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
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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
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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.
Chapter 1: Introduction
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.
Chapter 1: Introduction
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
Chapter 1: Introduction
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)
Chapter 1: Introduction
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
Chapter 1: Introduction
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).
Chapter 1: Introduction
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
Chapter 1: Introduction
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.
Chapter 1: Introduction
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
Chapter 1: Introduction
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.
Chapter 1: Introduction
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.
Chapter 1: Introduction
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
Chapter 1: Introduction
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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).
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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).
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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’.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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)
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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)
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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)
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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,
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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)
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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).
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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.
Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
20% increase in ‘c’
20% reduction in ‘k’
PZT patch, |Za|
20% increase in ‘m’ Structure, |Z|
20% increase in ‘c’
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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) 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).
(a)
(c) (d)
(e)
Pristine State
20% increase in ‘c’
20% reduction in ‘k’
20% increase in ‘m’
20% reduction in ‘k’
20% increase in ‘m’
Pristine State20% increase in ‘c’
20% reduction in ‘k’
PZT patch
20% increase in ‘m’
20% increase in ‘c’
Pristine State
(b)
Structure, x Structure, y
PZT patch, xa PZT patch, ya
|Za|
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
(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).
(c) (d)
(b)
(e)
Pristine State
500% increase in ‘c’
80% reduction in ‘k’1000% increase in ‘m’
80% reduction in ‘k’
1000% increase in ‘m’
Pristine State
500% increase in ‘c’
80% reduction in ‘k’
PZT patch, xa
PZT patch, ya
PZT patch1000% increase in ‘m’
Pristine State
500% increase in ‘c’
(a)
Structure, x
Structure, y
|Za|
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
(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).
(c) (d)
(a) (b)
(e)
Pristine State
20% increase in ‘c’
Pristine State20% increase in ‘c’
20% reduction in ‘k’
PZT patch, xa
PZT patch, ya
PZT patch20% increase in ‘m’
20% increase in ‘c’
Pristine State
Structure, x Structure, y
|Za|
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
(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).
(b)
(d)
(e)
Pristine State
20% increase in ‘c’
Pristine State
20% increase in ‘c’
50% reduction in the imaginary part
PZT patch, xa
PZT patch, ya
PZT patch
20% increase in ‘c’
Pristine State
(a)
(c)
Structure, x
Structure, y
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
(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 (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’
20% increase in ‘m’Pristine State
PZT patch, xa
PZT patch, ya
Pristine State
PZT patch
20% increase in ‘m’
Structure, x
Structure, y
y =-yay =-ya
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
(a) Conductance vs Frequency. (b) Susceptance vs Frequency.
(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)
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
(a) Conductance vs Frequency. (b) Susceptance vs Frequency.
(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
)
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
Chapter 3: PZT-Structure Electro-Mechanical Interaction
(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
)
20% reduction in ‘k’20% increase in ‘m’
20% increase in ‘c’
Pristine State
BA (S
)
Chapter 3: PZT-Structure Electro-Mechanical Interaction
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.
3.8 CONCLUDING REMARKS
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.
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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.
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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)
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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,
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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.
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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.
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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).
Physical Parameter Value
Density (kg/m3) 7800
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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.
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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)
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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)
Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance
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.
4.9 CONCLUDING REMARKS
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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,
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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’
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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.
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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.
Physical Parameter Value
Density (kg/m3) 2715
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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).
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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
)
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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.
Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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,
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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.
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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.
(a) Real part (x) vs frequency. (b) Imaginary part (y) vs frequency.
(b)(a)
Experimental Equivalent system
-350
-300
-250
-200
-150
25 30 35 40
Frequency (kHz)
y (N
s/m
)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
142
22
22*
−+
+=
ωω
ωkmc
cmCx and ω
ωω
ωω
ωω *
22
2
Kkmc
kmkcmy −
−+
−−
= (5.84)
Experimental Equivalent system
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*
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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
)
Experimental Equivalent system
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.
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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.
(a) Real part (x) vs frequency. (b) Imaginary part (y) vs frequency.
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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.
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
147
(b)
Fig. 5.23 Effect of damage on extracted mechanical impedance in 180-200 kHz range.
(a) Real part (x) vs frequency. (b) Imaginary part (y) vs frequency.
(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
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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)
Experimental Equivalent system
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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.
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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)
Experimental Equivalent system
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.
(a) Real part (x) vs frequency. (b) Imaginary part (y) vs frequency.
(b)(a)
-130
-125
-120
-115
-110120 124 128 132 136 140
Frequency (kHz)
y (N
s/m
)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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.
5.13 CONCLUDING REMARKS
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)
Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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)
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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)
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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) Conductance vs frequency. (b) Susceptance vs frequency.
(a)
(b)
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
165
Table 6.1 Averaged parameters of test sample of PZT patches.
Physical Parameter Value
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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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)
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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)
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete
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
% 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
(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
168Chapter 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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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)
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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.
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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).
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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
Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete
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.
6.8 CONCLUDING REMARKS
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
-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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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).
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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)
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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,
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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)
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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)
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
)
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
)
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
(a) Conductance vs frequency. (b) Susceptance vs frequency.
210
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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).
(e) Susceptance vs frequency.
G (S
)
B (S
)
G (S
)
G (S
)G
(S)
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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).
(e) Susceptance vs frequency.
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
)
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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
Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models
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.
Chapter 8: Practical Issues Related to EMI Technique
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
Chapter 8: Practical Issues Related to EMI Technique
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
Chapter 8: Practical Issues Related to EMI Technique
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.
Chapter 8: Practical Issues Related to EMI Technique
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)
Chapter 8: Practical Issues Related to EMI Technique
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
Chapter 8: Practical Issues Related to EMI Technique
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
Chapter 8: Practical Issues Related to EMI Technique
222
multiplexer module ensures much more stable and repeatable signatures due to
more secure connections.
8.5 CONCLUDING REMARKS
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
Chapter 9: Conclusions and Recommendations
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.
Chapter 9: Conclusions and Recommendations
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
Chapter 9: Conclusions and Recommendations
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
Chapter 9: Conclusions and Recommendations
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.
Chapter 9: Conclusions and Recommendations
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.
Chapter 9: Conclusions and Recommendations
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.
Author’s Publications
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.
Author’s Publications
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-
Author’s Publications
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.
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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 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 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 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;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);