between the

two using regression analysis291002.20021.02.1874 kxkA −−+= (5.85)

This demonstrates that it is possible to calibrate the damage sensitive system

parameters with damage and to employ them for damage diagnosis in real

scenarios.


147

(b)

Fig. 5.23 Effect of damage on extracted mechanical impedance in 180-200 kHz range.


(a)

-150

-130

-110

-90

-70

-50

-30

-10

10

30

50

180 185 190 195 200

Frequency (kHz)

y (N

s/m

)

Pristine stateIncipient damage

Moderate Damage

Severe damage

0

50

100

150

180 185 190 195 200

Frequency (kHz)

x (N

s/m

)Pristine stateIncipient damage

Moderate damage

Severe damage


148

0

50

100

150

180 185 190 195 200

Frequency (kHz)

x (N

s/m

)

(b)(a)

-150

-100

-50

0

50

180 185 190 195 200

Frequency (kHz)

y (N

s/m

)

-150

-100

-50

0

50

180 185 190 195 200

Frequency (kHz)

y (N

s/m

)

010203040506070

180 185 190 195 200

Frequency (kHz)

x (N

s/m

)

(c)

0

20

40

60

80

100

180 185 190 195 200

Frequency (kHz)

x (N

s/m

)

-150

-100

-50

0

50

180 185 190 195 200

Frequency (kHz)

y (N

s/m

)

(f)(e)


Fig. 5.24 Plot of mechanical impedance of aluminium block in 180-200 kHz for

various damage states.

(a) Incipient damage: Real part. (b) Incipient damage: Imaginary part.

(c) Moderate damage: Real part. (d) Moderate damage: Imaginary part.

(e) Severe damage: Real part. (f) Severe damage: Imaginary part.

(d)


149

3.05E-072.59E-07

2.18E-071.76E-07

0.00E+00

1.00E-07

2.00E-07

3.00E-07

4.00E-07

5.00E-07

PristineState

Incipientdamage

Moderatedamage

Severedamage

m (k

g)

12.4

13.614.5

16.5

10

11

12

13

14

15

16

17

18

PristineState

Incipientdamage

Moderatedamage

Severedamage

C* (N

s/m

)

7.45E+07 7.22E+07 7.15E+07 7.30E+07

0.00E+00

4.00E+07

8.00E+07

1.20E+08

1.60E+08

2.00E+08

PristineState

Incipientdamage

Moderatedamage

Severedamage

K *(

Ns/m

)1.10E-03

8.55E-04

1.30E-03 1.30E-03

8.00E-04

9.00E-04

1.00E-03

1.10E-03

1.20E-03

1.30E-03

1.40E-03

PristineState

Incipientdamage

Moderatedamage

Severedamage

c (N

s/m

)

Fig. 5.25 Effect of damage on equivalent system parameters in 180-200kHz range.

(a) Equivalent spring constant. (b) Equivalent mass. (c) Equivalent damping constant.

(d) Equivalent additional spring constant. (e) Equivalent additional damping constant.

(b)

(c) (d)

(e)

4.33E+053.60E+05

3.00E+052.27E+05

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

6.00E+05

7.00E+05

PristineState

Incipientdamage

Moderatedamage

Severedamage

k (N

s/m

)

(a)


150

The higher sensitivity of damage detection in the frequency range

180-200 kHz (as compared to 25-40 kHz range) is due to the fact that with increase

in frequency, the wavelength of the induced stress wave gets smaller and is

therefore more sensitive to any defects and damages. This is also due to the

presence of a damage sensitive anti-resonance mode in the frequency range

180-200 kHz (Fig. 5.19) and its absence in the 25-40 kHz range. This agrees with

the recommendation of Sun et al. (1995), that the frequency range must contain

prominent vibrational modes to ensure high sensitivity to damages. However, it

should be noted that in spite of the absence of any major resonance mode in the

frequency range 25-40 kHz, the damage is still effectively captured at the incipient

stage, although severe damages are not well differentiated from the incipient

damage.

This study demonstrates that the proposed method can evaluate structural

damages in aerospace components reasonably well. Besides miniature aerospace

gadgets, the methodology is also ideal for identifying damages in precision

machinery components, turbo machine parts and computer parts such as the hard

disks. These components are quite rigid and exhibit a dynamic behaviour similar to

the test structure described in this section. The piezo-impedance transducers,

because of their miniature characteristics, are unlikely to alter the dynamic

2200

2250

2300

2350

2400

2450

2.00E+05 3.00E+05 4.00E+05 5.00E+05

k (N/m)

A (m

m2 )

Pristine state

Incipient damage

Moderate damage

Severe damage

Fig. 5.26 Plot of residual specimen area versus equivalent spring constant.


characteristics of these miniature systems. They are thus preferable over other

sensor systems and techniques (Giurgiutiu and Zagrai, 2002).

As can be observed from Fig. 5.25(b), it is clear that by using this method, it

is possible to detect ‘mass loss’ in critical space shuttle components, such as the

RCC panels, in which this is a very common type of damage, as discussed in

Chapter 1. This type of damage is presently difficult to be identified by other

prevalent NDE techniques.

5.12 EXTENSION TO DAMAGE DIAGNOSIS IN CIVIL-STRUCTURAL

SYSTEMS

In order to demonstrate the feasibility of the proposed methodology for

monitoring large civil-structures, the data recorded during the destructive load test

on a prototype reinforced concrete (RC) bridge was utilized. The test bridge

consisted of two spans of about 5m, instrumented with several PZT patches,

10x10x0.2mm in size, conforming to grade PIC 151 (PI Ceramic, 2003). The bridge

was subjected to three load cycles so as to induce damages of increasing severity.

Details of the instrumentation as well as loading can be found in references- Soh et

al. 2000 and Bhalla, 2001. Root mean square deviation (RMSD) index was used to

evaluate damages in the previous study. In the present investigation, the evaluation

of damages is carried out using the newly developed approach.

1 2

4 3

7 6

5

8

Fig. 5.27 Damage diagnosis of a prototype RC bridge using proposed methodology.

151


152

Fig. 5.27 shows a view of the top surface of the deck after cycle II. The PZT

patches detected the presence of surface cracks much earlier than global condition

indicators, such as the load-deflection curve (Soh et al., 2000). Patch 4 was

typically selected as a representative PZT in the present analysis. Fig. 5.28 shows

the impedance spectra of the pristine structure as identified by the PZT patch 4 in

the frequency range 120-145 kHz. From this figure, it can be seen that the PZT

patch has ‘identified’ the structure as a parallel spring-damper combination, the

identified parameters being k = 9.76x107 N/m and c = 26.1823 Ns/m. The

equivalent parameters were also determined for the damaged bridge, after cycles I

and II.

Fig. 5.29 provides a look at the associated damage mechanism- ‘k’ can be

observed to reduce and ‘c’ to increase with damage progression. Reduction in the

stiffness and increase in the damping is well-known phenomenon associated with

crack development in concrete. Damping increased by about 20% after cycle I and

about 33% after cycle II. This correlated well with the appearance of cracks in the

vicinity of this patch after cycles I and II. Stiffness was found to reduce marginally

by about 3% only, after cycle II, indicating the higher sensitivity of damping to

damage as compared to stiffness.

0

10

20

30

40

50

120 124 128 132 136 140

Frequency (kHz)

x (N

s/m)


Fig. 5.28 Mechanical impedance of RC bridge in 120-140 kHz frequency range. The

equivalent system plots are obtained for a parallel spring damper combination.


(b)(a)

-130

-125

-120

-115

-110120 124 128 132 136 140

Frequency (kHz)

y (N

s/m

)


153

Thus, the proposed methodology can be easily extended to large civil-

structures as well. However, it should be noted that owing to the large size of the

typical civil-structures, the patch can only ‘identify’ a localized region of structure,

typically representative of the zone of influence of the patch. For large structures,

complete monitoring warrants an array of PZT patches. The patches can be

monitored on one-by-one basis and can effectively localize as well as evaluate the

extent of damages. The next chapter will present how the identified system

parameters can be calibrated with extent of damage.


This chapter has presented a new simplified PZT-structure interaction model

based on the new concept of ‘effective impedance’. As opposed to previous

impedance-based models, the new model condenses the two-directional mechanical

coupling between the PZT patch and the host structure into a single impedance

term, which can be determined from the measured admittance signatures. Hence

this model bridges the gap between the 1D impedance model of Liang et al. (1993,

1994) and the 2D model proposed by Zhou et al. (1995, 1996). Further, a detailed

26.1823

31.3414

34.6622

20

24

28

32

36

Pristine Cycle I Cycle II

c

9.76E+07 9.76E+07

9.53E+07

9.40E+07

9.60E+07

9.80E+07

1.00E+08

Pristine Cycle I Cycle II

kFig. 5.29 Effect of damage on equivalent system parameters of RC bridge.

(a) Equivalent damping constant. (b) Equivalent spring constant.

(b)(a)


154

step-by step procedure has been outlined to ‘update’ the model of the PZT patch

before it could be employed for ‘identifying’ the host structure. It is demonstrated

that this updating enables a much more accurate determination of system

parameters.

This chapter also presents a new diagnostic approach for identification and

NDE of structures based on the equivalent system ‘identified’ by means of the EMI

technique. It makes use of real as well as imaginary components of admittance

signature for determining damage sensitive equivalent structural parameters. As

proof-of-concept, the method was successfully applied to diagnose damages on a

representative aerospace/ mechanical structure and a prototype civil structure. In

order to make full utilization of the proposed methodology, we need to calibrate the

identified system parameters with damage progression. Presently, this was

demonstrated by developing an empirical relationship between the residual capacity

of the specimen and the equivalent stiffness as identified by the PZT patch. This

could serve as an empirical phenomenological model for the tested miniature

component. The piezo-impedance transducers can be installed on the inaccessible

parts of crucial machine components, aircraft main landing gear fitting or turbo-

engine blades, RCC panels of space shuttles and civil-structures to perform

continuous real-time SHM. The equivalent system is identified from the

experimental data alone. No analytical/ numerical model is required as a

prerequisite. The approach is not only simple to apply but at the same time provides

an essence of the associated damage mechanism. Besides NDE, the proposed model

can be employed in numerous other applications, such as predicting system’s

response, energy conversion efficiency and system power consumption.

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete

155

Chapter 6

CALIBRATION OF PIEZO-IMPEDANCE TRANSDUCERSFOR STRENGTH PREDICTION AND DAMAGEASSESSMENT OFCONCRETE

6.1 INTRODUCTION

In Chapter 5, a new method was developed to ‘identify’ system parameters and

predict system behaviour using the electro-mechanical admittance signatures of

surface bonded piezo-impedance transducers. However, it is equally important to

relate the identified impedance parameters with physical parameters such as

strength/ stiffness and to calibrate changes in the parameters with damage

progression in the host structure. This is the main objective of the present chapter.

Comprehensive tests were performed on concrete specimens up to failure in

order to empirically calibrate the ‘identified’ system parameters with damage

severity. Besides, a new experimental technique has been developed to determine

in-situ concrete strength non-destructively using the EMI method.

6.2 CONVENTIONAL NDE METHODS IN CONCRETE

In general, from the point of view of NDE, concrete technologists are interested

in (i) concrete strength determination, and (ii) concrete damage detection.

Special importance is attached to strength determination for concrete because

its elastic behaviour and to some extent service behaviour can be easily predicted

from strength characteristics. Although direct strength tests, which are destructive in

nature, are excellent for quality control during construction, their main shortcoming

is that the tested specimen may not truly represent the concrete in the actual

structure. The destructive tests reflect more the quality of the supplied materials

rather than that of the constructed structure. Delays in obtaining results, lack of


156

reproducibility and high costs of tests are some other drawbacks. The NDE

methods, on the other hand, aim to measure the strength of concrete in the actual

structures. However, these cannot be expected to yield absolute values of strength

since they measure some property of concrete from which an estimation of its

strength, durability and elastic parameters can be obtained. A detailed review of the

NDE methods for concrete strength prediction is covered by Malhotra (1976) and

Bungey (1982). A very brief description of the most common methods for concrete

strength estimation is presented below.

6.2.1 Surface Hardness Methods

These methods are based on the principle that strength of concrete is

proportional to its surface hardness. The surface hardness is measured using the

indentation test, which involves impacting the specimen surface which a standard

mass, activated by given energy, and measuring the size of the resulting indentation.

Although there is little theoretical relationship between indentation size and

strength, many empirical correlations have been established, which give a

reasonable estimation of strength within 20-30% error. Most common indentation

devices are William’s testing pistol, Frank spring hammer and Einbeck pendulum

hammer.

The main limitation of this method is that the devices need frequent

calibration. Also, the results are strongly dependent on the type of cement,

aggregate, age and moisture content of the specimen and are not very reproducible.

6.2.2 Rebound Method

The rebound method consists of predicting concrete strength based on the

rebound of a hardened steel hammer dropped on specimen surface. The rebound

hammer, known as Schmidt rebound hammer, was invented by Ernst Schmidt in

1948. Empirical correlations have since been established between rebound number

and concrete strength.

In spite of quick and inexpensive estimation of strength by Schmidt hammer,

the results are influenced by surface roughness, type of specimen (shape and size),

age, moisture content, type of cement and aggregate.


157

6.2.3 Penetration Techniques

These techniques are based on measuring the depth of penetration of a

standard probe, impacted on the surface of the specimen, with a standard energy.

The penetration is mechanically done in the case of Simbi hammer and by

gunpowder blast in the case of Spit pin hammer and Windsor probe.

The main drawback of the penetration techniques is that they leave a minor

damage on a small area (about 8mm diameter in the case of Windsor Probe) of

concrete. Further, the calibration is strongly dependent on the source and type of the

aggregate used. Besides, large variations in strength prediction are observed. Hence,

the main usefulness of the penetration techniques simply lies in determining the

relative quality of concrete in place rather than quantitatively predicting the

strength.

6.2.4 Pullout Test

This test measures the force required to pull out from concrete a specially

shaped steel rod, whose enlarged end has been cast into the concrete. A very high

degree of correlation exists between the pullout force and the compressive strength.

The pull out tests are therefore reproducible with a high degree of accuracy.

The major drawback, however, is that the test causes a small damage to the

concrete surface which must be repaired. Another drawback is that since the pullout

assemblies need to be incorporated into the form work before concreting, the tests

need to be planned in advance.

6.2.5 Resonant Frequency Method

This method is based on the principle that the velocity of sound through a

component is proportional to the natural frequency of the component, which is in-

turn proportional to the Young’s modulus of elasticity (and hence strength) of the

medium. This method has been standardized by the American Society for Testing

and Materials (ASTM). The velocity of sound in concrete is obtained by

determining the fundamental resonant frequency of vibration of the specimen,

which is usually a cylinder (150mm diameter and 300mm length) or a prism


158

(75x75x300mm). A schematic setup using a commercial device, called sonometer,

is shown in Fig. 6.1(a). In this device, an electronic audio oscillator generates

electrical audio frequency voltages, which are converted into mechanical pulses by

the transmitter. As the waves travel through the concrete, they are picked up by a

piezoelectric crystal acting as receiver at the other end of the specimen. The

frequency of the oscillator is turned until maximum deflection is displayed in the

meter, which indicates resonance. From the measured frequency, the dynamic

Young’s modulus is calculated using standard equations. The dynamic modulus of

elasticity of concrete is in-turn correlated empirically with concrete strength, as

shown in Fig. 6.1(b).

The main drawback of this method is that it can only be carried out on small

lab-sized specimens rather than the structural members in the field. Also, the shape

of the specimen needs to be limited to cylindrical or prismatic type only. Besides,

the results depend on the type of concrete under investigation. Last but not the least,

the test demands the availability of two opposite free surfaces on the specimen.

6.2.6 Ultrasonic Pulse Velocity Method

Test specimen

Transmitter

Receiver

Fig. 6.1 (a) Determining natural frequency of specimen using sonometer.

(b) Correlation between dynamic modulus and concrete strength.

Source: Malhotra (1976).

Sonometer

(a) (b)


159

This method works on the same principle as the resonant frequency method.

The only difference is that the velocity of sound is determined by measuring the

time of travel of electronically generated longitudinal waves (15-50 kHz) through

concrete, using a digital meter or a cathode ray oscilloscope. The pulse generation

and reception is carried out by using piezo crystals. This test has also been

standardized by ASTM. Some of the commercially available test equipment are

soniscope, ultrasonic concrete tester and PUNDIT (portable ultrasonic non-

destructive digital indicating tester). Fig. 6.2(a) shows the test setup using PUNDIT.

The pulse velocity measurements are correlated with strength, as shown in Fig.

6.2(b), and the error is typically less than 20%.

Because the velocity of the pulses is independent of the geometry of the

component and depends on its elastic properties alone, the method is suitable both

in the lab environment as well as in the field. It is typically used to test the quality

of concrete in bridge piers, road pavements and concrete hydraulic structures up to

15m thickness.

The main limitation of the method is that the transducers must always be

placed on the opposite faces of the structure for accurate results. Very often, this is

not possible and this sometimes limits the application of the technique. Also, the

Fig. 6.2 (a) Determining velocity of sound in concrete using PUNDIT.

(b) Correlation between ultrasonic pulse velocity and strength.

Source: Malhotra (1976).

Test Specimen

PUNDIT

Receiver

Transmitter

(a) (b)


160

correlation between strength and the velocity is strongly dependent on the type of

cement and the aggregate.

Besides strength determination, the other aspect of NDE, namely concrete

damage detection, is conventionally carried out by using techniques such as

ultrasonic methods, impact echo and acoustic emission, which have already been

described in Chapter 2.

6.3 CONCRETE STRENGTH EVALUATON USING EMI TECHNIQUE

In Chapter 5, Eq. (5.56) was derived to predict the electrical admittance across

the terminals of a square PZT patch, surface bonded to a structure possessing an

effective mechanical impedance Zs,eff. From this relationship, admittance spectra

can be obtained for a ‘free’ and ‘clamped’ PZT patch, by substituting Zs,eff equal to

0 and ∞ respectively. Figs. 6.3 displays the admittance spectra (0-1000 kHz),

corresponding to these boundary conditions, for a PZT patch 10x10x0.3mm in size,

conforming to grade PIC 151 (PI Ceramic, 2003). It is observed from this figure

that the three resonance peaks, corresponding to “free-free” planar PZT vibrations,

vanish upon clamping the patch. The act of bonding a PZT patch on the surface of

a structure also tends to similarly restrain the PZT patch. However, in real

situations, the level of clamping is expected to be intermediate of these two extreme

situations and therefore, the admittance curves are likely to lie in between the

curves corresponding to the extreme situations, depending on the stiffness (or

strength) of the component.

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 200 400 600 800 1000

Frequency (kHz)

B (S

)

0.00001

0.0001

0.001

0.01

0.1

0 200 400 600 800 1000

Frequency (kHz)

G (S

)

Free

Fully clamped

Free

Fig. 6.3 Admittance spectra for free and fully clamped PZT patches.

(a) Conductance vs frequency. (b) Susceptance vs frequency.

Fully clamped


161

In order to test the feasibility of predicting concrete strength using this

principle, identical PZT patches (measuring 10x10x0.3mm, grade PIC 151, key

parameters as listed in Table 4.1, page 100), were bonded on the surface of concrete

cubes, 150x150x150mm in size. At the time of casting, the proportions of various

constituents were adjusted such that different characteristic strengths would be

achieved. Same type of cement as well as aggregartes were used for all specimens.

After casting, a minimum curing period of 28 days was observed for all the

specimens, except two of the specimens, for which it was kept one week so as to

achieve a low strength at the time of the test. In order to achieve identical bonding

conditions, same thickness of epoxy adhesive layer (RS 850-940, RS Components,

2003) was applied between the PZT patches and the concrete cubes. In order to

ensure this, two optical fibre pieces, 0.125mm in diameter, were first laid parallel to

each other on the concrete surface, as shown in Fig. 6.4 (a). The layer of epoxy was

then applied on concrete surface and the PZT patch was placed on it. Light pressure

was maintained over the assembly using a small weight. The setup was left

undisturbed in this condition at room temperature for 24 hours to enable full curing

of adhesive. The optical fibre pieces were left permanently in the adhesive layer.

This procedure ensured a uniform thickness of 0.125mm of the bonding layer in all

the specimens tested.

Fig. 6.4 (a) Optical fibre pieces laid on concrete surface before applying adhesive.

(b) Bonded PZT patch.

PZT Patch

Wires

0.125 mm fibre

Wires

PZT patch


162

Fig. 6.5 shows the conductance and susceptance plots of the PZT patches bonded to

concrete cubes of five different strengths. The strengths indicated on the figure were

determined experimentally by subjecting the cubes to cyclic loading on a universal

testing machine (UTM). The test procedure will be covered in detail in the next

section. The figure also shows the analytical curves for PZT patch in free as well as

clamped conditions.

0

0.002

0.004

0.006

0.008

0.01

100 150 200 250 300 350 400

Frequency (kHz)

B (S

)

Free PZT

Strength = 17MPa

Strength = 43MPa

Strength = 54MPa

Strength = 60MPa

Strength = 86MPa

Fully clamped

0.00E+00

4.00E-03

8.00E-03

1.20E-02

1.60E-02

100 150 200 250 300 350 400

Frequency (kHz)

G (S

)

Free PZT

Strength = 17MPa

Strength = 43MPaStrength = 54MPa

Strength = 60MPa

Strength = 86MPa

Fully clamped

Fig. 6.5 Effect of concrete strength on first resonant frequency of PZT patch.


(a)

(b)


163

It is apparent from the figures that the first peak frequency (see Fig. 6.3a)

gradually shifts in the right direction as the strength of concrete is increased. This

shifting is on account of the additional stiffening action due to bonding with

concrete, the level of stiffening being related to the concrete strength. Fig. 6.6

shows a plot between the observed first resonant frequency and measured concrete

strength for data pertaining to a total of 17 PZT patches bonded to a total of 11

concrete cubes. At least two cubes were tested corresponding each strength and

average frequencies were worked out. Free PZT curve was used to obtain the data

point corresponding to zero strength.

From regression analysis, following empirical relationship was found between

concrete strength (S) and the observed first resonant frequency

94.1966657.20089.0)( 2 +−= ffMPaS (6.1)

where the resonant frequency, f, is measured in kHz. This empirical relationship can

be used to evaluate concrete strength non-destructively for low to high strength

concrete (10MPa < S < 100MPa).

It should be mentioned here that good correlation was not found between

concrete strength and the second and the third peaks (see Fig. 6.3). This is because

R2 = 0.9552

0

20

40

60

80

100

120

170 185 200 215 230 245 260

First peak frequency (kHz)

Stre

ngth

(MP

a)

Fig. 6.6 Correlation between concrete strength and first resonant frequency.

15% errorlimits


at frequencies higher than 500 kHz, the PZT patches become sensitive to their own

conditions rather than the conditions of the structure they are bonded with (Park et

al, 2003b).

Although the tests reported in this study were carried out on 150mm cubes,

the empirical relationship represented by Eq. 6.1 can be conveniently used for real-

life structures also since the zone of influence of the PZT patches is usually very

small in concrete. However, it should be noted that the strength considered in the

present study is obtained by cyclic compression tests, which is expected to be lower

than that obtained by the standard testing procedure. Also, the relationship will

depend on the type of the aggregates and the type of cement used. It will also

depend on the type and size of the PZT patches and type and thickness of the

bonding layer. Hence, Eq. (6.1) cannot be considered as a universal relationship.

Therefore, it is recommended that similar calibration should be first established in

the laboratory for the particular concrete under investigation before using the

method in the field.

The main advantage in the newly developed method is that there is no

requirement of the availability of two opposite surfaces, as in the case of the

resonant frequency method and the ultrasonic pulse velocity method. Also, no

expensive transducers or equipment are warranted.

6.4 EXTRACTION OF DAMAGE SENSITIVE CONCRETE PARAMETERS

FROM ADMITTANCE SIGNATURES

Consider the concrete cubes, 150x150x150mm in size, instrumented with

square PZT patches (10x10x0.3mm, PIC 151), as shown in Fig. 6.7. Using the

procedure outlined in Chapter 5, updated ‘models’ were obtained for five

PZTpatch

StrainGauge

Fig. 6.7 Concrete cube to be ‘identified’ by piezo-impedance transducer.

164


165

Table 6.1 Averaged parameters of test sample of PZT patches.


Electric Permittivity, T33ε (farad/m) 1.7785 x 10-8

Peak correction factor, Cf 0.898

)1(2 2

31

υ−=

EYdK (N/V2)5.35x10-9

Mechanical loss factor, η 0.0325

Dielectric loss factor, δ 0.0224

representative PZT patches of the set. Table 6.1 lists the key averaged PZT

parameters for the batch. For the other less important parameters, the values

supplied by the manufacturer, as shown in Table 4.1 (page 100) were used.

Using the computational procedure outlined in Chapter 5, the impedance

parameters of the concrete cubes were extracted out from the admittance signatures

of the bonded PZT patches in the frequency range 60-100 kHz. The MATLAB

program listed in Appendix E (with parameters listed in Table 6.1) was employed to

perform computations. The real and imaginary components of the extracted

mechanical impedance were found to exhibit a response similar to that of a parallel

spring damper combination, shown in Fig. 6.8. Typically, for concrete cube with a

strength of 43 MPa (designated as C43), the system parameters were identified to be

k = 5.269x107 N/m and c = 12.64 Ns/m. Fig. 6.9 shows a comparison between the

experimental impedance spectra and that corresponding to the parallel spring-

damper combination with k = 5.269x107 N/m and c = 12.64 Ns/m . A good

agreement can be observed between the two.

k

c

Fig. 6.8 Equivalent system ‘identified’ by PZT patch.


166

The concrete cubes were then subjected to cyclic loading in an experimental

set-up shown in Fig. 6.10. The PZT patches instrumented on the cubes were wired

to an impedance analyzer, which was controlled using the personal computer

labelled as PC1 in the figure. The strain gauge was wired to a strain recording data

logger, which was in-turn hooked to another personal computer marked PC2, which

also controlled the operation of the UTM. The cube was then loaded in compression

at a rate of 330 kN/min until the first predetermined load. It was then unloaded and

the conductance and susceptance signatures were acquired. In the next cycle, the

cube was loaded to the next higher level of load and the signatures were again

acquired after unloading. This loading, unloading and signature acquisition process

was repeated until failure. Thus, the damage was induced in a cyclic fashion.

Typical load histories for four cubes designated as C17 (Strength = 17MPa), C43

(Strength = 43MPa),, C54 (Strength = 54MPa) and C86 (Strength = 86MPa), are

shown in Figure 6.11.

0

10

20

30

40

50

60 70 80 90 100

Frequency (kHz)

x (N

s/m

)

-150

-130

-110

-90

-70

60 70 80 90 100

Frequency (kHz)

y (N

s/m

)

Fig. 6.9 Impedance plots for concrete cube C43.

(a) Real component of mechanical impedance (x) vs frequency.

(b) Imaginary component of mechanical impedance (y) vs frequency.

Equivalentsystem Experimental

Equivalentsystem

Experimental

(a) (b)


167

Fig. 6.10 Experimental set-up for inducing damage on concrete cubes.

Fig. 6.11 Load histories of four concrete cubes.

(a) C17 (b) C43 (c) C52 (d) C86

(a) (b)

(c)(d)

0

10

20

30

40

50

0 500 1000 1500 2000 2500

Microstrain

Stre

ss (M

Pa)

II

IIIIV V

VI

Failure

I

0

10

20

30

40

50

60

0 500 1000 1500 2000 2500 3000

Microstrain

Stre

ss (M

Pa)

Failure

II

IIIIV

VVI

I

0

4

8

12

16

20

0 500 1000 1500 2000 2500

Microstrain

Stre

ss (M

Pa)

FailureI, II, III

IVV

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000 3500

Microstrain

Stre

ss (M

Pa)

Failure

VI

V

I, II, III, IV

PC 1

PC 2

ImpedanceAnalyzer

Data loggerConcretecube

Load cell

UniversalTestingMachine(UTM)


From Fig. 6.11, it is observed that the secant modulus of elasticity

progressively diminishes as the number of load cycles is gradually increased. Loss

in secant modulus was worked out after each load cycle. At the same time, the

extracted equivalent spring stiffness, worked out from the recorded PZT signatures,

was found to diminish proportionally. Fig. 6.12 shows the plots of the loss of secant

modulus against the loss of equivalent spring stiffness for four typical cubes C17,

C43, C52 and C86. A good correlation can be observed between the loss in secant

modulus and the loss in equivalent stiffness as identified by the piezo-impedance

transducers. From these results, it is evident that equivalent spring stiffness can be

regarded as a damage sensitive parameter and can be utilized for quantitatively

predicting the extent of damage in concrete. It should be noted that the equivalent

spring stiffness is obtained solely from the signatures of the piezo-impedance

transducers. No information about concrete specimen is warranted a priori.

y = 7.9273x - 34.583R2 = 0.9124

0

20

40

60

80

100

4 6 8 10 12 14 16 18 20% loss in equivalent stiffness

% lo

ss in

sec

ant m

odul

us

y = 0.6776x + 41.607R2 = 0.8865

40

50

60

70

80

10 15 20 25 30 35 40 45 50

% loss in equivalent stiffness

% lo

ss in

sec

ant m

odul

us

y = 0.2495x + 17.7R2 = 0.9593

15

20

25

30

35

0 10 20 30 40 50 60


% lo

ss in

sec

ant m

odul

us

y = 0.6776x + 41.607R2 = 0.8865

40

50

60

70

80

10 15 20 25 30 35 40 45 50


% lo

ss in

sec

ant m

odul

us

(a) (b)

(c) (d)

Fig. 6.12 Correlation between loss of secant modulus and loss of

equivalent spring stiffness with damage progression.

(a) C17 (b) C43 (c) C52 (d) C86
168

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete

169

It should also be mentioned here the extracted equivalent damping was

found to increase with damage. This was as expected, since damping is known to

increase with the development of cracks in concrete. Fig. 6.13 shows typical plot of

increase in equivalent damping with damage progression for cube C43. Also shown

is the progressive loss in the equivalent stiffness with load ratio. However, in most

other cubes, no consistent pattern was found with respect to damping. Only a

phenomenal increase near failure was observed. For this reason, the equivalent

stiffness was selected as the damage sensitive parameter due to its progressive

decrement with damage progression and consistent performance. Section 6.7 covers

the development of an empirical damage model based on the equivalent spring

stiffness.

6.5 MONITORING CONCRETE CURING USING EXTRACTED

IMPEDANCE PARAMETERS

In order to evaluate the feasibility of the ‘identified’ spring stiffness in

monitoring curing of concrete, a PZT patch, 10x10x0.3mm in size (grade PIC 151,

PI Ceramic) was instrumented on a concrete cube, again measuring

150x150x150mm in size, as shown in Fig. 6.14. Again, a bond layer thickness of

0.125mm was achieved with the aid of optical fibre pieces. The instrumentation was

done three days after casting the cube. The PZT patch was periodically interrogated

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

Load ratio

% C

hang

e

Equivalent Damping

Equivalent Stiffness

Fig. 6.13 Changes in equivalent damping and equivalent stiffness for cube C43.


170

for the acquisition of electrical admittance signatures and this was continued for a

period of one year. Figs. 6.15 and 6.16 respectively show the short term and the

long term effects of ageing on the conductance signatures in the frequency range

100-150 kHz. It is observed that with ageing, the peak is shifting towards the right

and at the same time getting sharper. This trend is exactly opposite to the trend

observed during compression tests, where the peaks usually shift towards the left

(Bhalla, 2001). The shifting of the resonance peak towards the right in the present

case indicates that the stiffness (and hence the strength) is increasing with time. The

phenomenon of peak getting sharper with time suggests that the material damping is

reducing (concrete was initially ‘soft’). It is a well known fact that most damping in

concrete occurs mainly in the matrix, some in the interfacial boundaries and a very

small fraction in the aggregates. Moisture in the matrix is the major contributor to

damping (Malhotra, 1976). Hence, with curing, as moisture content drops, the

damping in concrete tends to fall down.

It should be noted here that the particular peak in this figure is the resonance

peak of the structure. It should not be confused with the resonance peak of the PZT

patch, such as that shown in Fig. 6.5. As concrete strength increases, the resonce

peak of the PZT patch subsides down due to the predominance of structural

interaction. However, the structural resonance peak (Figs. 6.15 and 6.16), on the

other hand, tends to get sharper. In other words, increasing structural stiffness tends

to ‘dampen’ PZT resonance and ‘sharpen’ the host structure’s resonance peak.

Fig. 6.14 Monitoring concrete curing using EMI technique.

PZT patch

Wire


171

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

0.0011

100 105 110 115 120 125 130 135 140 145 150

f (kHz)

G (S

)

Day 8

Day 4

Day 5

Day 10 Day 14

Fig. 6.15 Short-term effect of concrete curing on conductance signatures.

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.001

0.0011

100 105 110 115 120 125 130 135 140 145 150

f (kHz)

G (S

)

Day 50

Day 4

Day 8

Day 120 Day 365

Fig. 6.16 Long-term effect of concrete curing on conductance signatures.


172

In order to quantitatively describe the phenomenon, the equivalent spring

constant of the cube was worked out in the frequency range 60-100 kHz using the

signatures of the bonded PZT patch. Results are presented in Fig. 6.17. It can be

observed from the figure that as the curing progressed, the equivalent spring

stiffness increased, reaching an asymptotic value, of about 115% higher than the

first recorded value (four days after casting). After 28 days, the increase in the

equivalent spring stiffness was about 80%.

On comparison with similar monitoring using the ultrasonic pulse velocity

technique (Malhotra, 1976), it is found that the present approach is much better in

monitoring concrete curing. For example, Malhotra (1976) reported an increase of

only 7% in the ultrasonic pulse velocity between day 4 and day 10. On the other

hand, in the present experiment, a much higher increase of 60% was observed

between day 4 and day 10. This establishes the superior performance of the present

method for monitoring curing of concrete.

This method can be applied in the construction industry to decide the time of

removal of the form work. It can also be employed to determine the time of

commencement of prestressing operations in the prestressed concrete members.

Besides, numerous other industrial processes, which involve such curing (of

materials other than concrete, such as adhesives), can also be benefited.

0

25

50

75

100

125

150

0 50 100 150 200 250 300 350 400

Age (Days)

% In

crea

se in

k

Fig. 6.17 Effect of concrete curing on equivalent spring stiffness.


173

6.6 ESTABLISHMENT OF IMPEDANCE-BASED DAMAGE MODEL FOR

CONCRETE

6.6.1 Definition of Damage Variable

It has already been shown that in the frequency range 60-100 kHz, concrete

essentially behaves as parallel spring damper system. The equivalent stiffness ‘k’

has been established as a damage sensitive system parameter since it is found to

exhibit a reasonable sensitivity to any changes taking place in the system on

account of damages. This section deals with calibrating ‘k’ against damage using

the data from compression tests on concrete cubes of strengths ranging from

moderate to high values.

In general, any damage to concrete causes reduction in the equivalent spring

stiffness as identified by the piezo-impedance transducer surface-bonded to it. At ith

frequency, the associated damage variable, Di, can be defined as

oi

dii k

kD −= 1 (6.2)

where oiK is the equivalent spring stiffness at the jth measurement point in the

pristine state and diK is the corresponding value after damage. It may be noted that

0 < Di < 1. Thus Di measures the extent of ‘softening’ of the identified equivalent

stiffness due to damage. Di is expected to increase in magnitude with damage

severity. The host structure can be deemed to fail if D exceeds a critical value Dc.

However, from the comprehensive tests on concrete cubes, it was found that it is not

possible to define a unique value of Dc. This is due to unavoidable uncertainties

related to concrete, its constituents and the PZT patches. Therefore it is proposed to

define the critical value of the damage variable using the theory of fuzzy sets.

Since theory of statistics, probability and fuzzy sets is extensively employed for

analysing the comprehensive data pertaining to damage variable for concrete

specimens, the following sections give a brief introduction to these concepts before

formally addressing the problem of sensor calibration for damage evaluation.


174

6.6.2 Theory of Statistics and Probability

Mathematical statistics is mainly concerned with random variable, that is, a

variable which can assume different values due to unpredictable factors. For

example, the damage variable Di defined earlier is a random variable. Within a

given excitation frequency range (60-100 kHz in the present case), it usually carries

random values at different frequencies. In general, a random variable can be either

discrete or continuous. The mean value of a sample consisting of N values (x1, x2,

x3,.., xn) of a random variable ‘x’ is defined by

∑=

=N

jjx

N 1

1µ (6.3)

and the variance, s2 , is defined by

∑=

−−

=N

jjx

Ns

1

22 )()1(

1 µ (6.4)

The square root of s2 is the standard deviation and is denoted by σ.

Experience suggests that most random experiments (involving a random

variable) exhibit statistical regularity or ‘stability’. If D is a random event, there

exists a number p(D) (0 ≤ p(D) ≤ 1) called the probability of D. This means that if

the experiment is performed very often, it is practically certain that the relative

frequency of occurrences of D is approximately equal to p(D).

The probability density function, p(x), of a continuous random variable ‘x’

is a function which defines the probability of the variable over the possible range of

values the variable can attain. The function p(x) satisfies the following condition

1)( =∫∞=

−∞=dxxp

x

x (6.5)

The distribution function or cumulative distribution function, F(x), of such a

continuous variable is defined as

∫=

−∞==

xv

vdvvpxF )()( (6.6)

where the integrand is continuous, possibly except at finitely many values of ν.

Differentiating Eq. (6.6) with respect to x, we get

F’(x) = p(x) (6.7)

The mean of a continuous distribution is defined by


175

dxxxpx

x∫∞=

−∞== )(µ (6.8)

Eqs. (6.5) to (6.8) can be easily modified to suit a discrete random variable by

replacing the integration by summation (Kreyszig, 1993).

Given a data set x1, x2, x3,.., xN of independent observations, the empirical

cumulative distribution function can be obtained by

∑≤

=xx

ii

nN

xF 1)(ˆ (6.9)

where ni is the frequency of xi in the data set. This provides an empirical estimate

of F(x).

The distribution of a random variable encountered in real situations may

conform to any of the standard distributions, such as the normal, the Binomial, the

hypergeometric or the Poisson distribution. Table 6.2 lists the probability

distribution function for these distributions. Details of other standard distributions

are covered by Kreyszig (1993). Whether a given random variable has a distribution

confirming to a standard distribution can be ascertained by means of the

Kolmogorov-Smirnov goodness of fit test. For this purpose, the empirical

distribution, )(ˆ xFn , need to be worked out using Eq. (6.9). The unknown

distribution F(x) is said to fit the specified distribution Fo(x) with a confidence level

of (1-α) (where 0 ≤ α ≤ 1, typically 10 to 15%) if

α≤− )()(ˆmax xFxF on (6.10)

Table 6.2 Common probability distributions.

DISTRIBUTION PROBABILITY DENSITY

FUNCTION f(x)

Normal 2

2

2)(

21 σ

µ

πσ

−−

x

e

Poisson !xex µµ −

Binomial

xnxnx nn

C−

−

µµ 1

Note: nxC = Number of possible combinations of x objects out of n


176

6.6.3 Theory of Fuzzy Sets

Scientists and engineers describe complex physical systems by very simple

mathematical models, often making considerable idealizations in the process. A

practical approach to simplify a complex system is to tolerate a reasonable amount

of imprecision, vagueness and uncertainty during the modelling phase. It was this

logic which Zadeh (1965) employed when he introduced the notion of fuzzy sets.

This principle of scientifically accepting a certain loss of information has turned out

to be satisfactory in many knowledge based systems. Fuzzy systems are widely

used to model information that is afflicted with imprecision, vagueness, and

uncertainty.

A fuzzy set is defined as a class of objects with continuum grades of

membership. Such a set is characterized by a membership (or characteristic)

function, which assigns to each object, a grade of membership ranging from 0 to 1.

Let X be a space of objects with the generic element of X denoted by ‘x’. When A

is a set in space X in the ordinary sense of terms, its membership function can take

only two values 1 and 0, according as ‘x’ does or does not belong to X. On the other

hand, a fuzzy set (or class) Af in X is characterized by a membership (characteristic)

function fm(x), which associates with each object in ‘x’ a real number in the interval

[0,1], representing the “grade of membership of x” in A. The nearer the value of

fm(x) to unity, the higher the grade of membership of ‘x’ in A. For example, let X

be the real line R and let Af be a fuzzy set of numbers which are ‘much’ greater

than ‘1’. Then one can give a precise, albeit subjective values of characterization of

A by specifying fm(x) as a function on R. The representative values of such a

function might be fm(0) = 0, fm(10) = 0.1 and fm(100) = 1.0 and so on. In general,

fuzzy sets have merely an intuitive basis as a formal description of vague data.

Fuzzy sets are generally specified by experts directly in an intuitive way.

Fuzzy sets were first used in civil engineering in the late 1970s (e.g. Brown,

1979). Chameau et al. (1983) suggested many potential applications of fuzzy sets in

civil engineering. Typically in structural analysis, a number of basic variables are

involved such as geometry and dimensions, material parameters, boundary

conditions, loads and the methods of modelling and analysis. Some of these

variables show randomness, some show fuzziness and some are characterized by


177

both. The element of randomness is due to the uncertainty of the loads, the

modelling uncertainties and the statistical uncertainties (due to the use of limited

information). The fuzziness related uncertainty is due to the definition of internal

parameters such as structural performance. Many innovative applications of fuzzy

logic and fuzzy sets in civil engineering can be found in the literature, such as

Dhingra et al. (1992), Valliappan and Pham (1993), Soh and Yang (1996), Wu et al.

(1999, 2001) and Yang and Soh (2000).

The membership functions represent the subjective degree of preference of a

decision maker within a given tolerance. The determination of a fuzzy membership

function is the most difficult as well as the most controversial part of applying the

theory of fuzzy sets for solving engineering problems. In engineering applications,

the most commonly used shapes are linear, half concave, exponential, triangular,

trapezoidal, parabolic, sinusoidal and the extended π-shape (Valliappan and Pham,

1993; Wu et al., 1999, 2001), some of which are shown in Fig. 6.18. The choice of

the particular shape depends on the opinion of the expert, since there is no hard and

fast rule to ascertain which shape is more realistic than others.

If p(D) is the probability density function for describing a structural failure

event D, the failure probability may be expressed as

∫=S

f dDDpP )( (6.11)

where ‘S’ is the space of the structural failure event. However, by the use of fuzzy

set theory, a failure event can be treated as a ‘fuzzy failure event’. If the failure

0

1

0 1

Damage Variable (D)

fA (D

) Linear

Parabolic

Sinusoidal

Exponential

Fig. 6.18 Different types of membership functions for fuzzy sets.


178

space is a fuzzy set with a membership function fm(D), the fuzzy failure probability

can be defined as (Wu et al., 1999)

∫=S

mf dDDpDfP )()( (6.12)

This principle has been used presently in evaluating concrete damage presently.

6.6.4 Statistical Analysis of Damage Variable for Concrete

Coming back to damage diagnosis in concrete, Fig. 6.19 shows the

equivalent spring stiffness worked out at various load ratios (applied load divided

by failure load) for five cubes labelled as C17, C43, C52, C60 and C86. Damage

variables were computed at each frequency in the interval 60-100 kHz,

corresponding to each load ratio, for all the five cubes. Mean and standard deviation

of damage variable were then evaluated at each damage ratio. Statistical

examination of the data pertaining to the damage variable indicated that it followed

a normal probability distribution (see Table 6.2). To verify this, Fig. 6.20 shows the

empirical cumulative probability distribution of Di and also the theoretical normal

probability distribution for all the cubes at or near failure. It is found that the

distribution of the damage variables fits very well into the normal distribution. The

adequacy of the normal distribution was quantitatively tested by Kolmogorov-

Smirnov goodness-of-fit test technique and the normal distribution was found to be

acceptable under a 85% confidence limit for all the cubes. Similarly, damage

variables for all other damage states were also found to follow the normal

probability distribution fairly well.

6.6.5 Fuzzy Probabilistic Damage Calibration of Piezo-Impedance

Transducers

From the theory of continuum damage mechanics, an element can be deemed

to fail if D > Dc. As pointed out earlier, instead of defining a unique value of the

critical damage variable Dc, we are employing a fuzzy definition to take

uncertainties into account. Using the fuzzy set theory, a fuzzy region may be

defined in the interval (DL, DU) where DL and DU respectively represent the lower

and the upper limit of the fuzzy region (Valliappan and Pham, 1993; Wu et al.,

1999). D > DU represents a failure region with 100% failure possibility and D < DL


179

Fig. 6.19 Effect of damage on equivalent spring stiffness (LR stands for ‘Load ratio’).

(a) C17 (b) C43 (c) C52 (d) C60 (e) C86

(a) (b)

(c) (d)

(e)

0.00E+00

1.00E+07

2.00E+07

3.00E+07

4.00E+07

5.00E+07

6.00E+07

60 70 80 90 100

Frequency (kHz)

Equ

ivale

nt S

tiffn

ess

(N/m

) LR = 0LR = 0.268

LR = 0.670LR = 0.536

LR = 0.804

LR = 0.402

LR = 1.000

3.00E+07

3.50E+07

4.00E+07

4.50E+07

5.00E+07

5.50E+07

60 70 80 90 100

Frequency (kHz)

Equi

vale

nt s

tiffn

ess

(N/m

) LR = 0.311

LR = 0.726

LR = 0.519

LR = 0.830

LR = 1.000LR = 0.882

LR = 0

0.00E+00

1.00E+07

2.00E+07

3.00E+07

4.00E+07

60 70 80 90 100

Frequency (kHz)

Equi

vale

nt s

tiffn

ess

(N/m

) LR = 0LR = 0.172

LR = 0.517LR = 0.345

LR = 1.000

LR = 0.690LR = 0.862

1.50E+07

2.00E+07

2.50E+07

3.00E+07

3.50E+07

4.00E+07

80 84 88 92 96 100

Frequency (kHz)

Equ

ivale

nt S

tiffn

ess

(N/m

)

LR = 0 LR = 0.148

LR = 0.444LR = 0.296

LR = 0.592

LR = 1.000

LR = 0.741

LR = 0.963LR = 0.888

1.50E+07

2.00E+07

2.50E+07

3.00E+07

3.50E+07

4.00E+07

4.50E+07

60 70 80 90 100

Frequency (kHz)

Equ

ivale

nt s

tiffn

ess

(N/m

)

LR = 0LR = 0.206

LR = 0.774

LR = 0.413LR = 0.619

LR = 1.000LR = 0.929


180

0

0.2

0.4

0.6

0.8

1

0.35 0.4 0.45 0.5

D

Pro

babi

lity

Dis

tribu

tion

EmpiricalTheoretical

0

0.2

0.4

0.6

0.8

1

0.5 0.6 0.7 0.8 0.9 1

D

Prob

abili

ty D

istri

butio

n

Empirical

Theoretical

0

0.2

0.4

0.6

0.8

1

0.2 0.3 0.4 0.5 0.6

D

Prob

abili

ty D

istri

butio

n

Empirical

Theoretical

Fig. 6.20 Theoretical and empirical probability density functions near failure.

(a) C17 (b) C43 (c) C52 (d) C60 (e) C86

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

D

Prob

abili

ty D

istri

butio

n

Empirical

Theoretical

0

0.2

0.4

0.6

0.8

1

0.4 0.5 0.6 0.7 0.8 0.9 1

D

Prob

abili

ty D

istri

butio

n

Empirical

Theoretical

(a) (b)

(c) (d)

(e)


181

represents a safe region with 0% failure possibility. Within the fuzzy or the

transition region, that is DL< D < DU, the failure possibility could vary between 0%

and 100%. A characteristic or a membership function fm could be defined (0<

fm(D)<1) to express the grade of failure possibility within the region (DL, DU). The

fuzzy failure probability can then be determined, from Eq. (6.12), as

∫=

=

=≥=1

0

)()()(D

DmCf dDDpDfDDPP (6.13)

where p(D) is the probability density function of the damage variable D, which in

the present case complies with normal distribution. Based on observations during

concrete cube compression tests, DL and DU were chosen as 0.0 and 0.40

respectively. Further, sinusoidal membership function, given by following equation,

was adopted

−−

−+= )5.05.0(

)(sin5.05.0 LU

LUm DDD

DDf π

(6.14)

This function was chosen since it was found to reflect the observed trend in

transducer response (in terms of damage variable based on ‘identified’ equivalent

stiffness) with damage growth. From practical experience, it has been observed that

the damage variable typically follows the trend of an S-curve, i.e. initially rising

steeply with damage progression and then attaining saturation. This is represented

very well by the sinusoidal membership function.

Making use of this membership function, the fuzzy failure probability (FFP)

was worked out for the five concrete cubes at each load ratio. A MATLAB program

listed in Appendix F was used to perform the computations. It should be noted that

Wu et al. (1999) used similar principles to carry out fuzzy probabilistic damage

prediction of rock masses to explosive loads.

A load ratio of 0.4 can be regarded as incipient damage since concrete is

expected to under ‘working loads’. All concrete cubes were found to exhibit a fuzzy

failure probability of less than 30% at this load ratio. Similarly, after a load ratio of

0.8, the concrete cubes can be expected to be under ‘ultimate loads’. For this case,

all the cubes exhibited a fuzzy failure probability of greater than 80% irrespective

of strength. This is shown in Fig. 6.21. Fig. 6.22 shows the FFP of the cubes at

intermediate stages during the tests. Based on minute observations during the tests


182

on concrete cubes, following classification of damage is recommended based on

FFP.

(1) FFP < 30% Incipient Damage (Micro-cracks)

(2) 30% < FFP < 60% Moderate damage (Cracks start

opening up)

(3) 60% < FFP < 80% Severe damage (large visible cracks)

(4) FFP > 80% Failure imminent

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Load ratio

Fuzz

y Fa

ilure

Pro

babi

lity

% C17C43C52C60C86

Incipient Damage

Moderate Damage

Severe Damage

Failure Imminent

Micro cracks

Cracks opening up

Large visible cracks

Fig. 6.21 Fuzzy failure probabilities of concrete cubes at

incipient damage level and at failure stage.

0

20

40

60

80

100

C 17 C 43 C 52 C 60 C 86

Fuzz

y Fa

ilure

Pro

babi

lity

(%)

Incipient Damage

Severe Damage (Failure Imminent)

Fig. 6.22 Fuzzy failure probabilities of concrete cubes at

various load levels.


183

Thus, the fuzzy probabilistic approach quantifies the extent of damage on a uniform

0-100% scale. This can be employed to evaluate damage in real-life concrete

structures.

6.7 DISCUSSIONS

All the PZT patches exhibited more or less a uniform behaviour with damage

progression in concrete, although the strength of concrete cubes varied from as low

as 17 MPa to as high as 86 MPa. Hence, the PZT patches were subjected to a wide

range of mechanical stresses and strains during the tests. At a load ratio of 1.0,

almost same order of FFP is observed, irrespective of the absolute load or stress

level (for example 17 MPa for C17 and 86 MPa for C86). In general, the PZT

material shows very high compressive strength, typically over 500 MPa and it

essentially exhibits a linear stress-strain relation up to strains as high as 0.006. A

typical experimental plot for the PZT material is shown in Fig. 6.23 (Cheng and

Reece, 2001). In the experiments conducted on concrete cubes, the strain level

never exceeded 0.003 (50% of the linear limit). Also, it was observed that in all the

cubes tested, the damage typically initiated near the edges of the cube and migrated

to regions near the PZT patch with increasing load ratios. After failure of the cubes,

all the PZT patches were found intact. Fig. 6.24 shows close ups of the cubes after

the tests.

Fig. 6.23 Typical stress-strain plot for PZT (Cheng and Reece, 2001).


184

Fig. 6.24 Cubes after the test. (a) C17 (b) C43 (c) C52 (d) C60 (e) C86

(a) (b)

(c) (d)

(e)

C17

C43

C52

C86

C60


185

The results show that the sensor response reflected more the damage to the

surrounding concrete rather than damage to the patches themselves. In general, we

can expect such good performance in materials like concrete characterized by low

strength as compared to the PZT patches. Hence, damage to concrete is likely to

occur first, rather than the PZT patch. Further, though the cubes were tested in

compression, the same fuzzy probabilistic damage model can be expected to hold

good for tension also.


This chapter has covered the development of a new experimental technique

based on EMI technique for evaluating concrete strength non-destructively. Also, it

has shown the feasibility of monitoring concrete curing using piezo-impedance

transducers. It is found that the equivalent spring stiffness of concrete “identified”

by a surface bonded PZT patch can serve as a damage sensitive structural

parameter. It could be utilized for identifying and quantifying damages in concrete.

A fuzzy probability based damage model is proposed based on the extracted

equivalent stiffness to evaluate the extent of damage using the impedance data. This

has facilitated the calibration of the piezo-impedance transducers in terms of

damage severity and this can serve as a convenient empirical phenomenological

damage model for quantitatively estimating damage in concrete in the real-life

structures.

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

186

Chapter 7

INCLUSION OF INTERFACIAL SHEAR LAG EFFECT INIMPEDANCE MODELS

7.1 INTRODUCTION

The piezo-impedance transducers are bonded to the surface of the host

structures using an adhesive mix (such as epoxy), which forms a permanent finite

thickness interfacial layer between the structure and the patch. In the analysis

presented so far in this thesis, the effects of this layer were neglected. The force

transmission from the PZT patch to the host structure was assumed to occur at the

ends of the patch (1D model of Liang et al., 1994) or along the continuous boundary

edges of the patch (2D effective impedance model, Chapter 5). In reality, the force

transfer takes place through the interfacial bond layer via shear mechanism. This

chapter reviews the mechanism of force transfer through the bond layer and

presents a step-by-step derivation to integrate this mechanism into impedance

formulations, both 1D and 2D. The influence of various parameters (associated with

the bond layer) on the electro-mechanical admittance response are also investigated.

7.2 SHEAR LAG EFFECT

Fig. 7.1 A PZT patch bonded to a beam using adhesive bond layer.

τ

Tp+ ∂Tp∂x dxTp

dx

BEAM

ts

tpBond layer

PZT patch

l lx

ydx

DifferentialElement


187

Crawley and de Luis (1987) and Sirohi and Chopra (2000b) respectively

modelled the actuation and sensing of a generic beam element using an adhesively

bonded PZT patch. The typical configuration of the system is shown in Fig. 7.1.

The patch has a length 2l, width wp and thickness tp, while the bonding layer has a

thickness equal to ts. The adhesive layer thickness has been shown exaggerated to

facilitate visualization. The beam has depth tb and width wb. Let Tp denote the axial

stress in the PZT patch and τ the interfacial shear stress. Following assumptions

were made by Crawley and de Luis (1987) and Sirohi and Chopra (2000b) in their

analysis:

(i) The system is under quasi-static equilibrium.

(ii) The beam is actuated in pure bending mode and the bending strain is

linearly distributed across any cross section.

(iii) The PZT patch is in a state of pure 1D axial strain.

(iv) The bonding layer is in a state of pure shear and the shear stress is

independent of ‘y’.

(v) The ends of the segmented PZT actuator/ sensor are stress free, implying a

uniform strain distribution across the thickness of the patch.

A more detailed deformation profile is shown in Fig. 7.2, which shows the

symmetrical right half of the system of Fig. 7.1. Let ‘up’ be the displacement at the

interface between the PZT patch and the bonding layer and ‘u’ the corresponding

displacement at the interface between the bonding layer and the beam.

Fig. 7.2 Deformation in bonding layer and PZT patch.

A

B

A’

B’

x u

up

Bondinglayer

PZT patch

Beam

Afterdeformation

x

yupo

uo


188

7.2.1 PZT Patch as Sensor

Let the PZT patch be instrumented only to sense strain on the beam surface

and hence no external electric field be applied across it. Considering the static

equilibrium of the differential element of the PZT patch in the x-direction, as shown

in Fig. 7.1(a), we can derive

pp t

xT∂

∂=τ (7.1)

At any cross section of the beam, within the portion containing the PZT patch, the

bending moment is given by

)5.05.0( psbppp ttttwTM ++= (7.2)

Also, from Euler-Bernoulli’s beam theory,

−=

bb t

IM5.0

σ (7.3)

where σb is the bending stress at the extreme fibre of the beam and ‘I’ the second

moment of inertia of the beam cross-section. The negative sign signifies that

sagging moment and tensile stresses are considered positive. Comparing Eqs. (7.2)

and (7.3) and with 12/3bbtwI = , we get

0)2(3

2 =++

+ spb

bb

pppb ttt

twtwT

σ (7.4)

Assuming (tp+2ts )<<tb ,differentiating with respect to x, and substituting Eq. (7.1),

we get

03

=

+

∂∂

τσ

bb

pb

tww

x (7.5)

Further, from Hooke’s law,

bbb SY=σ (7.6)

pE

p SYT = (7.7)

γτ sG= (7.8)

where Yb and YE respectively denote the Young’s modulus of elasticity of the beam

and the PZT patch (at zero electric field for the patch) respectively and Sb and Sp the

corresponding strains. Gs denotes the shear modulus of elasticity of the bonding


189

layer and γ the shear strain undergone by it. Substituting Eqs. (7.6) to (7.8) into Eqs.

(7.1) and (7.5), we get Eqs. (7.9) and (7.10) respectively.

xS

tYG ppps ∂

∂=γ (7.9)

03

=

+

∂∂

γsbb

pbb G

tww

xSY (7.10)

From Fig. 7.2, the shear strain in the bonding layer can be determined as

s

p

tuu −

=γ (7.11)

Substituting Eq. (7.11) into Eqs. (7.9) and (7.10), differentiating with respect to x,

and simplifying, we get Eqs. (7.12) and (7.13) respectively

ξ

=

∂

∂

psp

bsp

ttYSG

xS

2

2

(7.12)

ξ

−=

∂∂

pbbb

bspb

ttwYSGw

xS 3

2

2

(7.13)

where

−= 1

b

p

SS

ξ (7.14)

Subtracting Eq. (7.13) from Eq. (7.12), we get

022

2

=Γ−∂∂

ξξ

x (7.15)

where

+=Γ

pbbb

ps

psp

s

ttwYwG

ttYG 32 (7.16)

This phenomenon of the difference in the PZT strain and the host structure’s

strain is called as shear lag effect. The parameter Γ (unit m-1) is called the shear lag

parameter. The ratio ξ is called as strain lag ratio. The ratio ξ is a measure of the

differential PZT strain relative to surface strain on the host substrate, caused by

shear lag. The general solution for Eq. (7.15) can be written as

xBxA Γ+Γ= sinhcoshξ (7.17)

Since the PZT patch is acting as sensor, no external field is applied across it. Hence,

free PZT strain = d31E3 = 0. Thus, following boundary conditions hold good:

(i) At x = -l , Sp = 0 ⇒ ξ = -1. (ii) At x = +l, Sp = 0 ⇒ ξ = -1.


190

Applying these boundary conditions, we can obtain the constants A and B as

lA

Γ−

=cosh

1 and B = 0 (7.18)

Hence, lxΓΓ

−=coshcosh

ξ (7.19)

Using Eq. (7.14), we can derive

ΓΓ

−=lx

SS

b

p

coshcosh1 (7.20)

Fig. 7.3 shows a plot of the strain ratio (Sp/Sb) across the length of a PZT patch (l =

5mm) for typical values of Γ = 10, 20, 30, 40, 50 and 60 (cm-1). From this figure, it

is observed that the strain ratio (Sp/Sb) is less than unity near the ends of the PZT

patch. The length of this zone depends on Γ, which in turn depends on the stiffness

and thickness of the bond layer (Eq. 7.16). As Gs increases and ts reduces, Γ

increases, and as can be observed from Fig. 7.3, the shear lag phenomenon becomes

less and less significant and the shear is effectively transferred over very small

zones near the ends of the PZT patch.

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

(x/l)

(Sp/Sb)

Γ = 10

20

30

5040

60

Fig. 7.3 Strain distribution across the length of PZT patch for

various values of Γ.


191

Thus, if the PZT patch is used a sensor, it would develop less voltage across

its terminals (than for perfectly bonded conditions) due to the shear lag effect. In

other words, it will underestimate the strain in the substructure. In order to quantify

the effect of shear lag, we can compute effective length of the sensor, as defined by

(Sirohi and Chopra, 2000b)

∫==

=

lx

xbpeff dxSSl

0)/( (7.21)

which is nothing but area under the curve (Fig. 7.3) between x = 0 and x = l. Hence,

this is a sort of ‘equivalent length’, which could be deemed to have a constant

strain, equal to Sb, the strain on the beam surface. Substituting Eq. (7.20) into Eq.

(7.21) and upon integrating, we can derive effective length factor as

ll

lleff

ΓΓ

−=tanh1 (7.22)

Fig. 7.4 shows a plot of the effective length (Eq. 7.22) for various values of the

shear lag parameter Γ. Typically, for Γ > 30cm-1, (leff / l) > 93%, suggesting that

shear lag effect can be ignored for relatively high (> 30 cm-1) values of Γ.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Γ (cm-1)

leff /l

Fig. 7.4 Variation of effective length with shear lag factor.


192

7.2.2 PZT Patch as Actuator

If the PZT patch is employed as an actuator for a beam structure, it can be

shown (Crawley and de Luis, 1987) that the strains Sp and Sb will be as given by

lxS p Γ+

ΓΛ+

+Λ

=cosh)3(

cosh)3(

3ψψ

ψ (7.23)

lxSb Γ+

ΓΛ−

+Λ

=cosh)3(

cosh3)3(

3ψψ

(7.24)

where Λ = d31E3 is the free piezoelectric strain and ψ = (Ybtb/YEtp) is the product of

modulus and thickness ratios of the beam and the PZT patch. Fig. 7.5 shows the

plots of (Sp / Λ) and (Sb / Λ) along the length of the PZT patch (l = 5mm) for ψ =

15. It is observed that like in the case of sensor, as Γ increases, the shear is

effectively transferred over small zone near the two ends of the patch. As Γ → ∞,

the strain is transferred over an infinitesimal distance near the ends of the PZT

patch. For the limiting case, as apparent from Fig. 7.5,

)3(3ψ+Λ

== pb SS (7.25)

which sets the maximum fraction of the piezoelectric free strain Λ that can be

induced into the beam. Further, as ψ → 0, Sb → Λ.Typically, for Γ > 30cm-1, the

strain energy induced in the substructure by PZT actuator is within 5% of the

perfectly bonded case. Therefore, for Γ > 30 cm-1, ignoring the effect of the bond

layer will provide sufficiently accurate results for most engineering models.

It should be noted here that the analysis carried out by Crawley and de Luis

(1987) as well as Sirohi and Chopra (2000b) is valid for static conditions only.

These researchers extended their formulations to dynamic problems under the

assumption that the operating frequency is small enough to ensure that the PZT

patch acts ‘quasi-statically’. However, in the EMI technique, the operational

frequencies are of the order of the resonant frequency of the PZT patch, warranting

that the actuator dynamics should not be neglected.


-1 -0.5 0 0.5 10

0.05

0.1

0.15

0.2

(x/l)

(Sb/Λ)

Γ = 10

20

3050

40

60

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

(x/l)

(Sp/Λ)

Γ = 10

20

30

50 40

60

(a)

(b)

Fig. 7.5 Distribution of piezoelectric and beam strains for various values of Γ.

(a) Strain in PZT patch. (b) Beam surface strain.

193


194

7.3 INTEGRATION OF SHEAR LAG EFFECT INTO IMPEDANCE

MODELS

It was observed in the previous section that both as an actuator as well as a

sensor, shear lag effect is associated with the mechanism of force transmission

between the PZT patch and the host structure via the adhesive bond layer. However,

till date, this aspect has not been thoroughly investigated with respect to the EMI

technique, where the same patch serves as a sensor as well as an actuator

concurrently. Abe et al. (2002) encountered large errors in their stress prediction

methodology using EMI technique. This error was attributed to imprecise modelling

of the interfacial bonding layer.

Xu and Liu (2002) proposed a modified impedance model in which the

bonding layer was modelled as a SDOF system, connected in between the PZT

patch and the host structure, as shown in Fig. 7.6. The bonding layer was assumed

to possess a dynamic stiffness bK (or mechanical impedance bK /jω) and the

structure a dynamic stiffness sK (or mechanical impedance, Zs = sK /jω). Hence,

the resultant mechanical impedance for this series system can be determined as

(using Eq. 3.5)

ssb

b

sb

sb

res ZKK

K

jK

jK

jK

jK

Z

+=

+

=

ωω

ωω (7.26)

Fig. 7.6 Modified impedance model of Xu and Liu (2002) including bond layer.

k

c

m

PZT patch

Z

Bonding layer

StructureDynamicstiffness = Kb


195

or sres ZZ ζ= (7.27)

where ( )bs KK /11

+=ζ (7.28)

Hence, the coupled electromechanical admittance, as measured across the terminals

of the PZT patch and expressed earlier by Eq. (2.24), can be corrected as

+

+−=l

lYdZZ

ZYdhwljY E

sa

aET

κκ

ζεω

tan)(2 231

23133 (7.29)

The term ζ in this equation modifies the dynamic interaction between the PZT patch

and the host structure, taking into consideration the effect of the bonding layer.

ζ = 1 implies a very stiff bonding layer where as ζ = 0 implies free PZT patch. Xu

and Liu (2002) demonstrated numerically that for a SDOF system, as ζ decreases

(i.e bond quality degrades), the PZT system would show an increase in the

associated resonant frequencies. The investigators further stated that bK depends

on the bonding process and the thickness of the bond layer. However, no closed

form solution was presented to quantitatively determine bK and hence ζ (From

Eq. 7.28). Besides, no experimental verification was provided. The fundamental

mechanism of force transfer was therefore nowhere reflected in their analysis.

Ong et al (2002) integrated the shear lag effect into impedance modelling

using the analysis presented by Sirohi and Chopra (2000b). The PZT patch was

assumed to possess a length equal to leff (Eq. 7.22) instead of the actual length.

However, since the effective length was determined by considering sensor effect

only, the method took care of the associated shear lag only partially. Also, the

formulation was valid for beam type structures only and not general in nature.

Besides, since the frequencies of the order of 100-150kHz are involved, quasi-static

approximation (for calculating leff) is strictly not valid.

This chapter presents a detailed step-by-step analysis for including the shear

lag effect, first into 1D model (Liang et al., 1994) and then its extension into 2D

effective impedance based model (covered in Chapter 5).


196

7.4 INCLUSION OF SHEAR LAG EFFECT IN 1D IMPEDANCE MODEL

Consider the PZT patch, shown in Figs. 7.1 and 7.2, to be driven by an

alternating voltage source and let it be attached to any host structure (not

necessarily beam). All the assumptions of Sirohi and Chopra (2000) and Crawley

and de Luis (1987) (except for static condition) still hold good. In addition, we

assume that the PZT patch is infinitesimally small as compared to the host structure.

This means that the host structure has constant mechanical impedance all along the

points of attachment of the patch. By D’Alembert’s principle, we can write

following equation for dynamic equilibrium of an infinitesimal element of the patch

dxwtx

Ttu

dmdxw pppp

p ∂

∂=

∂

∂+ 2

2

)(τ (7.30)

where ‘dm’ is the infinitesimal mass and up the displacement in the PZT patch at the

location of the infinitesimal element. Because of the dominance of shear stress term,

we can neglect the inertial term in Eq. (7.30). The inertial force term has been

separately considered in impedance model (Chapter 2 and 5), where as a matter of

fact, the shear lag effect was ignored. Hence, the two effects are individually

considered and then combined. With this assumption, Eq. (7.30) can be reduced to

or pp t

xT∂

∂=τ (7.31)

Further, assuming pure shear in the bonding layer,

s

ps

tuuG )( −

=τ (7.32)

where sG = Gs(1+η′ j) is the complex shear modulus of the bonding layer and η′

the mechanical loss factor associated with the bond layer. The axial stress in the

PZT patch is given by (from PZT constitutive relation, Eq. 2.14 )

)( ∧−= pE

p SYT (7.33)

or )( ∧−′= pE

p uYT (7.34)

where EY is the complex Young’s modulus of the PZT patch, pp uS ′= is the PZT

strain and ∧ = E3d31 is the free piezoelectric strain. Substituting Eqs. (7.32) and

(7.34) into Eq. (7.31) and simplifying, we get


197

ps

spE

p uG

ttYuu ′′

=− (7.35)

At any vertical section through the host structure (which includes PZT patch), the

force transmitted to the host structure is related to the drive point impedance Z of

the host structure by

uZF &−= (7.36)

where u& is the drive point velocity at the point in question on the surface of the host

structure. Since the PZT patch is infinitesimally small, Z is practically same along

the entire length of the PZT patch. Eq. (7.36) can be further simplified as (noting

that uju ω=& )

ωZujtwT ppp −= (7.37)

Substituting Eq. (7.34) and differentiating with respect to x (noting that Z is

constant), and rearranging, we get

uYtw

ZjuE

pp

p ′

−=′′

ω (7.38)

By rearranging various terms, Eq. (7.35) can be written as

( )uuttY

Gu p

spE

sp −

=′′ (7.39)

Substituting Eq. (7.39) into Eq. (7.38) and solving, we get

uwGjZt

uups

sp ′

−=−

ω (7.40)

Eqs. (7.35) and (7.40) are the fundamental equations governing the shear transfer

mechanism via the adhesive bonding layer. Differentiating Eq. (7.40) twice with

respect to x and rearranging, we can obtain

uGwjZt

uusp

sp ′′′

−′′=′′

ω (7.41)

Substituting Eqs. (7.40) and (7.41) into Eq. (7.35), differentiating with respect to x,

and rearranging, we get


198

0=′′

−′′′

−+′′′′ u

ttY

Gu

jZtGw

ups

Es

s

sp

ω (7.42)

Let ωjZt

Gwp

s

sp−= (7.43)

Substituting yjxZ += , ss GjG )1( η′+= and simplifying, we get

bjap += (7.44)

where )()(

22 yxtxyGw

as

sp

+

′−=

ω

ηand

)()(

22 yxtyxGw

bs

sp

+

′+=

ω

η (7.45)

Since η and η′ are very small in magnitude, the coefficient of u ′′ can be written as

psE

s

psE

s

ttYG

ttYGq ≈= (7.46)

It should be noted that p is a complex term whereas the term ‘q’ is approximated as

a pure real term. Hence, the resulting differential equation (Eq. 7.42) can be written

as

0=′′−′′′+′′′′ uqupu (7.47)

The characteristic equation for this differential equation is

0234 =−+ λλλ qp (7.48)

Solving, we get roots of the characteristic equation as

24

,2

4,0,0

2

4

2

321qppqpp +−−

=++−

=== λλλλ (7.49)

Hence, the solution of the differential equation Eq.(7.19) can be written asxx CeBexAAu 43

21λλ +++= (7.50)

The constants A, B, C and D are to be evaluated from the boundary conditions.

Differentiating with respect to x, we getxx eCeBAu 43

432λλ λλ ++=′ (7.51)

Substituting Eqs. (7.50) and (7.51) into Eq.(7.40), we get

( )xx

sp

sxxp eCeBA

GwjZtCeBexAAu 4343

43221 )( λλλλ λλω

++

−+++= (7.52)


199

Denoting

−

sp

s

GwjtZ ω = p/1 by n , and simplifying, we get

xxp enCenBxAAnAu 43 )1()1()( 43221

λλ λλ ++++++= (7.53)

Differentiating with respect to x, we can obtain the strain in the PZT patch asxx

p enCenBAS 43 )1()1( 44332λλ λλλλ ++++= (7.54)

At x = 0 (the mid point of the PZT patch, Figs. 7.1 and 7.2), u = 0, which leads to

following condition from Eq. (7.50)

A1 = -(B + C) (7.55)

Further, the boundary condition that at x = 0 up = 0 leads to (from Eq. 7.53)

A2 = -(Bλ3 + Cλ4) (7.56)

Making substitution for A2 from Eq. (7.56) into Eq. (7.54), we get

])1([])1([ 44433343 λλλλλλ λλ −++−+= xx

p enCenBS (7.57)

The third and the fourth boundary conditions are imposed by the stress free ends of

the PZT patch. At x = -l and at x = +l, the axial strain in the PZT patch is equal to

the free piezoelectric strain or Λ (Crawley and de Luis, 1987). The application of

these two boundary conditions in Eq. (7.57) result in following equations

[ ] [ ] Λ=−++−+ −−444333

43 )1()1( λλλλλλ λλ ll enCenB (7.58)

[ ] [ ] Λ=−++−+ 44433343 )1()1( λλλλλλ λλ ll enCenB (7.59)

After solving these equations, the constants B and C can be determined as

( )

−−

−Λ

=

31

24

3241 kkkk

kkkkCB

(7.60)

where

33313)1( λλλ λ −+= − lenk (7.61)

44424)1( λλλ λ −+= − lenk (7.62)

33333)1( λλλ λ −+= lenk (7.63)

44444)1( λλλ λ −+= lenk (7.64)

In general, the force transmitted to the host structure can be expressed

as )( lxuZjF =−= ω , where u(x=l) is the displacement at the surface of the host structure

at the end point of the PZT patch. Conventional impedance models (e.g. Liang and


200

coworkers) assume perfect bonding between the PZT patch and the host structure,

i.e. the displacement compatibility u(x=l) = up(x=l), thereby approximating the

transmitted force as )( lxpujZF =−= ω . However, due to the shear lag phenomenon

associated with finitely thick bond layer, u(x=l) ≠ up(x=l). Based on the analysis

presented in this section, we can obtain following relationship between u(x=l) and

up(x=)l using Eq. (7.40)

′+

=′

−

=

=

==

=

o

o

lx

lx

sp

slxp

lx

uu

puu

GwjZtu

u11

1

1

1

)(

)()(

)(

ω (7.65)

where ou is as shown in Fig. 7.2. The term )()( / lxlx uu ==′ can be determined by using

Eqs. (7.50) and (7.51). Making use of this relationship, the force transmitted to the

structure can be written as

)( lxZuF =−= (7.66)

or )()(11lxpeqlxp

o

o

ujZuj

uu

p

ZF == =

′+

−= ωω (7.67)

where

′+

=

o

oeq

uu

p

ZZ11

(7.68)

is the ‘equivalent impedance’ apparent at the ends of the PZT patch, taking into

consideration the shear lag phenomenon associated with the bond layer. In the

absence of shear lag effect (i.e. perfect bonding), Zeq = Z.

On comparing with the result of Xu and Liu (2002) (Eq. 7.27), we find that

o

o

uu

p′

+= 11

1ζ (7.69)

The interfacial shear stress can be calculated by using Eq. (7.32). Substituting Eq.

(7.40) into Eq. (7.32), we get

uwZj

p

′

−=

ωτ (7.70)

From Eqs.(7.51), (7.56) and (7.70) we get


201

( ) ( )[ ]p

xx

weCeBZj 11 43

43 −+−−=

λλ λλωτ (7.71)

7.5 EXTENSION TO 2D-EFFECTIVE IMPEDANCE BASED MODEL

The formulations derived above can be easily extended to the effective

impedance based electro-mechanical model developed in Chapter 5. For this

derivation, it is assumed that the PZT is square in shape with a length equal to 2l.

The strain distribution and the associated shear lag are determined along each

principal direction and the two effects are assumed independent, which means that

the effects at the corners are neglected.

Consider an infinitesimal element of the PZT patch in dynamic equilibrium, as

shown in Fig. 7.7. Since this shows planar view, the shear stresses τxz and τyz are

not visible in the figure. Considering equilibrium along x-direction we can write

(De Faria, 2003),

01 =−∂

∂+

∂∂

p

xzxy

tyxT ττ

(7.72)

Ignoring the terms involving rate of change of shear strains (consistent with the

observation by Zhou et al., 1996), we get

p

xz

txT τ

=∂∂ 1 (7.73)

Further, using Eqs. (5.11) and (5.12), we can derive

( )[ ])1()1( 2121 νν

ν+Λ−+

−= SSYT

E

(7.74)

x

ydx

dyT1

τxy

dxxTT

∂∂

+ 11

T2

dyyTT

∂∂

+ 22

dyyxy

xy

∂∂

+τ

τ

Fig. 7.7 Stresses acting on an infinitesimal PZT element.


202

where ν is the Poisson’s ratio of the PZT patch.

or [ ])1()()1( 21 νυ

ν+Λ−′+′

−= pypx

E

uuYT (7.75)

Differentiating with respect to x and ignoring the second order terms involving both

x and y (Zhou et al., 1996), we get

''2

1

)1( px

E

uYxT

ν−=

∂∂ (7.76)

Substituting Eq. (7.76) into Eq. (7.73) and expanding τxz, we get

ps

pxspx

E

ttuuG

uY )()1(

''2

−=

−ν (7.77)

On rearranging, we get

''2 )1( px

s

psE

xpx uG

ttYuu

ν−=− (7.78)

Similarly, we can write, for the other direction

''2 )1( py

s

psE

ypy uG

ttYuu

ν−=− (7.79)

Adding Eqs. (7.78) and (7.79) and dividing by 2, we get

( ) ( ) ( )2)1(22 2

pypx

s

spE

yxpypx uuG

ttYuuuu ′′+′′

−=

+−

+

ν (7.80)

From Eq. (5.19), based on the definition of ‘effective displacement’, we can write

effps

spE

effeffp uG

ttYuu ,2, )1(

′′

−=−

ν (7.81)

or effpeff

effheffp uq

uu ,,,1 ′′≈− (7.82)

where effq has been approximated as pure real number, as in the 1D case. Here,

effpu , , by definition, is the effective displacement at the interface between the PZT

patch and the bond layer and ueff is the corresponding effective displacement at the

interface between the structure and the bonding layer. Further, from the definition

of effective impedance, introduced in Chapter 5, we can write, for the host structure


203

ωjuZFF effeff−=+ 21 (7.83)

or ωjuZltTltT effeffpp −=+ 21 (7.84)

From Eq. (5.13), we get

ων

juZSSltY

effeffppp

E

−=−

Λ−+

)1()2( 21 (7.85)

Substituting for pxp uS ′=1 and pyp uS ′=2 , making use the definition of effective

displacement, and differentiating, we can derive

effp

pE

effeffp u

ltY

jZu ,,

2

)1(′

−−=′′

ων (7.86)

Substituting for effpu ,′′ from Eq. (7.81), we get

effs

seffeffeffp u

lGjtZ

uu ′

+−=−

)1(2, ν

ω (7.87)

or effeff

effeffp up

uu ′

=−

1, (7.88)

Eqs. (7.82) and (7.88) are the governing equations for 2D case. The parameters

effp and qeff are thus given by

+−=

ων

jtZlGp

seff

seff

)1(2

spE

seff ttY

Gq )1( 2ν−≈ (7.89)

The rest of the procedure is identical to the one outlined in the previous section for

1D case. The equivalent effective impedance can then be derived as

eff

lxeff

lxeff

eff

effeqeff Z

uu

p

ZZ ζ=

′+

=

=

=

)(,

)(,,

11

(7.90)

7.6 EXPERIMENTAL VERIFICATION

In order to verify the derivations outlined above, two PZT patches,

10x10x0.3mm and 10x10x0.15mm, conforming to grade PIC 151 (PI Ceramic,


204

2003), were bonded to two aluminium blocks, each 48x48x10mm in size. The

experimental set-up shown in Fig. 5.5 (page 117) was employed. The PZT patches were

bonded to the blocks using RS 850-940 two-part epoxy adhesive (RS Components,

2003). The adhesive layer thickness was maintained at 0.125 mm for both the

specimens using two optical fibre pieces of this diameter, by the procedure outlined

earlier in Chapter 6. The two specimens have (ts/tp) ratio equal to 0.417 and 0.833

respectively.

For obtaining the effective mechanical impedance of the host structure, the

numerical approach based on FEM, outlined earlier in chapter 5, was employed. The

shear modulus of elasticity of the epoxy adhesive was assumed as 1.0 GPa in

accordance with Adams and Wake (1984). The mechanical loss factor of commercial

adhesives shows a wide variation and is strongly dependent on temperature. It might

vary from 5% to 30% at room temperature, depending upon the type of adhesive

(Adams and Wake, 1984). For this study, a value of 10% has been considered. A

MATLAB program listed in Appendix G was used to perform the computations

automatically.

Fig. 7.8 shows the plot of normalized conductance (Gh/L2) worked out using the

integrated 2D model developed in this chapter for the two specimens. The plot for

perfectly bonded condition is also shown. It is observed that with increasing thickness

of the adhesive layer, the sharpness of peaks in the conductance plot tends to diminish.

This fact is confirmed by the experimental plots shown in Fig. 7.9 for the two

specimens. Fig. 7.10 shows the plot of normalized susceptance (Bh/L2), worked out

using the new model for three cases- no bond layer, (ts/tp) = 0.417 and (ts/tp) = 0.833.

Again, it is observed that an increase in thickness tends to flatten the peaks. Besides,

00.0050.01

0.0150.02

0.0250.03

0 50 100 150 200 250

Frequency (kHz)

G, N

orm

aliz

ed (S

/m)

00.0050.01

0.0150.02

0.0250.03

0 50 100 150 200 250

Frequency (kHz)

G, N

orm

aliz

ed (S

/m)

Fig. 7.8 Theoretical normalized conductance.

(a) Perfect bonding. (b) ts/tp = 0.417. (c) ts/tp = 0.834.

(c)

ts/tp = 0.834ts/tp = 0.417

00.0050.01

0.0150.02

0.0250.03

0 50 100 150 200 250

Frequency (kHz)

G, N

orm

aliz

ed (S

/m)

Perfect bonding

(a) (b)


205

average slope of the curve also reduces marginally. This is confirmed by Fig. 7.11,

which shows the curves determined experimentally for the two specimens. Thus, the

shear lag model has made reasonably accurate predictions.

0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200 250

Frequency (kHz)

B, N

orm

aliz

ed (S

/m)

ts/tp =0.834

ts/tp =0.417

0

0.005

0.01

0.015

0.02

0.025

0 50 100 150 200 250

Frequency (kHz)

G, n

orm

aliz

ed (S

/m)

ts/tp =0.417

ts/tp =0.834

Fig. 7.11 Experimental normalized susceptance for ts/tp = 0.417 and ts/tp = 0.834.

Fig. 7.9 Experimental normalized conductance for ts/tp = 0.417 and ts/tp = 0.834.

0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200 250

Frequency (kHz)

B, N

orm

aliz

ed (S

/m)

0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200 250

Frequency (kHz)

B, N

orm

aliz

ed (S

/m)

(c)

ts/tp = 0.834ts/tp = 0.417

0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200 250

Frequency (kHz)

B, N

orm

aliz

ed (S

/m)

Perfect bonding

Fig. 7.10 Theoretical normalized susceptance.

(a) Perfect bonding. (b) ts/tp = 0.417. (c) ts/tp = 0.834.

(a) (b)


206

That excessive bond layer thickness coupled with poor bond quality can

adversely affect the signatures can be investigated by considering the case the case

ts/tp = 1.5. An aluminum specimen, again 48x48x10mm, was instrumented with a

PZT patch 10x10x0.3mm, but with a bond layer thickness of 0.45mm, implying a

thickness ratio of 1.5. To achieve poor bond quality, many pieces of fisherman’s net

cord (0.45 mm thickness) were laid on the surface of the host structure prior to

applying the adhesive layer. Fig. 7.12 (a) and (b) show the plots of G and B, worked

out using the shear lag model developed in this chapter, considering a value of Gs =

0.2GPa. It is clearly evident that free PZT behaviour tends to dominate itself over

the structural characteristics. The peak around 150 kHz corresponds to the first PZT

resonance. This finding is confirmed by the experimental plots shown in Fig.

7.12(c) and (d). Hence, the newly developed model can accurately predict shear lag

effect from small thickness to large thickness of the adhesive bond layer.

0.00E+00

4.00E-03

8.00E-03

1.20E-02

0 50 100 150 200 250

Frequency (kHz)

G (S

)

0.00E+00

4.00E-03

8.00E-03

1.20E-02

0 50 100 150 200 250

Frequency (kHz)

B (S

)

Fig. 7.12 Analytical and experimental plots for ts/tp equal to 1.5.

(a) Analytical G. (b) Analytical B. (c) Experimental G. (d) Experimental B.

(a)(b)

(c) (d)

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

0 50 100 150 200 250

Frequency (kHz)

G (S

)

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

6.00E-02

0 50 100 150 200 250

Frequency (kHz)

B (S

)


207

7.7 PARAMETRIC STUDY ON ADHESIVE LAYER INDUCED

ADMITTANCE SIGNATURES

From the derivations in the preceding sections, it can be observed that the

extent to which the electro-mechanical admittance signatures are influenced by

bond layer depends on following parameters

+−=

ωυ

jtZlG

pseff

seff

)1(2

spE

seff ttY

Gq

)1( 2υ−= (7.91)

For this parametric study, we considered a PZT patch 10x10x0.3mm (grade PIC

151) bonded to an aluminum block (grade Al 6061T6), 48x48x10mm in size. The

various factors affecting shear lag are Gs (or the ratio YE/G), thickness of adhesive

layer (or the ratio ts/tp) and sensor length (l). The influence of all these parameters is

studied in depth using the 2D shear lag based effective impedance formulations.

The PZT parameters are considered as listed in Table 6.1 (page 165). For the bond

layer, it is assumed that ts = 0.125mm, G = 1.0GPa, η′ = 0.1 (i.e. 10%). The

MATLAB program listed in Appendix G was employed to perform all the

computations.

7.7.1 Influence of Bond Layer Shear Modulus (Gs)

Fig. 7.13 shows the influence of bond layer shear modulus on the

conductance and susceptance signatures. It is observed that as Gs decreases, the

peaks of conductance subside down and shift rightwards (i.e. the apparent resonant

frequencies undergo an increase). That the peaks shift rightwards was also observed

by Xu and Liu (2002). In the susceptance plot, it is observed that the average slope

of the curve falls down slightly, besides peaks subsiding down. The worst results

are observed for G = 0.05 GPa, where the PZT patch behaves more or less

independent of the host structure, as marked by a peak at its resonance frequency,

rather than identifying the host structure. In this regard, it should be noted that the

imaginary part undergoes more identifiable change. Hence, it could be utilized in

detecting problems related to the bond layer.

From these observations, it is recommended that for best structural

identification, an adhesive with high shear modulus should be used for bonding the

PZT patch with the structure.


208

0

0.002

0.004

0.006

0.008

0.01

0 50 100 150 200 250

Frequency (kHz)

B (S

)

0

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200 250

Frequency (kHz)

G (S

)0

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

Frequency (kHz)

G (S

)

0

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

Frequency (kHz)

G (S

)

Fig. 7.13 Influence of shear modulus of elasticity of bond layer.

(a) Conductance vs frequency (perfect bonding).(b) Conductance vs frequency (Gs = 1.0GPa).

(c) Conductance vs frequency (Gs = 0.5GPa). (d) Conductance vs frequency (Gs = 0.05GPa).

(e) Susceptance vs frequency.

0

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

Frequency (kHz)

G (S

)Gs = 1.0 GPa

Perfect bonding

Gs = 0.5 GPa

(c)

(a) (b)

Gs = 1.0 GPa

Perfect bonding

Gs = 0.5 GPa

193 kHz 198 kHz

(d)

Gs = 0.05 GPa

(e)

Gs = 0.05 GPa


209

7.7.2 Influence of Bond Layer Thickness (ts)

Fig. 7.14 shows the plots of conductance and susceptance corresponding to

ts = 0.05mm (thickness ratio, ts/tp = 0.17) and 0.1mm (thickness ratio, ts/tp = 0.33).

It is apparent that as bond layer thickness increases, the peaks subside down and

shift rightwards. Besides, the average slope of the susceptance curve falls down.

Hence, the overall effect is similar to that of reducing Gs. Exceptionally thick bond

layer (thickness ratio > 1.0) may lead to highly erroneous structural identification,

as illustrated in the preceding section. Hence, it is recommended that the bond layer

thickness be maintained minimum possible, preferably less than 1/3rd of the patch

thickness.

0

0.002

0.004

0.006

0.008

0.01

0 50 100 150 200 250

0

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 2500

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 2500

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

193 kHz 198 kHz

Perfect bonding ts/tp = 0.17 ts/tp = 0.33198 kHz

(c)(a) (b)

Perfect bonding

ts/tp = 0.17

ts/tp = 0.33

(d)

Fig. 7.14 Influence of bond layer thickness.

(b) Conductance vs frequency (perfect bonding). (b) Conductance vs frequency (ts/tp = 0.17)

(c) Conductance vs frequency (ts/tp = 0.33). (d) Susceptance vs frequency.

G (S

)

B (S

)


7.7.3 Influence of Damping of Bond Layer (η′ )

Fig. 7.15 shows the influence of the damping of the bonding layer on

conductance and susceptance signatures. It is observed from Fig. 7.15(a) that as the

damping increases, the slope of the baseline conductance tends to fall down.

However, susceptance, on the other hand, remains largely insensitive to damping

variations, as can be observed from Fig. 7.15(b).

(a)

(b)

0

0.0004

0.0008

0.0012

0.0016

0.002

0 50 100 150 200 250

Frequency (kHz)

G (S

)

Mech. Loss factor = 20%Mech. Loss factor = 10%

Mech. Loss factor = 5%

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

0 50 100 150 200 250

Frequency (kHz)

B (S

)

Mech. Loss factor = 20%Mech. Loss factor = 10%

Mech. Loss factor = 5%

Fig. 7.15 Influence of damping of bond layer.


210


211

7.7.4 Overall Influence of Parameter effp

Fig. 7.16 shows the influence of the parameter effp on conductance and

susceptance plots. This is achieved by multiplying effp by a constant factor. It can

be observed that as effp increases, the sensor response tends to reach the ideal

condition corresponding to perfect bonding. From Eq. (7.91), it can be observed that

it is the shear modulus Gs and bond layer thickness, which govern the value of

parameter effp . Higher effp implies higher Gs and lower ts, which, as observed

earlier, are beneficial in getting better admittance response from the PZT patch.

0

0.002

0.004

0.006

0.008

0.01

0 50 100 150 200 250

0

0.0005

0.001

0.0015

0.002

0.0025

0 100 200

0

0.0005

0.001

0.0015

0.002

0.0025

0 100 2000

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

0

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

peff = 0.5 times

Perfect bonding

(c)

(a) (b)

Perfect bonding

(d)

(e)

peff = 1 times

peff = 2.0 times

peff = 2.0 timespeff = 1.0 times

peff = 0.5 times

Fig. 7.16 Influence of peff .(a) Conductance vs frequency (peff = 0.5 times). (b) Conductance vs frequency (peff = 1.0 times).(c) Conductance vs frequency (peff = 2.0 times).(d) Conductance vs frequency (perfect bonding).


G (S

)

B (S

)

G (S

)

G (S

)G

(S)


212

7.7.5 Overall Influence of Parameter effq

Fig. 7.17 shows the influence of the parameter effq on the admittance

signatures. On comparing Figs. 7.17(a), (b) and (c), it is apparent that the influence

of effq alone is not sufficient to improve the quality of conductance signatures.

Rather, in the susceptance plots, an increase of effq alone might marginally degrade

the quality of signatures, as can be observed from Fig. 7.17(e).

0

0.002

0.004

0.006

0.008

0.01

0 50 100 150 200 250

0

0.0005

0.001

0.0015

0.002

0.0025

0 100 200

0

0.0005

0.001

0.0015

0.002

0.0025

0 100 2000

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

0

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

Fig. 7.17 Influence of qeff .

(a) Conductance vs frequency (qeff = 0.5 times).

(b) Conductance vs frequency (qeff = 1.0 times).

(c) Conductance vs frequency (qeff = 2.0 times).

(d) Conductance vs frequency (perfect bonding).


qeff = 0.5 times

Perfect bonding

(c)

(a) (b)

Perfect bonding

(d)

(e)

qeff = 1 times

qeff = 2.0 times

qeff = 0.5 timesqeff = 1.0 times

qeff = 2.0 times

G (S

)G

(S)

G (S

)G

(S)

B (S

)


7.7.6 Influence of Sensor Length (l)

Fig. 7.17 shows the influence of sensor length for two typical sizes of PZT

patch, l = 5mm and l = 20mm. It is observed that for small sensor lengths, the

presence of bond layer does not affect the signature as adversely as for long PZT

patches. Hence, small lengths of PZT patches are recommended for better structural

identification.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 50 100 150 200 250

Frequency (kHz)

B (S

)0

0.01

0.02

0.03

0.04

0 50 100 150 200 250

Frequency (kHz)

G (S

)

0

0.0021

0.0042

0.0063

0.0084

0 50 100 150 200 250

Frequency (kHz)

B (S

)

0

0.0005

0.001

0.0015

0.002

0.0025

0 50 100 150 200 250

Frequency (kHz)

G (S

)

(a) (b)

(c) (d)

Fig. 7.17 Influence of sensor length.

(a) Conductance vs frequency (l = 5 mm).

(b) Conductance vs frequency (l = 20 mm).

(c) Susceptance vs frequency (l = 5 mm).

(d) Susceptance vs frequency (l = 20 mm).

Perfect bonding

l = 5 mm

l = 5 mm

213

l = 20 mm

l = 20 mm

With bond layer


214

7.7.7 Quantification of Overall Influence of Bond Layer

The parametric study described in the previous subsections showed the

influence of various parameters related to the bond layer on the admittance

response. The overall influence of the bond layer can be quantified using the

parameter ζ defined by Eq. (7.90). For best results, this factor should preferably be

as close as possible to unity. It is important to include the shear lag effect into the

analysis if ζ < 0.8.

7.8 SUMMARY AND CONCLUDING REMARKS

This chapter has rigorously addressed the problem of incorporating the

influence of adhesive layer in the electro-mechanical impedance modelling. The

treatment presented is generic in nature and not restricted to beam structures alone,

as in the case of Crawley and de Luis (1987) and Sirohi and Chopra (2000).

Besides, dynamic equilibrium of the system has been considered rather than relying

on equivalent length static coefficients. The formulations have been extended to 2D

effective impedance based model and have been experimentally verified. Hence, the

treatment is more general, rigorous and accurate.

The study covered in this chapter showed that the bond layer can significantly

influence structural identification if not carefully accounted for. Useful parametric

study was also carried out to consider the influence of the various parameters

related to adhesive bond layer. It is found that in order to achieve best results, the

PZT patch should be bonded to the structure using an adhesive of high shear

modulus and smallest practicable thickness. Too low shear modulus of elasticity or

too large thickness of the bond layer can produce erroneous or misleading results,

such as overestimation of peak frequencies or the dominance of PZT patch’s own

frequencies. Further, in order to minimize the influence of the bond layer, small

sized PZT patches should be employed for structural identification. In addition, the

imaginary part of the admittance signature, so far considered redundant, can play

meaningful part in detecting any deterioration of the bond layer since it is more

sensitive to damages to the bond layer. It is therefore recommended to pay careful

attention to the imaginary component of the admittance signature while applying

the EMI technique for NDE or structural identification.

Chapter 8: Practical Issues Related to EMI Technique

215

Chapter 8

PRACTICAL ISSUES RELATED TO EMI TECHNIQUE

8.1 INTRODUCTION

In spite of key advantages such as cost-effectiveness and high sensitivity,

there are several impediments to the practical implementation of the EMI technique

for NDE of real-life structures, such as aerospace components, machine parts,

buildings and bridges. The main challenge lies in achieving a consistent behavior

from the surface bonded piezo-impedance transducers over sufficiently long

periods, typically of the order of few years, under ‘harsh’ environmental conditions.

Hence, protecting PZT patches from unfriendly environmental effects is very

crucial in ensuring reliability of the patches for SHM.

This chapter reports a dedicated investigation stretched over several months,

carried out to ascertain long-term consistency of the electro-mechanical admittance

signatures. Possible means of protecting the patches by suitable covering layer and

the effects of such layer on the sensitivity of the patch are also investigated. The

chapter also investigates on the possible use of multiplexing to optimize sensor

interrogation time.

8.2 EVALUATION OF LONG TERM REPEATIBILITY OF SIGNATURES

The PZT transducers are relatively new for SHM engineers, who are more

accustomed to using conventional sensors such as strain gauges and accelerometers.

They are often skeptical about the reliability of the signature based EMI technique.

It is often argued that if the signatures are not repeatable enough over long periods

of time, it could be very confusing for the maintenance engineers to make any

meaningful interpretation about damage. No study has so far been reported to

investigate this vital practical issue.


216

In this research, an experimental investigation, spanning over two months, was

carried out in order to ascertain the repeatability of the admittance signatures.

Fig. 8.1 shows the details of the specimen employed for this purpose. It was an

aluminum plate, 200x160x2mm in size, instrumented with two PZT patches, which

were periodically scanned for over two months. Very often, the wires from the

patches to the impedance analyzer were detached and reconnected during the

experiments. Fig. 8.2 shows the conductance signatures of patch #1 over the two-

month duration. Very good repeatability is clearly evident from this figure.

Standard deviation was determined for this set of signatures at each frequency step.

Average standard deviation worked out to be 4.36x10-6S (Seimens) against a mean

value of 2.68x10-4 S. Hence, the normalized standard deviation (average standard

deviation divided by mean) worked out to be 1.5% only, which shows that the

repeatability of the signatures was excellent over the period of experiments. Fig. 8.3

similarly shows the susceptance plots of patch #1 over the same period. From this

figure, it is observed that susceptance plots also exhibit good repeatability. Similar

repeatability was also observed for the signatures acquired from the PZT patch #2.

8.3 PROTECTION OF PZT TRANSDUCERS AGAINST ENVIRONMENT

If piezo-impedance transducers are to be employed for the NDE of real-life

structures, they are bound to be influenced by environmental effects, such as

temperature fluctuations and humidity. Temperature effects have been studied by

many researchers in the past (e.g. Sun et al., 1995; Park et al., 1999) and algorithms

for compensating these have already been developed. However, no study has so far

been undertaken to investigate the influence of humidity on the signatures.

In this research, an experiment was conducted on the specimen shown in Fig.

8.1. The PZT patch #2, which was not protected by any layer, was soaked in water

for 24 hours and its signatures were recorded before as well as after this exercise

(excess water was wiped off the surface before recording the signature). Figs. 8.4(a)

and (b) compare the conductance and susceptance signatures respectively for two

conditions. That humidity has exercised adverse effect on the signatures is clearly

evident by the substantial vertical shift in the conductance signature (Fig. 8.4a).

From Eq. (2.24), it is most probable that the presence of humidity has

significantly increased the


0.000

0.000

0.000

0.000

0.000

0.000

G (S

)

100 mm 100 mm

50 m

m11

0 m

m

= = = =

PZT patch #2

Hole (damage)

PZT patch #1

0.0

0.0

0.0

0.0

0.0

0.0

B (S

)

Fig. 8.1 Test specimen for evaluating repeatability of admittance signatures.

1

2

3

4

5

6

100 110 120 130 140 150

Frequency (kHz)

Day1 Day 9 Day 20 Day 26 Day 40 Day 49 Day 64

Fig. 8.2 A set of conductance signatures of PZT patch #1 spanning over two months.

0

01

02

03

04

05

06

100 110 120 130 140 150

Frequency (kHz)

Day1 Day 9 Day 20 Day 26 Day 40 Day 49 Day 64

Fig. 8.3 A set of susceptance signatures of PZT patch #1 spanning over two months.

217


218

electric permittivity of the patch. This experiment suggests that a protection layer is

necessary to protect the PZT patches against humidity in the actual field applications. It

should be mentioned here that upon drying by a stream of hot air, the signatures

subsided down, although still not recovering the original condition completely.

Silicon rubber was chosen as a candidate protective material since it is known

to be a good water proofing material, chemically inert and at the same time very good

electric insulator. Besides, it is commercially available as paste which can be solidified

by curing at room temperature. To evaluate the protective strength of silicon rubber,

PZT patch #1 (see Fig. 8.1) was covered with silicon rubber coating (grade 3140, Dow

Corning Corporation, 2003). The previous experiment carried out on patch #2 (i.e.

soaking with water for 24 hours) was repeated on patch #1. Figs. 8.4 (c) and (d)

compare the signatures recorded from this patch in the dry state as well as humid state.

It is found that there is very negligible change in the signatures even after long exposure

to humid conditions. Hence, silicon rubber is capable of protecting PZT patches against

humidity.

Although this experiment clearly establishes the suitability of silicon rubber in

providing protection against humidity, it is however likely that its presence could reduce

the damage sensitivity of the PZT patch. In order to ascertain this doubt, damage was

induced in the plate by drilling a 5mm-diameter hole equidistant from the two PZT

patches (damage location is shown in Fig. 8.1). Fig. 8.5 shows the effect of this damage

on the signatures of the two PZT patches- the protected patch (patch #1) and the

unprotected patch (patch #2). The damage was quantified using the root mean square

deviation of the signature from its baseline position using Eq. (2.28). The RMSD index

was worked out to be 5.6% for the unprotected patch (patch #2) and 5% for the

protected patch (patch #1). This shows that the silicon rubber covering layer has only

marginal effect on the damage sensitivity of the PZT patches. Hence, silicon rubber is

very suitable in protecting the PZT patches against environmental hazards, without

significantly diminishing their sensitivity.

It should be mentioned here that commercially available packaged QuickPack®

actuators (Mide Technology Corporation, 2004) are also likely to be robust against

humidity, though no study has been reported so far. However, the packaging itself

enhances the cost by at least 10 times. The proposed protection, using silicon rubber, on

the other hand, offers a simple and an economical solution to the problem of humidity.


219

0.0002

0.00024

0.00028

0.00032

112 114 116 118 120 122 124

Frequency (kHz)

Con

duct

ance

(S)

Fig. 8.5 Effect of damage on conductance signatures.

(a) Unprotected PZT patch (patch #2).

(b) PZT patch protected by silicon rubber (patch #1).

0.00025

0.0003

0.00035

0.0004

128 130 132 134 136 138 140

Frequency (kHz)

Con

duct

ance

(S) Pristine state After damage

Pristine state

After damage

(a) (b)

0.00015

0.00025

0.00035

0.00045

0.00055

0.00065

100 110 120 130 140 150

Frequency (kHz)

G (S

)

(a)

Fig. 8.4 Effect of humidity on signature.

(a) Unprotected patch: G-plot. (b) Unprotected patch: B-plot.

(c) Protected patch: G-plot. (d) Protected patch: B-plot.

0

0.001

0.002

0.003

0.004

0.005

100 110 120 130 140 150

Frequency (kHz)

B (S

)

(d)

0

0.001

0.002

0.003

0.004

0.005

100 110 120 130 140 150

Frequency (kHz)

B (S

)

(b)

Unprotected patch Unprotected patch

Protected patch

Dry condition Humid condition

0.00015

0.00025

0.00035

0.00045

100 110 120 130 140 150

Frequency (kHz)

G (S

)

Protected patch

(c)


220

8.4 MULTIPLEXING OF SIGNALS FROM PZT ARRAYS

The EMI technique employs PZT arrays, which, for NDE, must be scanned on

one-to one basis. In real-life applications, this could turn out to be a very time

consuming operation. For example, if a structure has been instrumented with 50

PZT patches which are intended to be scanned in the frequency range 100-120 kHz

at an interval of 100Hz, the entire operation would consume approximately one

hour on the standard HP 4192A impedance analyzer, operating in the normal mode

using PC interface. However, such a thorough scan may not be warranted most of

the time.

In this research, the feasibility of reducing PZT scanning time using a

multiplexing device was investigated. The test specimen was an aluminium plate,

600x500x10mm in size, instrumented with 20 PZT patches, as shown in Fig. 8.6.

The patches were not connected directly to the impedance analyzer. Rather, they

were first wired to the 40 channel N2260A multiplexer module housed inside

3499B switch control system (Agilent Technologies, 2001), that was in-turn

connected to the HP 4192A impedance analyzer. The entire set-up is shown in

Fig. 8.7. With this system, any number of PZT patches (from one PZT patch to all)

can be activated simultaneously for interrogation. Park et al. (2001) also reported

connecting multiple patches to the impedance analyzer simultaneously. But his

arrangement lacked the flexibility of scanning the patches individually should the

need arise, since the patches were connected permanently. However, in the present

system, the advantage is that both the options (individually or group or subgroup)

Fig. 8.6 Test specimen for evaluating signature multiplexing.

PZT patches

Damage (10mm φ hole)

6 x 100 mm

5 x

100

mm


ar

sw

es

on

re

wo

gr

da

pa

da

PZ

sh

re

HP 4192Aimpedance analyzer

3499B switch controlsystem housing N2260Amultiplexer module

Controlling personalcomputer

Fig. 8.7 Experimental set-up consisting of impedance analyzer, controller PC

and multiplexer.

221

e available at a button’s press. Any number of patches can be activated simply be

itching, thus offering great optimization flexibility. The multiplexer module is

pecially manufactured for low-current applications, as in the present case.

With this arrangement, there is no necessity to scan the patches on one-to-

e basis in the routine checks. All the patches can be simultaneously scanned

gularly. In the case of an unusual observation from the collective signature (which

uld be the case at the onset of damage), one-to-one basis (or scanning small

oups of PZT patches collectively) can be resorted back so as to localize the

mage location.

This idea of multiplexing PZT signatures was tested on the twenty PZT

tches instrumented on the plate shown in Fig. 8.6. Fig. 8.8 shows the effect of

mage (a 10mm diameter hole, shown in Fig. 8.6) on the collective signature of 20

T patches. Presence of damage can be easily inferred from the conductance plot

own in the figure. Hence, the multiplexing of PZT signals can enable the user to

duce the interrogation time substantially. Moreover, the presence of the


222

multiplexer module ensures much more stable and repeatable signatures due to

more secure connections.


This chapter has addressed key practical issues related to the implementation

of the EMI technique for NDE of real-life structures. The results of the repeatability

study, which extended over a period of two months, demonstrated that PZT patches

exhibit excellent repeatable performance and are reliable enough to be used for

monitoring real-life structures. However, are at the same time, the signatures are

highly sensitive to humidity. Silicon rubber has been experimentally found to be a

good covering material to impart sound protection against humidity. Hence the

presence of silicon rubber layer can enable the application of the method on real-

world civil-structures, where it is necessary that the transducer should serve for long

periods. The striking feature of the silicon rubber is that it does not adversely affect

the damage sensitivity of the PZT patch. In addition, the feasibility of reducing the

scanning time and effort using commercially available multiplexing system has also

been demonstrated in this chapter.

0.04

0.05

0.06

0.07

0.08

120 121 122 123 124 125Frequency (kHz)

B (S

)

0.003

0.004

0.005

0.006

0.007

120 121 122 123 124 125

Frequency (kHz)

G (S

)

Fig. 8.8 Effect of damage on collective signature of 20 PZT patches.

(a) Conductance. (b) Susceptance.

Pristine state

After damage

Pristine state

After damage

(a) (b)

Chapter 9: Conclusions and Recommendations

223

Chapter 9

CONCLUSIONS AND RECOMMENDATIONS

9.1 INTRODUCTION

This thesis embodies findings from the research carried out for structural

identification, health monitoring and non-destructive evaluation using structural

impedance parameters extracted using surface bonded piezo-impedance transducers.

Specifically, a major objective of the research was to upgrade the EMI technique

from its present state-of-the art of relying on statistical non-parametric damage

evaluation using raw signatures. This conventional approach lacked not only an

understanding of the inherent damage mechanism but also a rigorous calibration to

realistically estimate damage severity in real-life situations.

The major novelty in the present research is that for the first time, extraction of

mechanical impedance parameters has been attempted using piezo-impedance

transducers. This approach has been shown more realistic as well as more sensitive

to damage. Any ‘unknown’ structure can be ‘identified’ by the proposed method,

without warranting any a priori information governing the phenomenological

nature of the structure.

The following sections outline the major contributions, conclusions and

recommendations stemming out from this research.

9.2 RESEARCH CONCLUSIONS AND CONTRIBUTIONS

Major research conclusions and contributions can be summarized as follows

(i) The raw conductance signature (real component), which is conventionally

employed for SHM in the EMI technique, is mixed with a ‘passive’

component, arising out of the capacitance of the PZT patch. This passive


224

component, which is ‘inert’ to structural damages, tends to lower down the

damage sensitivity of the conductance signature. The raw susceptance

signature (imaginary component) is similarly ‘camouflaged’, somewhat

more heavily, thereby diminishing its usefulness for SHM to the extent of

redundancy.

In this research, a new concept of active-signatures has been

introduced to extract damage sensitive ‘active’ components by carrying out

signature decomposition. This filtering process is found to substantially

improve the sensitivity of the both the real as well as the imaginary

component. Rather, it has been found to raise the level of sensitivity of the

imaginary component as high as its real counterpart. Hence, together, these

can be employed to derive more pertinent information governing the

phenomenological nature of the host structure.

(ii) A new method of analyzing the electro-mechanical admittance signatures

has been developed for diagnosing damages in skeletal structures. This

method involves extracting the ‘apparent’ drive point structural impedance

from the active conductance and active susceptance signatures. Hence, both

real and imaginary components are utilized for damage assessment. A

complex damage metric has been proposed for quantifying damages using

the extracted structural parameters. The real part of the damage metric

indicates changes in the equivalent SDOF damping, whereas the imaginary

part indicates the changes in the equivalent SDOF stiffness-mass factor

resulting from damages.

As proof-of-concept, the new methodology was applied on a model

RC frame subjected to base vibrations on a shaking table. The proposed

methodology was found to perform much better than the existing damage

quantification approaches i.e. the low frequency vibration methods as well

as the traditional raw-signature based damage assessment using the EMI

technique. The instrumented PZT patches were also found to provide

meaningful insight into the changes taking place in the structural parameters

due to damages.


225

(iii) In order to extend the impedance based damage diagnosis method to the

general class of structures, a new PZT-structure interaction model has been

developed based on the concept of ‘effective impedance’. As opposed to the

previous impedance-based models, the new model condenses the two-

directional mechanical coupling between the PZT patch and the host

structure into a single impedance term. The model has been verified on a

representative aerospace structural component over a frequency range of 0-

200 kHz. To the author’s best knowledge, this has been the first ever

attempt to compare theoretical and experimental admittance signatures

relevant to EMI technique for such high frequencies. As a byproduct, a new

method has been developed to drive the EDP mechanical impedance of any

complex real-life structure by 3D dynamic harmonic analysis using any

commercial finite element software. The new model bridges gap between

the 1D impedance model of Liang et al. (1993, 1994) and the 2D model

proposed by Zhou et al. (1995, 1996).

The new impedance formulations can be conveniently employed to

extract the 2D mechanical impedance of any ‘unknown’ structure from the

admittance signatures of a surface-bonded PZT patch. Besides NDE, the

proposed model can be employed in numerous other applications, such as

predicting system’s response, energy conversion efficiency and system

power consumption.

(iv) A new experimental technique has been developed to ‘update’ the model of

the piezo-impedance transducer before it could be surface bonded for

‘identifying’ the host structure. This updating has been found to facilitate a

more accurate identification of structural system’s parameters. The new

impedance formulations, in conjunction with the ‘updated’ PZT model, can

be employed to ‘identify’ the host structure and to carry out parametric

damage assessment. Proof-of-concept applications of the proposed structural

identification and health monitoring methodology have been undertaken on

structures ranging from precision machine and aerospace components to

large civil-structures. Since the dynamic characteristics of the host structure

are not altered by small sized PZT patches, a very accurate structural


226

identification is therefore possible by the proposed method. The piezo-

impedance transducers can be installed on the inaccessible parts of crucial

machine components, aircraft main landing gear fitting, turbo-engine blades,

RCC panels of space shuttles and civil-structures to perform continuous

real-time SHM. The equivalent system is identified from the experimental

data alone. No analytical/ numerical model is required as a prerequisite. The

proposed NDE method has also demonstrated ability to detect damages

resulting from loss of mass, such as in the RCC panels of space shuttles due

to oxidation.

(v) After identifying the impedance parameters, it is equally important to relate

them with physical parameters such as strength/ stiffness and to calibrate the

changes in the parameters with damage progression in the component.

Towards this end, comprehensive tests were performed on concrete

specimens up to failure to empirically calibrate the ‘identified’ system

parameters with damage severity. It has been found that in the frequency

range 60-100 kHz, concrete essentially behaves as a parallel spring damper

combination. The equivalent spring stiffness has been found to reduce and

the damping found to increase with damage progression. However, in most

tests, the damping was found to undergo major changes towards specimen

failure only. The equivalent stiffness, on the contrary, showed a uniform and

consistent trend and was found more suitable for diagnosing damages

ranging from incipient types to very severe types.

A fuzzy probability based damage model has been proposed based

on the extracted equivalent stiffness from the tests conducted on concrete.

This has enabled the calibration of the piezo-impedance transducers in terms

of damage severity and can serve as a practical empirical phenomenological

damage model for quantitatively estimating damage severity in concrete.

(vi) A new experimental technique has been developed to determine in situ

concrete strength non-destructively using the EMI principle. The new

technique is much superior than the existing strength prediction techniques

such as ultrasonic methods. The new method demands only one free surface

of the specimen only whereas the ultrasonic methods (Chapter 6) warrant


227

two opposite surfaces. In addition, this research has shown the feasibility of

monitoring curing of concrete using the EMI technique, demonstrating much

higher sensitivity than the conventional methods. This method can be

applied in the construction industry to decide the appropriate time of

removal of the formwork and the time of commencement of prestressing

operations in the prestressed concrete members.

(vii) This research has minutely investigated the mechanism of force transfer

between the PZT patch and the host structure through the interfacial

adhesive bond layer and has presented a step-by-step derivation to integrate

its effects into impedance formulations, both 1D and 2D. The treatment

presented in this research is of general nature and not restricted to beam

structures alone as in the case of the analysis presented by Crawley and de

Luis (1987) and Sirohi and Chopra (2000b).

Useful parametric study has also been carried out to investigate the

influence of the various parameters related to the adhesive bond layer. It has

been found that a high shear modulus of elasticity and a small thickness of

bond layer is imperative in ensuring accurate structural identification.

Preferably, the bond layer should not be thicker than one-third of the PZT

patch. Also, the length of the PZT patch should be kept as small as possible

to minimize inaccuracies due to shear lag.

(viii) Finally, this research has addressed key practical issues related to the

implementation of the EMI technique for NDE of real-life structures. The

results of the repeatability study, which extended over a period of two

months, demonstrated that the PZT patches exhibit excellent repeatable

performance and are reliable enough for monitoring real-life structures.

However, the signatures are at the same time highly prone to contamination

by humidity. Silicon rubber has been experimentally shown to be a good

material to impart sound protection against humidity. The striking feature of

the silicon rubber is that it does not adversely affect the damage sensitivity

of the PZT patch. The feasibility of reducing the scanning time and effort

using commercially available multiplexing system has also been

demonstrated.


228

9.3 RECOMMENDATIONS FOR FUTURE WORK

From the experience of carrying out research in the field of EMI technique, the

author believes that the present research work can be further extended as follows

(i) In this research, methods have been developed to filter off the damage

insensitive inert components from the admittance signatures. However,

theoretically, it is possible, by adjusting the PZT properties (namely EY ,

T33ε and d31) that the passive component is automatically nullified (Eq. 3.28).

This means that the raw signature itself will be as good as the active

signature, thereby eliminating the requirements of filtering. Hence, research

should be directed so as to obtain a material composition where this could

be achieved. Besides, research could be focused on developing temperature

tolerant PZT material so that the requirements of temperature compensation

could also be eliminated.

(ii) The effective impedance based electro-mechanical formulations derived in

Chapter 5 should be further extended to embedded PZT patches, such as in

laminated beams. In this case, it could be necessary to consider vibrations in

the thickness direction also in the analysis.

(iii) This research has demonstrated the possibility of non-destructive concrete

strength assessment using PZT patches. However, more tests should be

conducted in order to take into consideration the effects of variables like

type of cement, type and size of the aggregates, type and size of the PZT

patches and their mechanical and electrical properties. All these issues

should be addressed before the technique could be standardized and

commercialized.

Further experiments should also be performed so that the material

strength estimation technique can be extended to other materials. A

universal calibration chart could also be developed.

(iv) The fuzzy probabilistic damage severity calibration methodology presented

in this thesis for concrete can be extended to other materials. It adequacy

should be tested for concrete subjected to tension and bending also.


229

(v) In reality, for an adhesively bonded piezo-impedance transducer, the

governing differential equation is

dxwtx

Ttu

dmdxw pppp

p ∂

∂=

∂

∂+ 2

2

)(τ (9.1)

In the present analysis, the inertial force term and the shear force

term have been considered separately and the two effects are superimposed.

However, this is only an approximation. It is recommended that ways and

means should be developed to solve this differential equation by considering

the two effects concurrently. The resulting impedance model would be, truly

speaking, the most realistic one.

The author strongly believes that there is great potential in developing the

EMI technique as a universal cost-effective NDE technique.

Author’s Publications

230

AUTHOR’S PUBLICATIONS

JOURNAL

1. Soh, C. K., Tseng, K. K.-H., Bhalla, S. and Gupta, A. (2000), “Performance of

Smart Piezoceramic Patches in Health Monitoring of a RC Bridge”, Smart

Materials and Structures, Vol. 9, No. 4, pp. 533-542. (Based on author’s

M. Eng. thesis).

2. Bhalla, S. and Soh, C. K. (2003), “Structural Impedance Based Damage

Diagnosis by Piezo-Transducers”, Earthquake Engineering and Structural

Dynamics, Vol. 32, No. 12, pp. 1897-1916. (Based on Chapter 4 of thesis).

3. Bhalla, S. and Soh, C.K. (2004), “High Frequency Piezoelectric Signatures for

Diagnosis of Seismic/ Blast Induced Structural Damages”, NDT&E

International, Vol. 37, No. 1, pp. 23-33. (Based on Chapter 4 of thesis).

4. Bhalla, S. and Soh, C.K. (2004), “Structural Health monitoring by Piezo-

Impedance Transducers: Modeling”, Journal of Aerospace Engineering,

ASCE, Vol. 17, No. 4, pp. 154-165. (Based on Chapter 5 of thesis).

5. Bhalla, S. and Soh, C.K. (2004), “Structural Health monitoring by Piezo-

Impedance Transducers: Applications”, Journal of Aerospace Engineering,

ASCE, Vol. 17, No. 4, pp. 166-175. (Based on Chapter 5 of thesis).

6. Bhalla, S. and Soh, C.K. and Liu, Z. (2005), “Wave Propagation Approach for

NDE Using Surface Bonded Piezoceramics”, NDT&E International, Vol. 38,

No. 2, pp. 143-150.

7. Bhalla, S. and Soh, C. K. (2004), “Impedance Based Modeling for Adhesively

Bonded Piezo-Transducers”, Journal of Intelligent Material Systems and

Structures, Vol. 15, No. 12, pp. 955-972.


231

8. Soh, C. K. and Bhalla, S. (2004), “Calibration of Piezo-Impedance Transducers

for Strength Prediction and Damage Assessment of Concrete”, Smart Materials

and Structures, tentatively accepted (Based on Chapter 6 of thesis).

CONFERENCE:

1. Tseng, K. K.-H., Soh, C. K., Gupta, A. and Bhalla, S. (2000), “Health

Monitoring of Civil Infrastructure Using Smart Piezoceramic Transducer

Patches”, Proceedings of 2nd International Conference on Computational

Methods for Smart Structures and Materials, edited by C. A. Brebbia and A.

Samartin, 19-20 June, Madrid, WIT Press (Southampton), pp.153-162. (Based

on author’s M. Eng. thesis).

2. Bhalla, S., Soh, C. K., Tseng, K. K.-H and Naidu, A. S. K. (2001), “Diagnosis

of Incipient Damage in Steel Structures by Means of Piezoceramic Patches”,

Proceedings of 8th East Asia-Pacific Conference on Structural Engineering and

Construction, 5-7 December, Singapore, paper no. 1598. (Based on author’s M.

Eng. thesis).

3. Bhalla, S., Naidu, A. S. K. and Soh, C. K. (2002), “Influence of Structure-

Actuator Interactions and Temperature on Piezoelectric Mechatronic Signatures

for NDE”, Proceedings of ISSS-SPIE International Conference on Smart

Materials, Structures and Systems, edited by B. Dattaguru, S. Gopalakrishnan

and S. Mohan, 12-14 December, Bangalore, Microart Multimedia Solutions

(Bangalore), pp. 213-219. (Based on Chapter 3 of thesis).

4. Naidu, A. S. K. and Bhalla, S. (2002), “Damage Detection in Concrete

Structures with Smart Piezoceramic Transducers”, Proceedings of ISSS-SPIE

International Conference on Smart Materials, Structures and Systems, edited by

B. Dattaguru, S. Gopalakrishnan and S. Mohan, 12-14 December, Bangalore,

Microart Multimedia Solutions (Bangalore), pp. 639-645.


232

5. Ong, C. W. , Yang Y., Wong, Y. T., Bhalla, S., Lu, Y. and Soh, C. K. (2002),

“The Effects of Adhesive on the Electro-Mechanical Response of a

Piezoceramic Transducer Coupled Smart System”, Proceedings of ISSS-SPIE

International Conference on Smart Materials, Structures and Systems, edited by

B. Dattaguru, S. Gopalakrishnan and S. Mohan, 12-14 December, Bangalore,

Microart Multimedia Solutions (Bangalore), pp. 191-197.

6. Bhalla, S., Naidu, A. S. K., Ong, C. W. and Soh, C. K. (2002), “Practical Issues

in the Implementation of Electro-Mechanical Impedance Technique for NDE”,

in Smart Structures, Devices and Systems, edited by E. C. Harvey, D. Abbott

and V. K. Varadan, SPIE’s International Symposium on Smart Materials,

Nano-, and Micro-Smart Systems, 16- 18 December, Melbourne, Proceedings of

SPIE Vol. 4935, pp. 484-494. (Based on Chapter 8 of thesis)

7. Naidu, A. S. K., Bhalla, S. and Soh, C. K. (2002), “Incipient Damage

Localization in Structures Using Smart Piezoceramic Patches” in Smart

Structures, Devices and Systems, edited by E. C. Harvey, D. Abbott and V. K.

Varadan, SPIE’s International Symposium on Smart Materials, Nano-, and

Micro-Smart Systems, 16- 18 December, Melbourne, Proceedings of SPIE Vol.

4935, pp. 495-502.

8. Naidu, A. S. K., Bhalla, S. and Soh, C. K. (2002), “Damage Location

Identification in Smart Structures Using Modal Parameters”, Proceedings of the

2nd International Conference on Structural Stability and Dynamics, edited by C.

M. Wang, G. R. Liu and K. K. Ang, 16- 19 December, Singapore, World

scientific Publishing Co. Pte. Ltd., pp. 737-742.

9. Bhalla, S., Naidu, A. S. K., Yang, Y. W. and Soh, C. K. (2003), “An

Impedance-Based Piezoelectric-Structure Interaction Model for Smart Structure

Applications”, Proceedings of Second MIT Conference on Computational Fluid

and Solid Mechanics, edited by K. J. Bathe, June 17-20, Cambridge, pp. 107-


233

110. (Based on Chapter 5 of thesis. This paper was awarded Young Researcher

Fellowship Award).

10. Naidu, A. S. K., Bhalla, S. and Soh, C. K. (2004), “Recent Developments in

Smart Systems Based Structural Health Monitoring”, National Conference on

Materials and Structures, 23-24 January, NIT-Warangal, pp. 273-278.

References

234

REFERENCES

Abe, M., Park, G. and Inman, D. J. (2002), “Impedance-Based Monitoring of Stress

in Thin Structural Members”, Proceeding of 11th International Conference on

Adaptive Structures and Technologies, October 23-26, Nagoya, Japan, pp. 285-292.

Adams, R. D., Cawley, P., Pye, C. J. and Stone, B. J. (1978), “A Vibration

Technique for Non-Destructively Assessing the Integrity of Structures”, Journal of

Mechanical Engineering Science, Vol. 20, pp. 93-100.

Adams, R. D and Wake, W. C. (1984), Structural Adhesive Joints in Engineering,

Elsevier Applied Science Publishers, London.

Agilent Technologies (2003), Test and Measurement Catalogue, USA.

Ahmad, I. (1988), “Smart Structures and Materials”, Proceedings of U.S. Army

Research Office Workshop on Smart Materials, Structures and Mathematical

Issues, edited by C. A. Rogers, September 15-16, Virginia Polytechnic Institute &

State University, Technomic Publishing Co., Inc, pp. 13-16.

Aktan, A. E., Helmicki, A. J. and Hunt, V. J. (1998), “Issues in Health Monitoring

for Intelligent Infrastructure”, Smart Materials and Structures, Vol. 7, No. 5, pp.

674-692.

Aktan, A. E., Catbas, F. N., Grimmelsman, K. A. and Tsikos, C. J. (2000), “Issues

in Infrastructure Health Monitoring for Management”, Journal of Engineering

Mechanics, ASCE, Vol. 126, No. 7, pp. 711-724.

ANSYS Reference Manual; Release 5.6 (2000), ANSYS Inc., Canonsburg, PA,

USA.

References

235

Ayres, J. W., Lalande, F., Chaudhry, Z. and Rogers, C. A. (1998), “Qualitative

Impedance-Based Health Monitoring of Civil Infrastructures”, Smart Materials and


AWST (2003), Aviation Week and Space Technology, 16 February, 2003,

http://www.aviationnow.com

Banan, M. R., Banan, M. R. and Hjelmstad, K. D. (1994), “ Parameter Estimation

of Structures from Static Response. I. Computational Aspects”, Journal of

Structural Engineering, ASCE, Vol. 120, No. 11, pp. 3243- 3258.

Barbosa, C. H., Vellasco, M., Pacheco, M. A., Bruno, A. C. and Camerini, C. S.

(2000), “Nondestructive Evaluation of Steel Structures Using a Superconducting

Quantum Interference Device Magnetometer and a Neural Network System”,

Review of Scientific Instruments, Vol. 71, No. 10, pp. 3806-3815.

Bathe, K. J. (1996), Finite Element Procedures, Prentice Hall, New Jersey.

Bhalla, S. (2001), “Smart System Based Automated Health Monitoring of

Structures”, M.Eng. Thesis, Nanyang Technological University, Singapore.

Bhalla, S., Soh, C. K., Tseng, K. K. H and Naidu, A. S. K. (2001), “Diagnosis of

Incipient Damage in Steel Structures by Means of Piezoceramic Patches”,

Proceedings of 8th East Asia-Pacific Conference on Structural Engineering and

Construction, 5-7 December, Singapore, paper no. 1598.

Boller C. (2002), “Structural Health Management of Ageing Aircraft and Other

Infrastructure”, Monograph on Structural Health Monitoring, Institute of Smart

Structures and Systems (ISSS), pp. 1-59.

Brown, C. B. (1979), “A Fuzzy Safety Measure”, Journal of Engineering

Mechanics Division, ASCE, Vol. 105, pp. 855-872.

http://www.aviationnow.com/

References

236

Bungey, J. H. (1982), The Testing of Concrete in Structures, Surrey University

Press.

CAIB (2003), Columbia Accident Investigation Board, http://www.caib.us (date of

access: 9 September, 2003)

Chameau, J. L. A, Alteschaeffl, A., Michael, H. L. and Yao, J. P. T. (1983),

“Potential Applications of Fuzzy Sets in Civil Engineering”, International Journal

of Man-Machine Studies, Vol. 19, pp. 9-18.

Cheng, B. L. and Reece, M. J. (2001), “Stress Relaxation and Estimation of

Activation Volume in a Commercial Hard PZT Piezoelectric Ceramic”, Bulletin of

Material Science, Indian Academy of Sciences, Vol. 24, No. 2, pp. 165–167.

Choi, K. and Chang, F. K. (1996), “Identification of Impact Force and Location

Using Distributed Sensors”, AIAA Journal, Vol. 34, No. 1, pp. 136-142.

Chopra, I. (2000), “Status of Application of Smart Structures Technology to

Rotocraft Systems”, Journal of the American Helicopter Society, Vol. 45, No. 4, pp.

228-252.

Crawley, E. F., de Luis, J. (1987), “Use of Piezoelectric Actuators as Elements of

Intelligent Structures”, AIAA Journal, Vol. 25, No. 10, pp. 1373-1385.

De Faria, A. R. (2003), “The Effect of Finite Stiffness Bonding on the Sensing

Effectiveness of Piezoelectric Patches”, Technical note, Smart Materials and

Structures, Vol. 12, pp. N5-N8.

Dhingra, A. K., Rao, S. S. and Kumar, V. (1992), “Non-linear Membership

Functions in Multi-Objective Fuzzy Optimization of Mechanical and Structural

Systems”, AIAA Journal, Vol. 30, No. 1, pp. 251-260.

http://www.caib.us/

References

237

Dosch, J. J., Inman, D. J. and Garcia, E. (1992), “A Self Sensing Piezoelectric

Actuator for Collocated Control”, Journal of Intelligent Material Systems and

Structures, Vol. 3, pp. 166-185.

Dow Corning Corporation (2003), http://www.dowcorning.com

Elkordy, M. F., Chang, K. C. and Lee, G. C. (1994), "A Structural Damage Neural

Network Monitoring System", Microcomputers in Civil Engineering, Vol. 9, No. 2,

pp. 83-96.

Esteban, J. (1996), “Analysis of the Sensing Region of a PZT Actuator-Sensor”,

Ph.D. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg,

VA.

Fairweather, J. A. (1998), “Designing with Active Materials: An Impedance Based

Approach”, Ph.D. Thesis, Rensselaer Polytechnic Institute, New York.

Farrar, C. R. and Jauregui, D. A. (1998), “Comparative Study of Damage

Identification Algorithms Applied to a Bridge: I. Experiment”, Smart Materials and


Flood, I. and Kartam, N. (1994a), “Neural Networks in Civil Engineering I:

Principles and Understanding”, Journal of Computing in Civil Engineering, ASCE,

Vol. 8, No. 2, pp. 131-148.

Flood, I. and Kartam, N. (1994b), “Neural Networks in Civil Engineering II:

Systems and Applications”, Journal of Computing in Civil Engineering, ASCE,

Vol. 8, No. 2, pp. 149-162.

Giurgiutiu, V. and Rogers, C. A., (1997), “Electromechanical (E/M) Impedance

Method for Structural Health Monitoring and Non-Destructive Evaluation”,

Proceedings of International Workshop on Structural Health Monitoring, edited by

http://www.dowcorning.com/

References

238

F. K. Chang, Stanford University, Stanford, California, September 18-20,

Technomic Publishing Co., pp. 433-444.

Giurgiutiu, V. and Rogers, C. A. (1998), “Recent Advancements in the Electro-

Mechanical (E/M) Impedance Method for Structural Health Monitoring and NDE”,

Proceedings of SPIE Conference on Smart Structures and Integrated Systems, San

Diego, California, March, SPIE Vol. 3329, pp. 536-547.

Giurgiutiu, V., Reynolds, A. and Rogers, C. A. (1999), “Experimental Investigation

of E/M Impedance Health Monitoring for Spot-Welded Structural Joints”, Journal

of Intelligent Material Systems and Structures, Vol. 10, No. 10, pp. 802-812.

Giurgiutiu, V., Redmond, J., Roach, D. and Rackow, K. (2000), “Active Sensors

for Health Monitoring of Ageing Aerospace Structures”, Proceedings of the SPIE

Conference on Smart Structures and Integrated Systems, edited by N. M. Wereley,

SPIE Vol. 3985, pp. 294-305.

Giurgiutiu, V. and Zagrai, A. N. (2000), “Characterization of Piezoelectric Wafer

Active Sensors”, Journal of Intelligent Material Systems and Structures, Vol. 11,

pp. 959-976.

Giurgiutiu, V. and Zagrai, A. N. (2002), “Embedded Self-Sensing Piezoelectric

Active Sensors for On-Line Structural Identification”, Journal of Vibration and

Acoustics, ASME, Vol. 124, pp. 116-125.

Giurgiutiu, V., Zagrai, A. N. and Bao, J. J. (2002), “Embedded Active Sensors for

In-Situ Structural Health Monitoring of Thin-Wall Structures”, Journal of Pressure

Vessel Technology, ASME, Vol. 124, pp. 293-302.

Halliday, D., Resnick, R. and Walker, J. (2001), Fundamentals of Physics, 6th ed.,

Wiley, New York, pp. 768-800.

References

239

Hewlett Packard (1996), HP LF 4192A Impedance Analyzer, Operation Manual,

Japan.

Hixon, E.L. (1988), “Mechanical Impedance”, Shock and Vibration Handbook,

edited by C. M. Harris, 3rd ed., Mc Graw Hill Book Co., New York, pp. 10.1-10.46.

Hung, S. L. and Kao, C. Y. (2002), “Structural Damage Detection Using the

Optimal Weights of the Approximating Artificial Neural Networks”, Earthquake

Engineering and Structural Dynamics, Vol. 31, No. 2, pp. 217-234.

IEEE (1987), IEEE Standard on Piezolectricity, Std. 176, IEEE/ANSI.

Ikeda, T. (1990), Fundamentals of Piezoelectricity, Oxford University Press,

Oxford.

Inman, D. J., Ahmadihan, M. and Claus, R. O. (2001), “Simultaneous Active

Damping and Health Monitoring of Aircraft Panels”, Journal of Intelligent Material

Systems and Structures, Vol. 12, No. 11, pp. 775-783.

Jones, R. T., Sirkis, J. S. and Freibele, E. J. (1997), “Detection of Impact Location

and Magnitude for Isotropic Plates Using Neural Networks”, Journal of Intelligent

Material Systems and Structures, Vol. 8, No. 1, pp. 90-99.

Kamada, T., Fujita, T., Hatayama, T., Arikabe, T., Murai, N., Aizawa, S. and

Tohyama, K. (1997), “Active Vibration Control of Frame Structures with Smart

Structures Using Piezoelectric Actuators (Vibration Control by Control of Bending

Moments of Columns), Smart Materials and Structures, Vol. 6, pp. 448-456.

Kawiecki, G. (1998), “Feasibility of Applying Distributed Piezotransducers to

Structural Damage Detection”, Journal of Intelligent Material Systems and

Structures, Vol. 9, pp.189-197.

References

240

Kawiecki, G. (2001), “Modal damping Measurement for Damage Detection”, Smart

Materials and Structures, Vol. 10, pp. 466-471.

Kessler, S. S., Spearing, S. M., Attala, M. J., Cesnik, C. E. S. and Soutis, C. (2002),

“Damage Detection in Composite Materials Using Frequency Response Methods”,

Composites, Part B: Engineering, Vol. 33, pp. 87-95.

Kolsky, H. (1963), Stress Waves in Solids, Dover Publications, New York.

Kreyszig, E. (1993), Advanced Engineering Mathematics, 7th Ed., Wiley, New

York.

Kumar, S. (1991), “Smart Materials for Accoustic or Vibration Control”, Ph.D.

Dissertation, Pennsylvania State University, PA.

Kuzelev, N. R., Maklashevskii, V. Y., Yumashev, V. M. and Ivanov, I. V. (1994),

“Diagnostics of the State of Aircraft Parts and Assemblies with Radionuclide

Computer Tomography”, Russian Journal of Non-Destructive Testing, Vol. 30, No.

8, pp. 614-621.

LAMSS (2003), Laboratory for Active Materials and Smart Structures, University

of South Carolina, http://www.me.sc.edu/Research/lamss (date of access: 9

September, 2003)

Liang, C., Sun, F. P. and Rogers, C. A. (1993), “An Impedance Method for

Dynamic Analysis of Active Material Systems”, Proceedings of AIAA/ ASME/

ASCE/ Material Systems, La- Jolla, California, pp. 3587-3599.

Liang, C., Sun, F. P. and Rogers, C. A. (1994), “Coupled Electro-Mechanical

Analysis of Adaptive Material Systems- Determination of the Actuator Power

Consumption and System Energy Transfer”, Journal of Intelligent Material Systems

and Structures, Vol. 5, pp. 12-20.

http://www.me.sc.edu/Research/lamss

References

241

Loh, C. H. and Tou, I. C. (1995), “A System Identification Approach to the

Detection of Changes in Both Linear and Non-linear Structural Parameters”,

Earthquake Engineering & Structural Dynamics, Vol. 24, No. 1, pp. 85-97.

Lopes, V., Park, G., Cudney, H. H. and Inman, D. J. (1999), “Smart Structures

Health Monitoring Using Artificial Neural Network”, Proceedings of 2nd

International Workshop on Structural Health Monitoring, edited by F. K. Chang,

Stanford University, Stanford, CA, September 8-10, pp. 976-985.

Lu, Y., Hao, H., and Ma, G. W. (2000), “Experimental Investigation of RC Model

Structures under Simulated Ground Shock Excitations and Associated Analytical

Studies (Part-2).” Technical Report No. 5, June, for NTU-DSTA (LEO) Joint R&D

Project on Theoretical and Experimental Study of Damage Criteria of RC Structures

Subjected to Underground Blast Induced Ground Motions.

Lu, Y., Hao, H., Ma, G. W. and Zhou,Y. X. (2001), “Simulation of Structural

Response Under High Frequency Ground Excitation”, Earthquake Engineering and

Structural Dynamics, Vol. 30, No. 3, pp. 307-325.

Lynch, J. P., Sundararajan, A., Law, K. H., Kiremidjian, A. S., Kenny, T. and

Carryer, E. (2003a), “Embedment of Structural Monitoring Algorithms in a

Wireless Sensing Unit”, Structural Engineering and Mechanics, Vol. 15, No. 3, pp.

285-297.

Lynch, J. P., Partridge, A., Law, K. H., Kenny, T. W., Kiremidjian, A. S. and

Carryer, E. (2003b), “Design of Piezoresistive MEMS-Based Accelerometer for

Integration with Wireless Sensing Unit for Structural Monitoring”, Journal of

Aerospace Engineering, ASCE, Vol. 16, No. 3, pp. 108-114.

Manning, W. J., Plummer, A. R. and Levesley, M. C. (2000), “Vibration Control of

a Flexible Beam with Integrated Actuators and Sensors”, Smart Materials and


References

242

Makkonen, T., Holappa, A., Ella, J. and Salomaa, M. M. (2001), “Finite Element

Simulations of Thin-Film Composite BAW Resonators”, IEEE Transactions on

Ultrasonics, Ferroelectrics and Frequency Control, Vol. 48, No. 5, pp. 1241-1258.

Malhotra, V. M. (1976), Testing Hardened Concrete: Nondestructive Methods,

American Concrete Institute (ACI) Monograph No. 9.

Masri, S. F., Nakamura, M., Chassiakos, A. G. and Caughey, T. K. (1996), “Neural

Network Approach to Detection of Changes in Structural Parameters”, Journal of

Engineering Mechanics, ASCE, Vol. 122, No. 4, pp. 350-360.

Mide Technology Corporation (2004), http://www.mide.com

Nakamura, M., Masri, S. F., Chassiakos, A. G. and Caughey, T. K. (1998), “A

Method for Non-parametric Damage Detection Through the Use of Neural

Networks”, Earthquake Engineering & Structural Dynamics, Vol. 27, No. 9, pp.

997-1010.

NASA (2003), National Astronautics and Space Administration,

http://www.nasa.gov/columbia/media/index.html/COL_wing_diagrams.html (date

of access: 10 September 2003)

Ong, C. W., Yang Y., Wong, Y. T., Bhalla, S., Lu, Y. and Soh, C. K. (2002), “The

Effects of Adhesive on the Electro-Mechanical Response of a Piezoceramic

Transducer Coupled Smart System”, Proceedings of ISSS-SPIE International

Conference on Smart Materials, Structures and Systems, edited by B. Dattaguru, S.

Gopalakrishnan and S. Mohan, 12-14 December, Bangalore, Microart Multimedia

Solutions (Bangalore), pp. 191-197.

Oreta, A. W. C. and Tanabe, T. (1994), "Element Identification of Member

Properties of Framed Structures", Journal of Structural Engineering, ASCE, Vol.

120, No. 7, pp. 1961-1976.

http://www.mide.com/

http://www.nasa.gov/columbia/media/index.html/COL_wing_diagrams.html

References

243

Pandey, A. K., Biswas, M. and Samman, M. M. (1991), “Damage Detection from

Changes in Curvature Mode Shapes”, Journal of Sound and Vibration, Vol. 145,

No. 2, pp. 321-332.

Pandey, A. K. and Biswas, M. (1994), “Damage Detection in Structures Using

Changes in Flexibility”, Journal of Sound and Vibration, Vol. 169, No. 1, pp. 3-17.

Pardo De Vera, C. and Guemes, J. A. (1997), “Embedded Self-Sensing

Piezoelectric for Damage Detection”, Proceedings of International Workshop on

Structural Health Monitoring, edited by F. K. Chang, Stanford University, Stanford,

California, September 18-20, pp. 445-455.

Park, G., Kabeya, K., Cudney, H. H. and Inman, D. J. (1999), “Impedance-Based

Structural Health Monitoring for Temperature Varying Applications”, JSME

International Journal, Vol. 42, No. 2, pp. 249-258.

Park, G. (2000), “Assessing Structural Integrity Using Mechatronic Impedance

Transducers with Applications in Extreme Environments”, Ph.D. Dissertation,

Virginia Polytechnic Institute and State University, Blacksburg, VA.

Park, G., Cudney, H. H. and Inman, D. J. (2000a), “Impedance-Based Health

Monitoring of Civil Structural Components”, Journal of Infrastructure Systems,

ASCE, Vol. 6, No. 4, pp. 153-160.

Park, G., Cudney, H. H. and Inman, D. J. (2000b), “An Integrated Health

Monitoring Technique Using Structural Impedance Sensors”, Journal of Intelligent

Material Systems and Structures, Vol. 11, pp.448-455.

Park, G., Cudney, H. H. and Inman, D. J. (2001), “Feasibility of Using Impedance-

Based Damage Assessment for Pipeline Structures”, Earthquake Engineering and

Structural Dynamics, Vol. 30, No. 10, pp. 1463-1474.

References

244

Park, G., Inman, D. J., Farrar, C. R. (2003a), “Recent Studies in Piezoelectric

Impedance-Based Structural health Monitoring”, Proceedings of 4th International

Workshop on Structural Health Monitoring, edited by F. K. Chang, September 15-

17, Stanford University, Stanford, California, DES Tech Publications, Inc.,

Lancaster, PA, pp. 1423-1430.

Park, G., Sohn, H., Farrar, C. R. and Inman, D. J. (2003b), “Overview of

Piezoelectric Impedance-Based Health Monitoring and Path Forward”, The Shock

and Vibration Digest, Vol. 35, No. 5, pp. 451-463.

Peairs, D. M., Park, G. and Inman, D. J. (2002), “Self-Healing Bolted Joint

Analysis”, Proceedings of 20th International Modal Analysis Conference, February

4-7, Los Angles, CA.

Peairs, D. M., Park, G. and Inman, D. J. (2003), “Improving Accessibility of the

Impedance-Based Structural Health Monitoring Method”, Journal of Intelligent

Material Systems and Structures, Vol. 15, No. 2, pp. 129-139.

PI Ceramic (2003), Product Information Catalogue, Lindenstrabe, Germany,

http://www.piceramic.de.

Pines, D. J. and Lovell, P. A. (1998), “Conceptual Framework of a Remote Wireless

Health Monitoring System for Large Civil Structures, Smart Materials and


Popovics, S., Bilgutay, N. M., Karaoguz, M. and Akgul, T. (2000), “High-

Frequency Ultrasound Technique for Testing Concrete”, ACI Materials Journal,

Vol. 97, No. 1, pp. 58-65.

http://www.piceramic.de/

References

245

Pretorius, J., Hugo, M. and Spangler, R. (2004), “A c Comparison of Packaged

Piezoactuators for Industrial Applications”, Technical Note, Mide Technology

Corporation (http://www.mide.com)

Raju, V. (1998), “Implementing Impedance-Based Health Monitoring Technique”,

Master’s Dissertation, Virginia Polytechnic Institute and State University,

Blacksburg, VA.

Ramuhalli, P., Slade, J., Park, U., Xuan, L. and Udpa, L (2002), “Modeling and

Signal Processing of Magneto Optic Images for Aviation Applications”,

Proceedings of ISSS-SPIE International Conference on Smart Materials, Structures

and Systems, edited by B. Dattaguru, S. Gopalakrishnan and S. Mohan, 12-14

December, Bangalore, Microart Multimedia Solutions (Bangalore), pp. 198-205.

Reddy, J. N. (2001), Recent Developments in Smart Structures, Seminar, July 17,

National University of Singapore, Singapore.

Reddy, J. N. and Barbosa, J. I. (2000), “On Vibration Suppression of

Magnetostrictive Beams”, Smart Materials and Structures, Vol. 9, No. 1, pp. 49-58.

Rhim, J. and Lee, S. W., (1995), "A Neural-Network Approach for Damage

Detection and Identification of Structures", Computational Mechanics, Vol. 16, No.

6, pp. 437-443.

Richter, F., (1983), Physical Properties of Steels and their Temperature

Dependence, Report of Research Institute of Mannesmann, Germany.

Rogers, C. A. (1988), “Workshop Summary”, Proceedings of U.S. Army Research

Office Workshop on Smart Materials, Structures and Mathematical Issues, edited

by C. A. Rogers, September 15-16, Virginia Polytechnic Institute & State

University, Technomic Publishing Co., Inc., pp. 1-12.

http://www.mide.com/

References

246

Rogers, C. A., Barker, D. K. and Jaeger, C. A. (1988), “Introduction to Smart

Materials and Structures”, Proceedings of U.S. Army Research Office Workshop on

Smart Materials, Structures and Mathematical Issues, edited by C.A. Rogers,

September 15-16, Virginia Polytechnic Institute & State University, Technomic

Publishing Co., Inc., pp. 17-28.

Rogers, C. A. (1990), “Intelligent Material Systems and Structures”, Proceedings of

U.S.-Japan Workshop on Smart/ Intelligent Materials and Systems, edited by I.

Ahmad, A. Crowson, C. A. Rogers and M. Aizawa, March 19-23, Honolulu,

Hawaii, Technomic Publishing Co., Inc., pp. 11-33.

RS Components (2003), Northants, UK, http://www.rs-components.com.

Saffi, M. and Sayyah, T. (2001), “Health Monitoring of Concrete Structures

Strengthened with Advanced Composite Materials Using Piezoelectric

Transducers”, Composites Part B: Engineering, Vol. 32, No. 4, pp. 333-342.

Samman, M. M. and Biswas, M. (1994a), "Vibration Testing for Non-Destructive

Evaluation of Bridges. I: Theory", Journal of Structural Engineering, ASCE, Vol.

120, No. 1, pp. 269-289.

Samman, M. M. and Biswas, M. (1994b), "Vibration Testing for Non-Destructive

Evaluation of Bridges. II: Results", Journal of Structural Engineering, ASCE, Vol.

120, No. 1, pp. 290-306.

Sanayei, M. and Saletnik, M. J. (1996), “Parameter Estimation of Structures from

Static Strain Measurements. I: Formulation”, Journal of Structural Engineering,

ASCE, Vol. 122, No. 5, pp. 555-562.

Sensor Technology Limited (1995), Product Catalogue, Collingwood.

http://www.rs-components.com/

References

247

Shah, S. P., Popovics, J. S., Subramaniam, K. V. and Aldea, C. M. (2000), “New

Directions in Concrete Health Monitoring Technology”, Journal of Engineering

Mechanics, ASCE, Vol. 126, No. 7, pp. 754-760.

Singhal, A. and Kiremidjian, A. S. (1996), “Method for Probabilistic Evaluation of

Seismic Structural Damage”, Journal of Structural Engineering, ASCE, Vol. 122,

No. 12, pp. 1459-1467.

Sirohi, J. and Chopra, I. (2000a), “Fundamental Behaviour of Piezoceramic Sheet

Actuators”, Journal of Intelligent Material Systems and Structures, Vol. 11, No. 1,

pp. 47-61.

Sirohi, J. and Chopra, I. (2000b), “Fundamental Understanding of Piezoelectric

Strain Sensors”, Journal of Intelligent Material Systems and Structures, Vol. 11,

No. 4, pp. 246-257.

Skjaerbaek, P. S., Nielsen, S. R. K., Kirkegaard, P. H. and Cakmak, A. S. (1998),

"Damage Localization and Quantification of Earthquake Excited R.C-Frames",

Earthquake Engineering and Structural Dynamics, Vol. 27, No. 9, pp. 903-916.

Soh, C. K., Tseng, K. K. H., Bhalla, S. and Gupta, A. (2000), “Performance of

Smart Piezoceramic Patches in Health Monitoring of a RC Bridge”, Smart

Materials and Structures, Vol. 9, No. 4, pp. 533-542.

Soh, C. K., Yang, J. P. (1996), “Fuzzy Controlled Genetic Algorithm Search for

Shape Optimization”, Journal of Computing in Civil Engineering, ASCE, Vol. 10,

No. 2, pp. 143-150.

Song, G., Qiao, P. Z., Binienda, W. K. and Zou, G. P. (2002), “Active Vibration

Damping of Composite Beam Using Smart Sensors and Actuators”, Journal of

Aerospace Engineering, ASCE, Vol. 15, No. 3, pp. 97-103.

References

248

Storoy, H., Saether, J. and Johannessen, K. (1997), “Fiber Optic Condition

Monitoring During a Full Scale Destructive Bridge Test”, Journal of Intelligent

Material Systems and Structures”, Vol. 8, No. 8, pp. 633-643.

Stubbs, N. and Kim, J. T. (1994), “Field Verification of a Nondestructive Damage

Localization and Severity Estimation Algorithm”, Texas A & M University Report

prepared for New Mexico State University.

Sun, F. P., Chaudhry, Z., Rogers, C. A., Majmundar, M. and Liang, C. (1995)

“Automated Real-Time Structure Health Monitoring via Signature Pattern

Recognition”, edited by I. Chopra, Proceedings of SPIE Conference on Smart

Structures and Materials, San Diego, California, Feb.27-Mar1, SPIE vol. 2443, pp.

236-247.

Szewczyk, Z. P. and Hajela, P. (1994), "Damage Detection in Structures Based on

Feature-Sensitive Neural Networks", Journal of Computing in Civil Engineering,

ASCE, Vol. 8, No. 2, pp. 163-178.

Takagi, T. (1990), “A Concept of Intelligent Materials”, Proceedings of U.S.-Japan

Workshop on Smart/ Intelligent Materials and Systems, edited by I. Ahmad, A.

Crowson, C. A. Rogers and M. Aizawa, March 19-23, Honolulu, Hawaii,

Technomic Publishing Co., Inc., pp. 3-10.

Tseng, K. K. H and Naidu, A. S. K. (2001), “Non-Parametric Damage Detection

and Characterization Using Piezoceramic Material”, Smart Materials and


Ugural, A. C. and Fenster, S. K. (1995), Advanced Strength and Applied Elasticity,

3rd ed., Prentice Hall.

References

249

USDT (2003), United States Department of Transportation, Federal Highway

Administration, http://www.tfhrc.gov/pubrds/summer95/p95su23.htm (date of

access: 10 September, 2003)

Valliappan, S. and Pham, T. D. (1993), “Fuzzy Finite Element Analysis of a

Foundation on an Elastic Soil Medium”, International Journal for Numerical and

Analytical Methods on Geomechanics, Vol. 17, pp. 771-789.

Vardan, V. K. and Vardan, V. V. (2002), “Microsensors, Micoelectromechanical

Systems (MEMS) and Electronics for Smart Structures and Systems”, Smart

Materials and Structures, Vol. 9, No. 6, pp.953-972.

Varadan (2000), “Nanotechnology, MEMS and NEMS and their Applications to

Smart Systems and Devices”, Proceedings of ISSS-SPIE International Conference

on Smart Materials, Structures and Systems, edited by B. Dattaguru, S.

Gopalakrishnan and S. Mohan, 12-14 December, Bangalore, Microart Multimedia

Solutions (Bangalore), pp. 909-932.

Wang, B. T. and Chen, R. L. (2000), “The Use of Piezoceramic Transducers for

Smart Structural Testing”, Journal of Intelligent Material Systems and Structures,

Vol. 11, No. 9, pp. 713-724.

Wang, M. L., Heo, G. and Satpathi, D. (1998), “A Health Monitoring System for

Large Structural Systems”, Smart Materials and Structures, Vol. 7, No. 5, pp. 606-

616.

Winston, H. A., Sun, F. and Annigeri, B. S. (2001), “Structural Health Monitoring

with Piezoelectric Active Sensors”, Journal of Engineering for Gas Turbines and

Power, ASME, Vol. 123, No. 2, pp. 353-358.

http://www.tfhrc.gov/pubrds/summer95/p95su23.htm

References

250

Wu, C. Q., Hao, H. and Zhou, Y. X. (1999), “Fuzzy-Random Probabilistic Analysis

of Rock Mass Responses to Explosive Loads”, Computers and Geotechnics, Vol.

25, No. 4, pp. 205-225.

Wu, C. Q., Hao, H., Zhao, J. and Zhou, Y. X. (2001), “Statistical Analysis of

Anisotropic Damage of the Bukit Timah Granite”, Rock Mechanics and Rock

Engineering, Vol. 34, No. 1, pp. 23-38.

Xu, Y. G. and Liu, G. R. (2002), “A Modified Electro-mechanical Impedance

Model of Piezoelectric Actuator-Sensors for Debonding Detection of Composite

Patches”, Journal of Intelligent Material Systems and Structures, Vol. 13, No. 6,

pp. 389-396.

Yamakawa, H., Iwaki, H., Mita, A. and Takeda, N. (1999), “Health Monitoring of

Steel Structures Using Fiber Bragg Grating Sensors”, Proceedings of 2nd

International Workshop on Structural Health Monitoring, edited by F. K. Chang,

Stanford University, Stanford, CA, September 8-10, pp. 502-510.

Yang, Y. W. and Soh, C. K. (2000), “Fuzzy Logic Integrated Genetic

Programming for Optimization and Design”, Journal of Computing in Civil

Engineering, ASCE, Vol. 14, No. 4, pp. 249-254.

Yao, J. T. P. (1985), Safety and Reliability of Existing Structures, Pitman

Publishing Programme, London.

Zadeh, L. A. (1965), “Fuzzy Sets”, Information Control, Vol. 8, pp. 338-353.

Zhou, S., Liang, C. and Rogers, C. A. (1995), “Integration and Design of

Piezoceramic Elements in Intelligent Structures”, Journal of Intelligent Material

Systems and Structures, Vol. 6, No. 6, pp. 733-743.

References

251

Zhou, S. W., Liang, C. and Rogers C. A. (1996), “An Impedance-Based System

Modeling Approach for Induced Strain Actuator-Driven Structures”, Journal of

Vibrations and Acoustics, ASME, Vol. 118, No. 3, pp. 323-331.

Zhu, W. (2003), Sensors and Actuators (E6614), Lecture Notes, Nanyang

Technological University, Singapore.

Zienkiewicz, O. C. (1977), The Finite Element Method, McGraw Hill Book

Company (UK) Limited, London.

Zimmerman, D. C. and Kaouk, M. (1994), “Structural Damage Detection Using a

Minimum Rank Update Theory”, Journal of Vibration and Accoustics, Vol. 116,

pp. 222-231.

Appendix A

252

APPENDIX (A)

Visual Basic program to derive conductance and susceptance plots from ANSYSoutput. This program is based on 1D impedance model of Liang et al. (1994), Eq.2.24.

All units in the SI system‘Inputs: Frequency (Hz) Fr (N) Fi (N) Ur (m/s) Ui (m/s)’

‘Declaration of variables’Public Const LA As Double = 0.005 ‘Length of PZT patch’Public Const WA As Double = 1# ‘Width of PZT patch’Public Const HA As Double = 0.0002 ‘Thickness of PZT patch’Public Const RHO As Double = 7650# ‘Density of PZT’Public Const D31 As Double = -0.000000000166 ‘Piezoelectric strain coefficient’Public Const Y11E As Double = 63000000000# ‘Young’s modulus of PZT’Public Const E33T As Double = 0.000000015 ‘Electric permittivity of PZT’Public Const ETA As Double = 0.001 ‘Mechanical loss factor’Public Const DELTA As Double = 0.012 ‘Electric loss factor’Dim f As Double ‘Frequency in Hz’Dim k_real As Double ‘Real component of wave number’Dim k_imag As Double ‘Imaginary component of wave number’Dim x, y As Double ‘Real and imaginary components of structural mechanical impedance’Dim xa, ya As Double ‘Real and imaginary components of PZT mechanical impedance’Dim r, t As Double ‘Real and imaginary components of tankl/kl’Dim G, B As Double ‘Real and imaginary components of admittane’Dim Fr, Fi, Ur, Ui As Double ‘Real and imaginary components of force and displacement’

‘Main program’Sub main()Dim index As IntegerDim kl_real, kl_imag As Double ‘Real and imaginary components of kl’For index = 8 To 508

f = Cells(index, 1)Fr = Cells(index, 6)Fi = Cells(index, 7)Ur = Cells(index, 4)Ui = Cells(index, 5)Call calc_str_impdCall calc_k(f)kl_real = k_real * LAkl_imag = k_imag * LACall tanz_by_z(kl_real, kl_imag)Call Z_actuatorCall calc_YCells(index, 9) = xCells(index, 10) = yCells(index, 11) = xaCells(index, 12) = yaCells(index, 13) = GCells(index, 14) = BNext index

End Sub

‘Subroutine to calculate (tankl/kl)’Sub tanz_by_z(rl, im As Double)

Appendix A

253

Dim a, b, c, d, u, v, q As Double

a = (Exp(-im) + Exp(im)) * Sin(rl)b = (Exp(-im) - Exp(im)) * Cos(rl)c = (Exp(-im) + Exp(im)) * Cos(rl)d = (Exp(-im) - Exp(im)) * Sin(rl)

u = c * rl - d * imv = d * rl + c * imq = u * u + v * vr = (a * u - b * v) / qt = (-1#) * (a * v + b * u) / qEnd Sub

‘Subroutine to calculate kl’Sub calc_k(freq)Dim w, cons As Doublew = 2# * 3.14 * freqcons = Sqr(RHO / (Y11E * (1 + ETA * ETA)))k_real = cons * wk_imag = cons * w * (-0.5 * ETA)End Sub

‘Subroutine to calculate complex admittance’Sub calc_Y()Dim p, q, Big_p, Big_q, Big_R, Big_T, Big_pq As Double ‘Temporary variables’Dim temp_r, temp_i As Double ‘Temporary variables’p = x + xaq = y + yaBig_p = xa * p + ya * qBig_q = ya * p - xa * qBig_R = r - ETA * tBig_T = ETA * r + tBig_pq = p * p + q * qtemp_r = (Big_p * Big_T + Big_q * Big_R) / Big_pqtemp_i = (Big_p * Big_R - Big_q * Big_T) / Big_pqt_r = ETA - temp_rt_i = temp_i - 1multi = (WA * LA * 2# * 3.14 * f) / HAG=2* multi * (DELTA * E33T + t_r * D31 * D31 * Y11E)B =2* multi * (E33T + t_i * D31 * D31 * Y11E)End Sub

‘Subroutine to calculate actuator impedance’Sub Z_actuator()Dim multi As Doublemultia = (WA * HA * Y11E) / (2 * 3.14 * LA * f)Big_rt = r * r + t * txa = multi * (ETA * r - t) / Big_rtya = multi * (-1#) * (r + ETA * t) / Big_rtEnd Sub

‘Subroutine to calculate structure impedance’Function calc_str_impd()Dim div As Doublediv = 2# * 3.14 * fBig_U = Ur * Ur + Ui * Uix = (Fi * Ur - Fr * Ui) / (div * Big_U)y = (-1#) * (Fr * Ur + Fi * Ui) / (div * Big_U)End Function

Appendix B

254

APPENDIX (B)

Visual Basic program to derive real and imaginary components of structuralimpedance from admittance signatures. This program is based on 1D impedancemodel of Liang et al. (1994), Eq. 2.24.

All units in the SI system‘Inputs: Frequency (kHz) G (S) B (S)’

‘Declaration of variables’Public Const LA As Double = 0.005 ‘Half-length of PZT patch’Public Const WA As Double = 0.01 ‘Width of PZT patch’Public Const HA As Double = 0.0002 ‘Thickness of PZT patch’Public Const RHO As Double = 7800# ‘Density of PZT’Public Const D31 As Double = -0.00000000021 ‘Piezoelectric strain coefficient’Public Const Y11E As Double = 66700000000# ‘Young’s modulus of PZT’Public Const E33T As Double = 0.00000002124 ‘Electric permittivity of PZT’Public Const ETA As Double = 0.001 ‘Mechanical loss factor’Public Const DELTA As Double = 0.015 ‘Electric loss factor’Dim f As Double ‘Frequency in Hz’Dim k_real, k_imag As Double ‘Real and imaginary components of wave number’Dim kl_real, kl_imag As Double ‘Real and imaginary components of kl’Dim Ga As Double ‘Active conductance'Dim Ba As Double ‘Passive susceptance'Dim rgx, k, c As Double ‘Temporary variables’Dim x, y As Double ‘Real and imaginary components of structural mechanical impedance’Dim xa, ya As Double ‘Real and imaginary components of actuator mechanical impedance’Dim xt, yt As Double ‘xt = x + xa and yt = y + ya’Dim r, t As Double ‘Real and imaginary components of tankl/kl’Dim Big_R, Big_T As Double ‘Temporary variables’Dim multi As Double ‘Temporary variable’

‘Main program’Sub main()Dim Index As Integer ‘For loop index’For Index = 7 To 507 f = Cells(Index, 1) * 1000 ‘Conversion to Hz’

multi = (WA * LA * 2# * 3.14 * f) / HA G = 0.5*Cells(index, 2) Gp = multi * (E33T * DELTA + D31 ^ 2 * Y11E * ETA) Ga = G - Gp Cells(index, 4) = Ga

B = 0.5*Cells(index, 3)Bp = multi * (E33T - D31 ^ 2 * Y11E)

Ba = B - Bp Cells(index, 5) = Ba

rgx = Ga / Ba Call calc_k(f) kl_real = k_real * LA kl_imag = k_imag * LA

Appendix B

255

Call tanz_by_z(kl_real, kl_imag) Call Z_actuator k = D31 * D31 * Y11E * (WA * LA / HA) Big_R = r - ETA * t Big_T = t + ETA * r c = (Big_T + rgx * Big_R) / (rgx * Big_T - Big_R) ct = (ya - c * xa) / (c * ya + xa) xt = (-1#) * (2 * 3.14 * f) k * (ya * ct + xa) * (Big_T + Big_R * c) / (Ga * (1 + ct * ct)) yt = ct * xt x = xt - xa y = yt - ya Cells(Index, 6) = x Cells(Index, 7) = y Next rowNext nEnd Sub

‘Subroutine to calculate wave number’Sub calc_k(freq)Dim w, cons As Doublew = 2# * 3.14 * freqcons = Sqr(RHO / (Y11E * (1 + ETA * ETA)))k_real = cons * wk_imag = cons * w * (-0.5 * ETA)End Sub

‘Subroutine to calculate (tankl/kl)’Sub tanz_by_z(rl, im As Double)Dim a, b, c, d, u, v, q As Double

a = (Exp(-im) + Exp(im)) * Sin(rl)b = (Exp(-im) - Exp(im)) * Cos(rl)c = (Exp(-im) + Exp(im)) * Cos(rl)d = (Exp(-im) - Exp(im)) * Sin(rl)

u = c * rl - d * imv = d * rl + c * imq = u * u + v * v

r = (a * u - b * v) / qt = (-1#) * (a * v + b * u) / qEnd Sub

‘Subroutine to calculate mechanical impedance of actuator’Sub Z_actuator()Dim multia As Doublemultia = (WA * HA * Y11E) / (2 * 3.14 * LA * f)Big_rt = r * r + t * txa = multia * (ETA * r - t) / Big_rtya = multia * (-1#) * (r + ETA * t) / Big_rtEnd Sub

Appendix C

256

APPENDIX (C)

MATLAB program to derive elecro-mechanical admittance signatures fromANSYS output. The program is based on the new 2D model based on effectiveimpedance, covered in Chapter 5 (Eq. 5.30)

All units in the SI system%Inputs: Frequency (Hz) Fr (N) Fi (N) Ur (m/s) Ui (m/s)

data=dlmread('output250.txt','\t'); %Data-matrix, stores ANSYS output% The symbols declared below carry same meaning as in Appendices A, B

LA=0.005; HA= 0.0003; RHO=7800; D31= -0.00000000021;mu=0.3;Y11E= 66700000000; E33T=1.7919e-8; ETA= 0.035; DELTA= 0.0238;

f = data(:,1); %Frequency in HzFr = data(:,2); %Real component of effective forceFi = data(:,3); %Imaginary component of effective forceUr = data(:,4); %Real component of effective displacementUi = data(:,5); %Imaginary component of effective %displacement

N=size(f); %No of data points

for I = 1:N,%Calculation of structural impedanceomega(I) = 2* pi * f(I); %Angular frequency in rad/sBig_U(I)= Ur(I)*Ur(I) + Ui(I)*Ui(I);x(I) = 2*(Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I));y(I) = 2*(-1.0) * (Fr(I)*Ur(I)+Fi(I)*Ui(I))/(omega(I) * Big_U(I));

%Calculation of wave numbercons = (RHO *(1-mu*mu)/ (Y11E * (1 + ETA * ETA)))^0.5;k_real(I) = cons * omega(I);k_imag(I) = cons * omega(I) * (-0.5 * ETA);rl(I) = k_real(I) * LA;im(I) = k_imag(I) * LA;

%Calculation of tan(kl)/kla(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));u(I) = c(I) * rl(I) - d(I) * im(I);v(I) = d(I) * rl(I) + c(I) * im(I);h(I) = u(I)^2 + v(I)^2;r(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);t(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);

%Calculation of actuator impedancemultia(I) = (HA * Y11E) / (pi * (1-mu)* f(I));Big_rt(I) = r(I) * r(I) + t(I) * t(I);

Appendix C

257

xa(I) = multia(I) * (ETA * r(I) - t(I)) / Big_rt(I);ya(I) = multia(I) * (-1.0) * (r(I) + ETA * t(I)) / Big_rt(I);

%Calculation of conductance and susceptancep(I) = x(I) + xa(I);q(I) = y(I) + ya(I);Big_p(I) = xa(I) * p(I) + ya(I) * q(I);Big_q(I) = ya(I) * p(I) - xa(I) * q(I);Big_R(I) = r(I) - ETA * t(I);Big_T(I) = ETA * r(I) + t(I);Big_pq(I) = p(I) * p(I) + q(I) * q(I);temp_r(I) = (Big_p(I)*Big_T(I)+ Big_q(I)* Big_R(I)) / Big_pq(I);temp_i(I) = (Big_p(I)*Big_R(I)- Big_q(I)* Big_T(I)) / Big_pq(I);t_r(I) = ETA - temp_r(I);t_i(I) = temp_i(I) - 1;multi(I) = (LA * LA * omega(I)) / HA;K = 2.0 * D31 * D31 * Y11E /(1 - mu);G(I) = 4*multi(I) * (DELTA * E33T + K * t_r(I));B(I) = 4*multi(I) * (E33T + K * t_i(I));

end

subplot(2,1,1);plot(f,G);subplot(2,1,2);plot(f,B);

Appendix D

258

APPENDIX (D)

MATLAB program to derive PZT signatures from ANSYS output, using updatedPZT model (twin-peak). The program is based on the new 2D model based oneffective impedance, covered in Chapter 5. (Eq. 5.56)

NOTE: Single peak case can also be dealt with by using cf1 = cf2All units in the SI system%Inputs: Frequency (Hz), Fr (N), Fi (N), Ur (m), Ui (m)

%PZT parameters- based on measurement.data=dlmread('output250.txt','\t');%Data-matrix, stores the ANSYS output

%PZT parameters based on updated model derived by experiment%Symbols for following variables carry same meaning as Appendices A,BLA=0.005; HA= 0.0003; RHO=7800; D31= -2.1e-10;mu=0.3;Y11E= 6.67e10; E33T=1.7919e-8; ETA= 0.03; DELTA= 0.0238; K =5.16e-9;

f = data(:,1); %Frequency in HzFr = data(:,2); %Real component of effective forceFi = data(:,3); %Imaginary component of effective forceUr = data(:,4); %Real component of effective displacementUi = data(:,5); %Imaginary component of effective displacementN=size(f); %No of data pointscf1 = 0.94; %Correction factors for PZT peakscf2 = 0.883; %For single peak case, Cf1 = cf2

for I = 1:N,

%Calculation of structural impedanceomega(I) = 2* pi * f(I); %Angular frequency in rad/sBig_U(I)= Ur(I)*Ur(I) + Ui(I)*Ui(I);x(I) = 2*(Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I));y(I) = 2*(-1.0) * (Fr(I)*Ur(I) + Fi(I)*Ui(I))/(omega(I)* Big_U(I));

%Calculation of wave numbercons = (RHO *(1-mu*mu)/ (Y11E * (1 + ETA * ETA)))^0.5;k_real(I) = cons * omega(I);k_imag(I) = cons * omega(I) * (-0.5 * ETA);

%Calculation of tan(kl)/klrl(I) = k_real(I) * LA * cf1;im(I) = k_imag(I) * LA * cf1;

a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));u(I) = c(I) * rl(I) - d(I) * im(I);v(I) = d(I) * rl(I) + c(I) * im(I);

Appendix D

259

h(I) = u(I)^2 + v(I)^2;r1(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);t1(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);

rl(I) = k_real(I) * LA * cf2;im(I) = k_imag(I) * LA * cf2;

a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));u(I) = c(I) * rl(I) - d(I) * im(I);v(I) = d(I) * rl(I) + c(I) * im(I);h(I) = u(I)^2 + v(I)^2;r2(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);t2(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);

r(I) = 0.5 * (r1(I)+r2(I));t(I) = 0.5 * (t1(I)+t2(I));

%Calculation of actuator impedancemultia(I) = (HA * Y11E) / (pi * (1-mu)* f(I));Big_rt(I) = r(I) * r(I) + t(I) * t(I);xa(I) = multia(I) * (ETA * r(I) - t(I)) / Big_rt(I);ya(I) = multia(I) * (-1.0) * (r(I) + ETA * t(I)) / Big_rt(I);

%Calculation of conductance and susceptancep(I) = x(I) + xa(I);q(I) = y(I) + ya(I);Big_p(I) = xa(I) * p(I) + ya(I) * q(I);Big_q(I) = ya(I) * p(I) - xa(I) * q(I);Big_R(I) = r(I) - ETA * t(I);Big_T(I) = ETA * r(I) + t(I);Big_pq(I) = p(I) * p(I) + q(I) * q(I);temp_r(I) = (Big_p(I) * Big_T(I)+ Big_q(I) * Big_R(I)) / Big_pq(I);temp_i(I) = (Big_p(I)* Big_R(I) - Big_q(I) * Big_T(I)) / Big_pq(I);t_r(I) = ETA - temp_r(I);t_i(I) = temp_i(I) - 1;multia(I) = (LA * LA * omega(I)) / HA;G(I) = 4*multia(I) * (DELTA * E33T + K *t_r(I));B(I) = 4*multia(I) * (E33T + K *t_i(I));

endsubplot(2,1,1);plot(f,G);subplot(2,1,2);plot(f,B);

Appendix E

260

APPENDIX (E)

MATLAB program to derive structural mechanical impedance from experimentaladmittance signatures, using updated PZT model (twin-peak). The program isbased on the new 2D model based on effective impedance, covered in Chapter 5(Eq. 5.56).

NOTE: For single peak case, cf1 = cf2All units in the SI system%Inputs: Frequency (kHz), G (S), B (S)

%PZT parameters- based on measurement.data=dlmread('gb.txt','\t'); %Data-matrix,%The symbols for variables carry same meaning as in Appendices A,BLA=0.005; HA= 0.0003; RHO=7800; D31= -2.1e-10;mu=0.3;Y11E= 6.67e10; E33T=1.7919e-8; ETA= 0.03; DELTA= 0.0238;cf1 = 0.94; cf2 = 0.883; %Correction factors for PZT peaks

%For single peak case, cf1 = cf2f = 1000*data(:,1); %Frequency in HzG = data(:,2); %ConductanceB = data(:,3); %Susceptance

K = 5.16e-9; %K = 2*D31*D31*Y11E/(1-mu);no=size(f); %No of data points

for I = 1:no,

%Calculation of active signaturesomega(I) = 2*pi*f(I);multi(I) = 4*(LA * LA * omega(I)) / HA;Gp(I) = multi(I) * (E33T * DELTA + K * ETA);GA(I) = G(I)- Gp(I);Bp(I) = multi(I) * (E33T - K);BA(I) = B(I) - Bp(I);

%Calculation of M and NM(I) = (BA(I)*HA)/(4*omega(I)*K*LA*LA);N(I) = (-GA(I)*HA)/(4*omega(I)*K*LA*LA);

%Calculation of wave numbercons = (RHO * (1-mu*mu)/ (Y11E * (1 + ETA * ETA)))^0.5;k_real(I) = cons * omega(I);k_imag(I) = cons * omega(I) * (-0.5 * ETA);rl(I) = k_real(I) * LA;im(I) = k_imag(I) * LA;

%Calculation of tan(kl)/klrl(I) = k_real(I) * LA * cf1;im(I) = k_imag(I) * LA * cf1;

a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));

Appendix E

261

b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));u(I) = c(I) * rl(I) - d(I) * im(I);v(I) = d(I) * rl(I) + c(I) * im(I);h(I) = u(I)^2 + v(I)^2;r1(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);t1(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);

rl(I) = k_real(I) * LA * cf2;im(I) = k_imag(I) * LA * cf2;

a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));u(I) = c(I) * rl(I) - d(I) * im(I);v(I) = d(I) * rl(I) + c(I) * im(I);h(I) = u(I)^2 + v(I)^2;r2(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);t2(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);

r(I) = 0.5 * (r1(I)+r2(I));t(I) = 0.5 * (t1(I)+t2(I));


%Calculation of structural impedanceR(I) = r(I) - ETA * t(I);S(I) = ETA * r(I) + t(I);P(I) = xa(I) * R(I) - ya(I) * S(I);Q(I) = xa(I) * S(I) + ya(I) * R(I);MN(I)= M(I)^2+N(I)^2;x(I) = (P(I)*M(I)+Q(I)*N(I))/MN(I) - xa(I);y(I) = (Q(I)*M(I)-P(I)*N(I))/MN(I) - ya(I);

enddlmwrite('f.txt',f,'\t');dlmwrite('x.txt',x,'\t');dlmwrite('y.txt',y,'\t');

Appendix F

262

APPENDIX (F)

MATLAB program to compute fuzzy failure probability

All units in the SI system

x=sym('x');mu = 0.3314; % Mean damage variablesigma= 0.0466; % Standard deviation of damage variable

Dl = 0; % Lower limit of fuzzy intervalDu = 0.4; % Upper limit of fuzzy interval

fuzzy = 0.5 + 0.5*sin((pi/(Du-Dl))*(x-0.5*Du-0.5*Dl));

pow = (-0.5)*(x-mu)^2/(sigma^2);f = exp(pow)/(sqrt(2*pi)*sigma);

ans = double(int(f*fuzzy,0,0.4)+ int(f,0.4,1))

Appendix G

263

APPENDIX (G)

MATLAB program to derive electro-mechanical admittance signatures fromANSYS output, taking shear lag in the adhesive layer into account. The programis based on the new 2D model based on effective impedance, covered in Chapter 5(Eqs. 5.56 and 7.90).

NOTE: Single peak case can also be dealt with by using cf1 = cf2‘All units in the SI system’%Inputs: Frequency (Hz), Fr (N), Fi (N), Ur (m), Ui (m)

%PZT parameters- based on measurement.data=dlmread('output.txt','\t'); %Data-matrix, stores the ANSYS output

%Parameters of PZT (averaged, as in Chapter 6)%Symbols for following variables carry same meaning as Appendices A,BLA=0.005; HA= 0.0003; RHO=7800; D31= -0.00000000021;mu=0.3;Y11E= 66700000000; E33T=1.7785e-8; ETA= 0.0325; DELTA= 0.0224;K =5.35e-9;GE = 1.0e9; HE = 0.000125; BETA = 0.1;V=1.4;cf=0.898;%BETA represents mechanical loss factor of bonding material

f = data(:,1); %Frequency in HzFr = data(:,2); %Real component of effective forceFi = data(:,3); %Imaginary component of effective forceUr = data(:,4); %Real component of effective displacementUi = data(:,5); %Imaginary component of effective displacementN=size(f); %No of data points

for I = 1:N,%Calculation of structural impedanceomega(I) = 2* pi * f(I); %Angular frequency in rad/sBig_U(I)= Ur(I)*Ur(I) + Ui(I)*Ui(I);x(I) = 2*(Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I));y(I) = 2*(-1.0) * (Fr(I) * Ur(I) + Fi(I) * Ui(I)) / (omega(I) * Big_U(I));

%Consideration of shear lag effectPS = D31*V/HA; %Free PZT strainqe = GE*(1-mu*mu)/(Y11E*HA*HE); % qeffae = 2*LA*GE*(1+mu)*(y(I)-BETA*x(I))/(omega(I)*HE*(x(I)^2+y(I)^2));be = 2*LA*GE*(1+mu)*(x(I)+BETA*y(I))/(omega(I)*HE*(x(I)^2+y(I)^2));

pe=complex(ae,be); %peffroot3=(-pe/2)+sqrt(pe*pe/4+qe);root4=(-pe/2)-sqrt(pe*pe/4+qe);ne=1/pe;E3=exp(root3*LA);E4=exp(root4*LA);E3m=exp(-1*root3*LA);E4m=exp(-1*root4*LA);

%Determination of Constants from boundary conditionsk1 = (1+ne*root3)*root3*E3m-root3;k2 = (1+ne*root4)*root4*E4m-root4;k3 = (1+ne*root3)*root3*E3-root3;k4 = (1+ne*root4)*root4*E4-root4;

Appendix G

264

Be=PS*(k4-k2)/(k1*k4-k2*k3);Ce=PS*(k1-k3)/(k1*k4-k2*k3);A1=-Be-Ce;A2=-Be*root3-Ce*root4;ue=A1+A2*LA+Be*E3+Ce*E4; %End displacement on surface of host structurese=A2+Be*root3*E3+Ce*root4*E4; %End strain surface of host structureZ=complex(x(I),y(I));Zeq=Z/(1+ne*se/ue);x(I)=real(Zeq);y(I)=imag(Zeq);

%Calculation of wave numbercons = (RHO *(1-mu*mu)/ (Y11E * (1 + ETA * ETA)))^0.5;k_real(I) = cons * omega(I);k_imag(I) = cons * omega(I) * (-0.5 * ETA);

%Calculation of tan(kl)/klrl(I) = k_real(I) * LA * cf;im(I) = k_imag(I) * LA * cf;

a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));u(I) = c(I) * rl(I) - d(I) * im(I);v(I) = d(I) * rl(I) + c(I) * im(I);h(I) = u(I)^2 + v(I)^2;r(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);t(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);


%Calculation of conductance and susceptancep(I) = x(I) + xa(I);q(I) = y(I) + ya(I);Big_p(I) = xa(I) * p(I) + ya(I) * q(I);Big_q(I) = ya(I) * p(I) - xa(I) * q(I);Big_R(I) = r(I) - ETA * t(I);Big_T(I) = ETA * r(I) + t(I);Big_pq(I) = p(I) * p(I) + q(I) * q(I);temp_r(I) = (Big_p(I) * Big_T(I) + Big_q(I) * Big_R(I)) / Big_pq(I);temp_i(I) = (Big_p(I) * Big_R(I) - Big_q(I) * Big_T(I)) / Big_pq(I);t_r(I) = ETA - temp_r(I);t_i(I) = temp_i(I) - 1;multi(I) = (LA * LA * omega(I)) / HA;G(I) = 4*multi(I) * (DELTA * E33T + K *t_r(I));B(I) = 4*multi(I) * (E33T + K *t_i(I));Gnor(I)=G(I)*HA/(LA*LA); %Normalized conductanceBnor(I)=B(I)*HA/(LA*LA); %Normalized susceptance

end

plot(f,G);figure;plot(f,B);

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