Damage detection in plates based on electromechanical impedance measurements with piezoelectric wafer active sensors

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piezoelectric wafer active sensors

electromechanical impedance measurements with

Damage detection in plates based on

Academic year 2019-2020

Master of Science in Electromechanical Engineering

Master's dissertation submitted in order to obtain the academic degree of

Counsellor: Mojtaba Khayatazad

Supervisors: Prof. dr. ir. Wim De Waele, Prof. dr. ir. Mia Loccufier

Student number: 01500022

Wannes Janssens

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piezoelectric wafer active sensors

electromechanical impedance measurements with

Damage detection in plates based on

Academic year 2019-2020

Master of Science in Electromechanical Engineering

Master's dissertation submitted in order to obtain the academic degree of

Counsellor: Mojtaba Khayatazad

Supervisors: Prof. dr. ir. Wim De Waele, Prof. dr. ir. Mia Loccufier

Student number: 01500022

Wannes Janssens

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The author gives permission to make this master’s dissertation available for consultation and to copy parts of this master’s dissertation for personal use. In all cases of other use, the copyright terms have to be respected, in particular with regard of the obligation to state explicitly the source when quoting results from this master’s dissertation.

Ghent, May 2020 Janssens Wannes

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Preface and acknowledgements

This work contributes to the research activities of Soete Laboratory (fatigue & fracture) in mechanical systems. The research is part of the safe life project that aims to asses the lifetime of fatigue loaded structures. This master thesis gave me the opportunity to apply the theoretical knowledge I acquired during my studies electromechanical engineering in practice. Although this work is the result of many long and hard days, I enjoyed being able to contribute to the research of Soete Laboratory.

In particular, I would like to thank everyone who helped directly and indirectly realize this master thesis. My gratitude goes to my supervisors Prof. dr. ir. Wim De Waele, who was always there when I had problems or needed guidance and Prof. dr. ir. Mia Loccufier for the support. I would also like to thank my counsellor Mojtaba Khayatazad, for the time he spent helping me design the test matrix of this thesis, answering my questions and making sure I could continue my research at home despite the covid-19 epidemic. The contribution of Soete Laboratory, the use of their data acquisition equipment and sensors, is also highly appreciated. Last but not least, I would like to thank everyone at home. My parents for their unconditional support during my studies and thesis, and my friends for their motivational speeches.

Ghent, May 2020 Janssens Wannes

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Damage detection in plates based on electromechanical

impedance measurements with piezoelectric wafer active

sensor

Wannes Janssens

Supervisors: Prof. dr. ir. Wim De Waele, Prof. dr. ir. Mia Loccufier Counsellor: Mojtaba Khayatazad

Master’s dissertation submitted in order to obtain the academic degree of Master of Science in Electromechanical Engineering

Department of Electromechanical, Systems and Metal Engineering Chair: Prof. dr. ir. Luc Dupr´e

Faculty of Engineering and Architecture Academic year 2019-2020

Abstract

An experimental set-up is build to characterize the effect of the bonding quality on the electromechanical impedance measurements and investigate the ability of both the RMSD and the frequency shift as damage metric to detect and quantify damage based on raw EMI signatures. A first polycarbonate plate is equipped with four PZT patches to characterize bonding defects and different damage types. A second aluminium beam specimen is used to prove the capability of both RMSD and frequency shift as damage metric. From the bonding experiment it was concluded that consistent bonding is crucial for retrieving comparable electromechanical signatures. An added mass experiment on the polycarbonate plate showed only sensitivity for an initial increase of mass at the center and damage experiments on the polycarbonate plate confirmed that both RMSD and frequency shift are suitable for raw-signature damage assessment. It also showed that the frequency shift has a slight advantage for determining the extend of the damage. From the damage experiments on the aluminium beam, it was clear that the RMSD damage metric is simple to implement and is capable to detect damage in a large area. Although the frequency shift gave good results, it was hard to select the right peaks in the conductance signature of the aluminium beam. More research is needed regarding the use of frequency shift as damage metric in the EMI technique. As last experiment, the magnitude squared coherence function between two PZTs on the aluminium beam is measured and used for damage detection and showed the potential for damage assessment with this alternative transfer impedance approach.

Keywords: Electromechanical impedance (EMI) technique; Piezoelectric wafer active sensor (PWAS); Damage detection; RMSD; Frequency shift

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Damage detection in plates based on

electromechanical impedance measurements with

piezoelectric wafer active sensors

Wannes Janssens

Supervisor(s): Prof. dr. ir. Wim De Waele, Prof. dr. ir. Mia Loccufier Counsellor: Mojtaba Khayatazad

Abstract— The research to the electromechanical impedance (EMI) technique is essential to exploit its potential and become a mature non-destructive evaluation method. The technique requires a piezoelectric transducer (PZT) bonded to the surface of the structure under investi-gation. This PZT employs high frequency vibrations in the range of 30 to 400kHz to derive the electromechanical characteristic signature of the structure that can provide key information about the structural parame-ters of this structure. An experimental set-up is build to characterize the effect of the bonding quality on the measurements and investigate the abil-ity of both the RMSD and the frequency shift as damage metric to detect and quantify damage. From the bonding experiment it was concluded that consistent bonding is crucial for retrieving comparable measurement. The added mass experiment on an polycarbonate plate showed only sensitivity for an initial increase of mass at the center and damage experiments on the polycarbonate plate confirmed that both RMSD and frequency shift are suitable for raw-signature damage assessment, where the frequency shift has a slight advantage for determining the damage extend. From the age experiments on the aluminium beam, it was clear that the RMSD dam-age metric is simple to implement and is capable to detect damdam-age in a large area. Although the frequency shift gave good results, it was hard to select the right peaks in the conductance signature of the aluminium beam. More research is needed regarding the use of frequency shift as damage metric in the EMI technique.

Keywords— Electromechanical impedance (EMI) method; Piezoelectric wafer active sensor (PWAS); Piezoelectric transducer (PZT); Damage de-tection; NI DAQ; RMSD; Frequency shift.

I. INTRODUCTION

The aim of this work is to investigate the potential of electromechanical impedance measurements with piezoelectric transducers for damage detection in plate-like structures. Two different experimental set-ups are developed to investigate dif-ferent effects, including bonding quality and structural damage, on the electromechanical impedance signature of the PZT. In the first set-up, two polycarbonate plates have been instrumented with four PWASs type LF-W50E10BC. A first set of experi-ments is conducted to evaluate the variation in electromechani-cal impedance measurements due to differences in PZT proper-ties and bonding quality. An other set of experiments is done to evaluate the effect of different damage types and damage serveries on the EMI signatures. The second set-up is made from an aluminium beam equipped with three PWASs type LF-W50E10BC in line. The experiments conducted on the alu-minium specimen intent to evaluate the potential of local and transfer electromechanical impedance measurements for dam-age detection. In general, the electromechanical impedance of a PZT is measured with an impedance anlyzer. As an impedance analyzer comes with a high cost and size, an alternative measur-ing system is build to measure the impedance of the PZTs for

the experiments. The multifunctional DAQ device NI 6356 from Texas Instruments forms the base for the used measurement sys-tem. Analizing the measured impedances is the hardest part of the electromechanical impedance method. This work adresses some of the main methods used in literature to analyze changes in impedance signatures.

II. ELECTROMECHANICAL IMPEDANCE TECHNIQUE

The electromechanical impedance technique is a relatively new method for non-destructive evaluation and structural health monitoring [1, 2]. This technique requires the use of a piezo-electric transducer bonded to the surface of the structure. The transducer employs high frequency vibrations in the range of 30 to 400 kHz to derive a characteristic electrical signature of the structure. The measured electrical parameter is the electrome-chanical admittance of the piezoelectric transducer in frequency domain and provides key information on the structural charac-teristics of the host structure [3]. Piezoelectric materials are in-corporated in these piezoelectric transducers or patches. As lead zirconate titanate (PZT) is the most used piezoelectric material, the transducers are often referred to as PZT patches. In this work the piezoelectric wafer active sensor (PWAS) patch type is selected to conduct all the experiments. Piezoelectric mate-rial generates an electric charge in response to an applied me-chanical stress (piezoelectric effect) and conversely a mechan-ical deformation of the material by applying an electrmechan-ical field (converse piezoelectric effect), therefore piezoelectric patches facilitates the electromechanical impedance technique [4]. In the local electromechanical impedance technique, a PZT patch is bonded to the structure under investigation and due to an ap-plied alternating voltage and the converse piezoelectric effect, mechanical deformations are induced in both the patch and in the surrounding area of the host structure. The local response to the imposed deformations or mechanical vibrations is trans-ferred back to the patch and converted to an electrical response as a change in the admittance of the PZT wafer. Damage as-sessment in the EMI technique is done based on changes in the mechanical impedance of the structure, evaluated with elec-tromechanical impedance measurements of the bonded PZT [3, 5]. Liang et al. [6] proposed a model for the PZT-structure elec-tromechanical interaction. In this 1-D model, the PZT patched is condensed to a thin bar undergoing axial vibrations in direction ’1’ caused by the applied electric fieldE3in direction ’3’. The

bonding layer is replaced by a simple connection at the outer

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ends of the bar. The PZT host-structure system is modelled as a mechanical impedance Zmcoupled to the axial vibrating thin

bar, as illustrated in figure 1. The vibrating PZT patch has half-length l, width w and thickness h, and is negligible in terms of mass and stiffness compared to the host structure. Zmis called

the drive point mechanical impedance of the host structure at the end points of the PZT patch. Bhalla et al. [7] stated that the complex admittance ¯Y of the PZT for the 1-D approach is given by equation 1. Where Za is the mechanical impedance of the

PZT patch itself, ¯EEthe complex Young’s modulus of the PZT patch at constant electric field and κ the wave number, related to the angular frequency of excitation ω.

Zm l PZT patch Host structure 1 2 3 Ɛ3 ~ _w h _Z m

Fig. 1: 1-D interaction model of PZT patch and host structure

¯ Y = 2ωjwl h ¯ T 33− d231E¯T + _Z a Zm+ Za d2 31E¯T tan(κl) κl (1) This electromechanical coupling between the mechanical impedance Zm of the host structure and the electrical

admit-tance ¯Y of the PZT is used for damage detection with the EMI technique. The real part of the admittance is conductance G and changes in conductance can be related to the condition of the structure; the imaginary part of the admittance is the sus-ceptance B and changes in the sussus-ceptance is can be related to the bonding condition of the PZT. In this work, damage as-sessment is done with the raw signature approach. Sun et al. showed that the appearance of new peaks in the electromechan-ical signature and lateral and vertelectromechan-ical shift of peaks are promi-nent effects of structural damage [2]. As a result, damage as-sessment can be done quickly and easily without having to cal-culate the mechanical impedance from the electromechanical admittance signature. However, damage assessment based on the raw-signature will be less informative than the mechanical impedance method [3]. Beside the single peak approach, where changes in the location of the peaks are used as damage metric, statistical techniques are employed for damage assessment in the electromechanical impedance technique. A frequently used sta-tistical methods for comparing different EMI signatures is the root mean square deviation or RMSD [8].The RMSD index is defined as RM SD = v u u tPNi=1(G1i − G0i)2 PN i=1G0i (2) Where G1_{is the conductance is damage state and G}0_{is the}

cor-responding pre-damage conductance.

III. MEASUREMENT SYSTEM AND PROCEDURE

As an impedance analyzer is expensive and bulky, an alter-native measurement system to measure the electrical impedance Ze(f )of a PZT patch is build. Figure 2 shows a schematic

rep-resentation of the alternative measurement system used in the experiments. The measurement system is based on the concept introduced by Peairs et al. [9]. This is a low-cost method for actuating the PZT patch and simultaneously measuring its elec-tromechanical impedance. PZT Ze(f) Rs DAC Bus interface y(t) x(t) Excitation Response Bus interface DFT DFT Calculation of Z x[n] y[n] Excitation signal DAQ Device PC / Matlab Auxiliary circuit Ze[f] AO AI2 FRF X[f] Y[f] H[f] ADC AI1 x’(t) x’[n]

Fig. 2: System for PZT electromechanical impedance measure-ment

The measurement system uses a simple inexpensive auxiliary circuit build with a resistor Rs(197.6Ω) in series with the PZT.

The resistor is used to measure the current IP ZT through the

PZT indirectly by measuring the voltage drop over it. Beside the auxiliary ciruit, a NI USB-6356 from National Instruments is selected as the DAQ device for the excitation signal genera-tion and voltage measurements. Where the proposed system of Pears et al. only measured the voltage across the PZT y(t), the measurement system used for the experiments measures both the voltage across the PZT and the effective excitation voltage x0_[n].

Furthermore, for an accurate calculation of the impedance of the PZT, the input impedance of the analog inputs ¯Zinof the DAQ

device and the connecting wires r can’t be ignored. The equiva-lent electrical schematic of the measurement system is shown in figure 3.

The impedance of the PZT ¯Ze is calculated by combining

Ohm’s law and Kirchhoff’s law as follows ¯ Ze= ¯ VP ZT ¯ IP ZT = y(t) x0(t)RsZ¯in 1₋_xy(t)0(t) Rs+ ¯Zin − r (3)

As stated before, the electromechanical impedance technique is based on frequency response function (FRF) of the piezoelec-tric transducer’s impedance. To obtain the impedance in fre-quency domain, the discrete Fourier transforms (DFTs) of the

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Rp Cp Zin _PZT _Z_e x(t) Rp Cp Zin r Rs x’(t) y(t) I1(t) I2(t) I3(t)

Fig. 3: Equivalent electrical scheme of electromechanical impedance measurement system

sampled excitation signal ¯x0_[n]_{and response signal ¯y[n] are}

cal-culated.

The frequency response function or FRF, denoted as ¯H[f ], can be obtained as

H[f ] =DFT (x0[n])

DFT (y[n]) =

Y [f ]

X[f ] (4)

Combining equations 3 and 4 gives the electromechanical impedance of the PZT in the discretized frequency domain as

¯

Ze[f ] =DFT ¯Ze[n]

H[f ]RsZ¯in[f ]

1_{− H[f] R}s+ ¯Zin[f ] − r

(5) There are several measurement techniques for determining the FRF of the impedance. The main difference lies in the signal for exciting the PZT at different frequencies. In the measure-ment system, a linear chirp is used as dynamic excitation signal. This is a sinusoidal signal where the frequency increases lin-early with respect to time. The excitation signal contains all the frequency components of the desired frequency range, allowing the frequency response function of the impedance to be obtained quickly.

IV. EXPERIMENTAL INVESTIGATION

A. Bonding layer characterisation

The quality of the adhesive bonding layer is important as the deformation of the PZT is transferred through this layer to the host structure. The main goal of this set of experiments is to evaluate the impact of the quality of the bonding between the PZT and the host structure on the electromechanical impedance measurements. Two polycarbonate plates (160mm x 160mm x 8mm) are equipped with four PZT’s symmetrically. The poly-carbonate enables visual inspection of the bonding layer without damaging it. The PZTs on the first specimen (PLX 1) are glued according to the general installation procedure used in this work, where the PZTs on the second specimen (PLX 2) are glued with an insufficient amount of adhesive, resulting in a partial bonding of these PZTs. The conductance and susceptance signatures of the PZTs on both polycarbonate specimens are shown in figure 4. As expected, the four conductance signatures of PLX 1 are very similar. This is not the case for the signatures of the PZTs on PLX 2 due to the bad bonding. Park et al. [10] stated that bonding defects (quality) have a major influence on the mag-nitude of the susceptance signature, what can be seen for the

measurements of specimen with bad bonded PZTs, although re-markable large variations are in susceptance are also observed for the good bonded PZTs.

0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 50 100 150 0 0.01 0.02 0.03 0.04 0.05 50 100 150 0 0.01 0.02 0.03 0.04 0.05 PZT1 PZT2 PZT3 PZT4 PZT5 PZT6 PZT7 PZT8

Fig. 4: Conductance and susceptance plot of PZTs on polycar-bonate plate

B. Polycarbonate plate with added mass and damages B.1 Adding mass

A set of experiments is conducted to investigates the sensi-tivity of the EMI technique to changes in mass of the structure. This is done by adding mass to the center of the polycarbonate plate (PLX 1). The electromechanical impedance of PZT 1 on the specimen is measured for different added masses. The added mass is increased from 10.0g to 96.5g in several steps. Dif-ferences in conductance signature of the PZT due to the added mass are evaluated in terms change of the RMSD value for sub-frequency intervals. This is done for the zero added mass sig-nature as reference and by using the previous stage as reference (relative RMSD) as shown in figure 5.

The RMSD values are very low compared to those in the literature. This can be explained by the fact that the experi-ments have been conducted on polycarbonate, which has a much higher material damping than aluminium or steel that is pub-lished about. Looking at the relative RMSD, higher sensitivity is observed for the initial increases of mass. Beside the RMSD as damage metric, also the frequency shift of the peaks in the conductance signature can be used for damage assessment. The frequency shift for the four peaks is given in figure 6.

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0.000 0.010 0.020 0.030 0.040 [49kHz - 83kHz] [83kHz - 105kHz] [105kHz - 130kHz] [130kHz - 160kHz] 0.000 0.005 0.010 0.015 0.020 10g 46.5g 56.5g 76.5g 86.5g 96.5g

Fig. 5: RMSD and relative RMSD plot for added mass at center of polycarbonate plate [49kHz - 83kHz] [83kHz - 105kHz] [105kHz - 130kHz] [130kHz - 160kHz] 0 50 100 150 200 _10g 46.5g 56.5g 76.5g 86.5g 96.5g

Fig. 6: Frequency shift plot for added mass at center of polycar-bonate plate

Clearly, the frequency shift is higher for higher frequencies, what implies that the frequency shift approach is more sensitive to the added mass at higher frequencies than RMSD.

B.2 Central hole

This research focuses on damage assessment using the EMI technique. A first damage is introduced to the polycarbonate plate (PLX 1) as a drilled hole of 1.5mm in the middle. The damage is increased in several steps by increasing the hole’s di-ameter to 3mm. Again very small changes in conductance are observed, this results in small RMSD and frequecy shift values as shown in figure 7. 0 0.005 0.01 0.015 0.02 0.025 _{hole 1.5mm} hole 2.0mm hole 2.5mm hole 3.0mm [49kHz - 83kHz] [83kHz - 105kHz] [105kHz - 130kHz] [130kHz - 160kHz] 0 20 40 60 80 100

Fig. 7: Frequency shift plot for added mass at center of polycar-bonate plate

B.3 Holes in middle and at the side

Next, two rows of 3mm holes are drilled in the middle and at the side of the plate to simulate a crack. The first hole of 3mm is already present and is the new reference state for the baseline measurement. In figure 8 the different damage stages are numbered.

PZT1 PZT2

PZT3 PZT4

5 3 1 2 4 9 8 7 6

Fig. 8: Different damage stages by drilling holes in the polycar-bonate plate

First holes 2, 3, 4 and 5 are drilled. The electromechanical impedance of each damage stage is measured with PZT 1 and the conductance signatures is plotted for a part of the frequency range in figure 9.

Clearly, the peak is shifted to the left. The RMSD and the frequency shift with respect to the baseline (1 central hole of 3mm) are calculated and shown in figure 10.

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94 94.5 95 95.5 96 6 7 8 9 10 11 10-3 Baseline 2 holes 3 holes 4 holes 5 holes

Fig. 9: Part of conductance signatures of polycarbonate plate for holes 1-5 0 0.01 0.02 0.03 0.04 2 holes 3 holes 4 holes 5 holes [49kHz - 83kHz] [83kHz - 105kHz] [105kHz - 130kHz] [130kHz - 160kHz] -200 -150 -100 -50 0

Fig. 10: RMSD and frequency shift plot for row of holes at the center polycarbonate plate

Looking at the sub-frequency ranges (figure 10), the RMSD value and the absolute value of the frequency shift increases as the damage accumulates. On average, higher RMSD values are observed at lower frequency ranges, where the absolute quency shift is higher at higher frequencies. The negative fre-quency shift can be expected as a positive shift is observed for adding mass to the plate and drilling a hole reduces the mass. This reasoning does not take changes in damping into account, yet it seems to hold up.

Furthermore, the damage to the polycarbonate plate is ex-tended (holes 6-9). Again the RMSD and frequency shift are calculated from the conductance signatures for sub-frequency ranges. The frequency shift is not stable, meaning it fluctuates without any trend and low RMSD values are observed. This could indicate that the damage is outside the effective sensing area of PZT 1, but more experiments related to the sensing area are needed to prove this.

C. Aluminium beam with damage introduced

The ability of damage detection on an aluminium beam is investigated with both the point and transfer electromechanical impedance technique.

Fig. 11: Aluminium beam for damage experiments

The experimental setup with the aluminium beam is shown in figure 11. The beam has a length, width and thickness of re-spectively 350mm, 70mm and 8mm, and is instrumented with three PZTs in a line. As for all experiments, LF-W50E10B-C type PWASs are used for EMI measurements. The damage is introduced to the aluminium beam as a saw cut, representing a crack-like damage. The cut is made with a hacksaw at 115mm from the short side, right in-between PZT 1 and PZT2. Dam-age is increased gradually by increasing the cutting length. The first cut is 1.9mm, where after it is increased to 3.2mm, 5.3mm, 7.9mm and the last measurement is done for a cutting length of 12.4mm.

Fig. 12: Conductance signatures for different damage stages (PZT 1 - AL 1)

In figure 12 some peaks of the conductance signature of PZT 1 on the aluminium beam (AL 1) are plotted. It can be observed that the peaks shift to the right with increasing the damage or cutting length. The RMSD is calculated for ten sub-frequency intervals and the frequency shift for five well selected peaks

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[30kHz - 43kHz][43kHz - 56kHz][56kHz - 69kHz][69kHz - 82kHz][82kHz - 95kHz]_{[95kHz - 108kHz]}_{[108kHz - 121kHz]}_{[121kHz - 134kHz]}_{[134kHz - 147kHz]}_{[147kHz - 160kHz]} 0.0 0.5 1.0 1.5 D1 D2 D3 D4 D5

Fig. 13: RMSD plot for different saw cut lengths in aluminium beam (PZT 1 - AL 1)

The highest sensitivity of the RMSD to this type of damage is at lower frequencies and the RMSD increases with increasing damage. Also the frequency shift for some peaks is plotted in figure 14.

peak 37101Hz peak 55690Hz peak 72738Hz peak 111413Hz peak 149230Hz

-150 -100 -50 0 50 D1 D2 D3 D4 D5

Fig. 14: Frequency shift plot for different saw cut lengths in aluminium beam (PZT 1 - AL 1)

This plot shows that the selected peaks clearly shift as the damage increases. The frequency shift can both serve as damage metric for damage detection and as a first indicator of the extent of the damage (for an aluminum beam).

V. CONCLUSIONS

The effect of a consistent bonding of the PZT to the impedance signatures and the feasibility of both the RMSD and frequency shift damage metric are demonstrated. The RMSD and frequency shift were able to detect different pattern of holes in a polycarbonate plate and saw cuts in an aluminium beam. RMSD is already a widley used damage metric, although the frequency shift results were quite promising, more research is needed to determine the right peaks and interpret the changes.

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List of Figures

1.1 Conductance signature of an aluminium beam in healthy and damaged

condi-tion . . . 1

2.1 Piezoelectric sensor/actuator . . . 5

2.2 Non-centro-symmetrical crystal . . . 5

2.3 Centro-symmetrical crystal . . . 6

2.4 PWASLF-W50E10B-C . . . 7

2.5 Determination of mechanical impedance of a PZT patch . . . 10

2.6 Simplified model for PZT-structure interaction . . . 11

2.7 An infinitesimal element of PZT patch under dynamic equilibrium . . . 12

2.8 2D modelling of the mechanical impedance coupling for the PZT-structure interaction . . . 15

2.9 Conductance and susceptance plots of a PWAS patch bonded to aluminium beam . . . 17

2.10 Drive point and transfer electromechanical impedance . . . 19

2.11 Auxiliary circuit for electromechanical impedance measurements . . . 20

2.12 Linear chirp signal (20 to 200kHz) in time and frequency domain . . . 21

2.13 Complete system for PZT electromechanical impedance measurement . . . . 22

2.14 System for the calibration of the excitation signal . . . 23

2.15 Complete system for PZT electromechanical impedance measurement with cal-ibrated excitation signal . . . 24

2.16 Complete system for PZT electromechanical impedance measurement with con-tinuous excitation signal measurement . . . 24

2.17 Alternative measurement system for the transfer impedance based on the co-herence function . . . 25

3.1 NI USB-6356 from National Instruments . . . 28

3.2 Measurements systems auxiliary circuit with DAQ connections . . . 28

3.3 Equivalent electrical scheme of electromechanical impedance measurement sys-tem . . . 29

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LIST OF FIGURES xiii 3.4 Connections of the NI USB-6356 for the PZT impedance measurement system 30 3.5 Comparison of theoretical impedance, resistance and reactance of RC circuit

with measured values recorded with the measurement system. . . 31

3.6 Setup for impedance measurements of PZT in free-free condition . . . 34

3.7 Deformation bonding layer and PZT patch . . . 35

3.8 Polycarbonate plate for experiments . . . 36

3.9 Setup for free support condition . . . 37

3.10 Gluing of PZT’s on PLX1 . . . 37

3.11 Gluing of PZT’s on PLX2 . . . 38

3.12 Gluing mask of PZT’s on PLX2 . . . 39

3.13 Setup for clamped plate . . . 40

3.14 Setup for four point support . . . 40

3.15 Adding mass to center of the plate . . . 41

3.16 Damage stages by drilling holes . . . 42

3.17 Aluminium beam for experiments . . . 43

3.18 Experimental setup for aluminium beam with installed PZTs . . . 43

3.19 Damage stages aluminium beam . . . 44

4.1 Admittance signatures for five free PWASs . . . 45

4.2 Zoom of admittance signatures for five free PZTs . . . 46

4.3 Conductance and susceptance of undamaged polycarbonate plates (PLX 1 -PLX 2) . . . 47

4.4 Conductance signatures for different boundary conditions (PZT 1 - PLX 1) . 48 4.5 RMSD and frequency shift for different boundary conditions (PZT 1 - PLX 1) 49 4.6 Conductance signatures for different added masses (PZT 1 - PLX 1) . . . 50

4.7 Zoomed conductance signatures for different added masses (PZT 1 - PLX 1) 50 4.8 RMSD and Frequency Shift plot for added mass experiment . . . 51

4.9 Relative RMSD plot for added masses (PZT 1 - PLX 1) . . . 52

4.10 Conductance signatures for different holes in middle of plate (PZT 1 - PLX 1) 53 4.11 RMSD and frequency shift plot for central hole (PZT 1 - PLX 1) . . . 54

4.12 Conductance signatures for row of holes in middle of plate (PZT 1 - PLX 1) 55 4.13 RMSD and frequency shift plot for row of holes at the center (PZT 1 - PLX 1) 56 4.14 Conductance signatures for row of holes at one side of plate (PZT 1 - PLX 1) 57 4.15 RMSD and frequency shift plot for row of holes at one side (PZT 1 - PLX 1) 58 4.16 Relative RMSD plot for row of holes at one side (PZT 1 - PLX 1) . . . 58

4.17 Conductance signatures of PZT 1 on aluminium beam for different cutting lengths . . . 59

4.18 Conductance signatures of PZT 3 on aluminium beam for different cutting lengths . . . 60

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LIST OF FIGURES xiv 4.19 Zoom conductance signatures of PZT 1 on aluminium beam for different cutting

lengths . . . 60

4.20 Zoom conductance signatures of PZT 3 on aluminium beam for different cutting lengths . . . 61

4.21 RMSD for different cutting lengths (PZT 1 - AL 1) . . . 61

4.22 Frequency shift for different cutting lengths (PZT 1 - AL 1) . . . 62

4.23 RMSD for different cutting lengths (PZT 3 - AL 1) . . . 62

4.24 Frequency shift for different cutting lengths (PZT 3 - AL 1) . . . 63

4.25 Baseline coherence functions for the aluminium beam . . . 64

4.26 Coherence function for different damage stages (PZT 2-1 - AL 1) . . . 65

4.27 Coherence functions for different damage stages (PZT 2-3 - AL 1) . . . 65

4.28 Standard deviation based damage metric plot for damaged aluminium beam (PZT 2 actuator - AL 1) . . . 66

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Nomenclature

Abbreviations

ADC Analog-to-digital converter AI Analog input

AL Aluminium beam specimen AO Analog output

AWG American wire gauge CC Correlation coefficient DAC Digital-to-analog converter DAQ Data acquisition

DFT Discrete Fourier transform DIO Digital input/output

EMI Electromechanical impedance FRF Frequency response function MAPD Mean absolute percent deviation MFC Micro-fiber composite (transducer) MSC Mean square coherence

NDE Non-destructive evaluation NDT Non-destructive testing PLX Polycarbonate plate specimen PWAS Piezoelectric wafer active sensor PZT Lead (Pb) zirconate titanate RC Resistor-capacitor

RMSD Root mean square deviation SDOF Single degree of freedom (system) SHM Structural health monitoring USB Universal serial bus

Symbols

ij Dielectric constant F/m

γ Dielectric loss factor −

κ Wave number ₋

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NOMENCLATURE xvi

E Electric field V /m

Ei Electric field component V /m

µ Electric dipole C· m

ν Mechanical loss factor ₋

ν Poisson’s ratio ₋

ω Angular frequency rad/s

ΦxxΦyy Auto power spectrum −

Φxy Crossed power spectrum −

ρ Material density kg/m3 σ Standard deviation ₋ a Radius of disk m B Susceptance S C Capacitor F c Damping coefficient N s/m C1, C2 Constants −

D Electric displacement tensor c/m2

Di Electric displacement component C/m2

dim, djk Piezoelectric constant m/V or C/N E Young’s modulus kg/(m· s) err Error ₋ F Force N f Frequency Hz fs Sample frequency Hz G Conductance S

H Frequency response function ₋

h Height m

I Electric current A

k Spring constant N/m

l Length m

M Mass kg

M SC Magnitude squared coherence ₋

p0 Perimeter of PZT patch in undeformed condition m

R Resistor ω

S Strain tensor −

Si Strain component −

skm Elastic compliance constant m2/N

T Stress tensor N/m2

t time s

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NOMENCLATURE xvii

u Displacement m

V Electric potential difference V

w Width m

X, Y DFT of signals ₋

x, y Voltage signals V

Y Admittance S

Z Electrical impedance Ω

Za Mechanical impedance of PZT patch N s/m

Zi Mechanical impedance component N s/m

Zm Mechanical impedance N s/m

Superscripts

0 Pre-damage state 1 Post-damage state E At constant electric field c converse piezoelectric effect D At constant displacement d direct piezoelectric effect S At constant strain T At constant stress Subscripts 0 Amplitude of quantity 1, 2, 3 or x, y, z Coordinate axes a PZT patch avg Average c Ideal damper e Electrical ef f Effective eq Equivalent F Final I Initial i, j, k Direction

in Input of DAQ device

k Ideal spring

M Ideal mass

m Mechanical

nyq Nyquist

p For parallel connection point Drive point approach

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NOMENCLATURE xviii s For series connection

trans Transfer approach Others

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Chapter 1

Introduction

The development of smart materials such as lead zirconate titanate ensured a revolution in the field of structural health monitoring (SHM) depending on non-destructive testing (NDT). These smart materials form the foundation for so called piezoelectric transducers or PZT’s. In literature a lot of promising research has already been done on health monitoring with PZT’s. Today, smart piezoelectric sensors are used for monitoring the integrity of rock bolts. The rock bolts health’s monitoring covers the critical status, such as the occurrence of corrosion, the axial force and resin quality and delamination [1]. In 2001, some researchers investigated the potential of using piezoelectric transducers to monitor the integrity of a pipeline system after an earthquake. This research demonstrated the capabilities of a mechanical impedance based technique to detect imminent damage in advance of actual failure [2]. These research examples are just a few of the many applications of piezoelectric transducers. In the last decades, the electromechanical impedance method or EMI has emerged as an novel non-destructive testing (NDT) method based on the mechanical impedance of a structure and is capable of diagnosing almost all types of materials and structures, such as aircraft components [3], steel truss bridges [4] and reinforced concrete structures [5].

90 91 92 93 94 95 96 97 98 99 100 0 1 2 3 4 5 6 7 10 -3 Healthy Damaged

Figure 1.1: Conductance signature of an aluminium beam in healthy and damaged condition

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CHAPTER 1. INTRODUCTION 2 The EMI method uses the electromechanical impedance of a to a structure bonded piezoelec-tric transducer to evaluate changes in the mechanical impedance of a structure. It is proven that the conductance of the transducer is related to the mechanical impedance of the struc-ture. In figure 1.1, a part of the measured conductance of a bonded PZT to an aluminium beam (300mm x 70mm x 8mm) is given in both a healthy and damaged condition. Although the many promising publications, the electromechanical impedance method is still in it early stage to become fully commercialized as structural health monitoring method.

1.1 Structural health monitoring

For structures under service, it is very relevant to monitor the load and/or the occurrence of damages. The load spectrum and its reciprocal deflections or strains can corroborate key design decisions, whereas damage monitoring is essential to ensure safety. Structural health monitoring (SHM) is defined as the continuously measuring of parameters and characteristics of a structure in order to estimate the integrity of these structures [6]. It is proven that a well-designed SHM system minimizes the overall maintenance cost and risk of premature failure of a structure. Structural health monitoring is a promising technique that can contribute to a shift in maintenance philosophy. SHM allows shifting from the traditional time-based maintenance to condition-based maintenance, where a sensing system will monitor the structure and notify when damage is detected. High accuracy and reliability are crucial for the monitoring system to ensure the life-safety and economic benefits associated with condition based maintenance. Besides these beneficial tendencies, it is worth looking to the extra hardware investment and the required sophisticated data analysis procedure [7].

The process of structural health monitoring is typically split into four major steps. The oper-ational evaluation defines the damage and how it can be monitored. It is recommended that this damage has unique features that can be measured unambiguously because it has a direct impact on the reliability of the SHM system. In order to give a little more clarity, NO WE! we’re getting a little ahead of the fact. To give an example, as this work investigates the po-tential of local electromechanical impedance measurements for structural health monitoring, it is proven that the temperature during the measurement plays a significant role [8]. Be-cause this external parameter, the structural feature cannot indicate damage unambiguously, an extensive study on all influences, in order to be able to design a reliable and accurate SHM system, is done in upcoming chapters.

The second step of the SHM process is the data acquisition and cleansing step. With a vision on how and which structural features should be monitored, which are defined in the first step, the hardware is selected. More specifically, the number, type and location of the sensors and the DAQ device(s) are determined. Often economic considerations will be decisive for the choice of hardware. In addition, it will also have to be decided how often measurements

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CHAPTER 1. INTRODUCTION 3 should be carried out and how quickly processing needs to be performed. Data cleansing can be defined as the process of selecting the right part of raw data for the further processing. The data acquisition and cleansing step is flexible and can be improved with information of later steps in the SHM process.

Most of the publications in the field of structural health monitoring are about the third step in the SHM process, feature selection. The identification of features in the extracted data can make it possible to distinguish damaged and undamaged structures from each other, which is the aim of a well designed SHM system. The identification of features for dam-age detection is often done with experiments in different damdam-age stdam-ages. Today, numerical simulations are gaining popularity for finding new identifying features for damage detection. The PhD supported by this thesis also investigates the added value of numerical simulations to feature selection for the electromechanical impedance based structural health monitoring systems. Identifying damage-sensitive features can also be done by fitting physical-based or non-physical-based models of the structures response to the measured data [9].

In the fourth and last step, statistical models are developed to enhance damage detection. These models can be used to answer the key questions in structural health monitoring: is the structure damaged, what is the severity of the damage and what is its location,...? The answers on these questions help determine the remaining useful life of the structure [10]. The ultimate goal of the research into the electromechanical impedance technique is to develop a reliable SHM system. This work only focuses on a small part of the EMI technique as the potential of electromechanical impedance measurements with piezoelectric transducers for damage detection. The feasibility of electromechanical impedance measurements with PZT’s bonded to a structure has already been demonstrated in previous master dissertations [11], this for different types of PZT transduceres bonded to both aluminium and steel.

The first part of the experiments conducted in this work aims to gain more insight into the characterisation of the PZTs and their bonding, in order to improve the numerical simulations of the EMI method. The second section encloses damage detection using both the local and transfer electromechanical impedance, and comparing the strengths and shortcomings of the RMSD and frequency shift approach.

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Chapter 2

Electromechanical impedance

technique based on piezoelectric

transducers

The electromechanical impedance (EMI) technique is a relatively new method for non-destructive evaluation (NDE) and structural health monitoring [2][4]. This technique requires the use a piezoelectric transducer bonded to the surface of the structure under investigation. The transducer employs high frequency vibrations in the range of 30 to 400kHz to derive a char-acteristic electrical signature of the structure. The measured electrical parameter is the elec-tromechanical admittance of the piezoelectric transducer in frequency domain and provides key information on the structural characteristics of the host structure [12].

2.1 Piezoelectric transducers

Since the discovery of piezoelectricity in 1880, the applications for these smart materials have increased exponentially. Lead zirconate titanate Pb(ZrxTi1−x)O3 is world’s most used

piezoelectric ceramic material. Ceramic materials are widely used in industry, from daily life applications as in diesel injectors and inkjet printers to new emerging technologies and tech-niques as for smart vibration energy harvesting techtech-niques [13] and piezoelectric transducers for the electromechanical impedance technique [14].

2.1.1 Piezoelectric effect

Smart or responsive materials are designed to have physical properties that can be controlled by external stimulation. Some examples of external stimuli of these smart materials are moisture level, stress, temperature or electrical and magnetic field. Piezoelectric materials are classified as smart materials since they are able to generate an electric charge in

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 5 sponse to applied mechanical stress. The duality of this characteristic makes piezoelectric material unique, meaning that the material beside the piezoelectric effect also exhibits the converse piezoelectric effect. This converse piezoelectric effect implies a mechanical deforma-tion (strain) of the material that is caused by an applied electrical field. The electrical field is coupled with the electrical displacement through its permittivity and the strain is linked by stress through it compliance (inverse of elasticity). This duality is shown in figure 2.1. Divers materials own piezoelectric properties, such as quartz, berlinite, gallium orthophos-phate, zinc oxide, aluminium nitrat and lead zirconate titanate (PZT). The piezoelectric effect can be used as sensing principle, where mechanical stress can be measured, and as actuating principle, where a mechanical displacement can be induced by an electrical field [15].

External Voltage Tout Tin Vin Vout External Force

Figure 2.1: Piezoelectric sensor/actuator

The aim of the development of piezoelectric materials is to amplify a small mechanical de-flection into a useful electrical signal. The microscopic working principle of a piezoelectric material is illustrated with the crystal unit cell shown in figure 2.2 (left). In the initial state, without an external mechanical load, both the center of the positive and the center of the negative charges of the crystal unit cell coincide. By applying mechanical stress to the crystal, the structure deforms and causes separation of the positive and negative centers and generates a dipole µ as shown in figure 2.2 (right). The group of crystals that exhibits this phenomenon are called the non-centro symmetric crystals.

+ -+ -+ - μ=0 + -+ -+ - μ≠0

Figure 2.2: Non-centro-symmetrical crystal

However, not all crystals possess this quality. Most crystals are centro-symmetrical, where a deformation does not induce any dipole moment, as shown in figure 2.3

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 6 + -+ - μ=0 + -+ - μ=0

Figure 2.3: Centro-symmetrical crystal

The piezoelectric effect is frequently used for converting mechanical into electrical energy. This transformation is based on the direct relation between the electric field generated by the polarisation of the crystal and the mechanical deformation of the material [16].

The piezoelectric effect can also be described as a tensor relation between the mechanical variables, stress T and strain S, and the electric variables, electric field_{E and electric} displace-ment D. Supposing linear piezoelectric material behaviour, the following tensor expression is obtained:

Di = TijEj+ ddimTm (2.1)

Sk= dcjkEj+ sEkmTm (2.2)

Equation 2.1 represents the direct effect. Piezoelectric sensor applications are based on this direct piezoelectric effect, whereas actuator applications are based on the converse piezoelec-tric effect described in equation 2.2. A general tensor form of equations 2.1 and 2.2 is given by: " D S # = " T _dd dc _sE # · " E T # (2.3)

In equation 2.3 D represents the electric displacement vector (3x1), S the strain tensor (3x3), E the applied electric field vector (3x1) and T the stress tensor (3x3). T _{represents the}

dielelectric permittivity tensor under constant stress, dd _{the piezoelectric strain coefficient}

tensors of the direct piezoelectric effect, dc _{the piezoelectric strain coefficient tensors of the}

converse piezoelectric effect and sE _{the elastic compliance tensor under constant electric field}

[17].

2.1.2 Applications of piezoelectric material

Piezoelectric materials are used in many applications and are fundamental for numerous sen-sors such as pressure transducers, force sensen-sors and accelerometers. This material is shown

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 7 to be promising in active vibration control of lab-sized structures and machines. Current re-search is also exploring applications in the bio-medical field, the development of piezoelectric immunosensors for the detection of antibiotics [18] or a monitoring capsule for monitoring the biological response to an implantation in soft tissue [19]. The most relevant application of piezoelectric materials with respect to structural health monitoring are the piezoelectric transducers used for the electromechanical impedance technique. Many different piezoelec-tric transducers are commercially available, from lightweight, inexpensive piezoelecpiezoelec-tric wafer active sensors (PWAS) to complex and expensive macro-fiber composite transducers (MFC). In this work, the potential of piezoelectric wafer active sensors for the electromechanical impedance method is investigated.

2.1.3 Piezoelectric wafer active sensors

Smart materials are incorporated in the so-called piezoelectric transducers or patches. This patches facilitates the electromechanical impedance technique. As lead zirconate titanate (PZT) is the most used piezoelectric material, the transducers are often referred to as PZT patches. This section is limited to piezoelectric wafer active sensors as only this type is used in the experiments. These PWASs are small buzzers, composed of two circular electrodes with PZT material in between. The PWAS type used for further experiments is named LF-W50E10B-C and is shown in figure 2.4.

Figure 2.4: PWASLF-W50E10B-C

The LF-W50E10B-C has an installation diameter of 50mm and an active PZT diameter of 30mm. Its rated impedance is 800Ω and the capacity has a value of 50000pF_{±30%. The} specific operating temperature range is between -20◦C to 60◦C which is important for inves-tigating the effect of the temperature and finally the maximal operating voltage has a value of 30V peak-to-peak.

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 8 For the circular-shaped PWAS, the piezoelectric constitutive equations 2.1 and 2.2 can be written in cylindrical coordinates [20, 21]

Srr = sE11Trr+ s12E Tθθ+ dc31Ez (2.4)

Sθθ = sE12Trr+ s11E Tθθ+ dc31Ez (2.5)

DZ = dd31(Trr+ Tθθ) + T33Ez (2.6)

It can be noted that for axisymmetric motion, the mechanical strains can be written as

Srr = ∂ur ∂r (2.7) Sθθ = ur r (2.8)

2.2 The electromechanical impedance technique

The electromechanical impedance is defined as the local mechanical impedance of a structure, measured with a piezoelectric transducer. This local mechanical impedance of a structure is affected by damage and has the potential to be a damage feature in a structural health mon-itoring system (see 1.1). Beside the potential as damage feature, the mechanical impedance also forms the physical concept behind the electromechanical impedance technique.

The local electromechanical impedance is measured by means of a PZT patch. The PZT patch is bonded to the structure under investigation and due to an applied alternating voltage and the converse piezoelectric effect, mechanical deformations are induced in both the patch and in the surrounding area of the host structure. The local response to the imposed deformations or mechanical vibrations is transferred back to the PZT patch and converted to an electrical response as a change in the admittance of the PZT patch. Local structural characteristics of the host structure are reflected in the admittance signature of the PZT [12, 22].

Mechanical impedance and the interaction between PZT patch and structure forms the back-bone of the electromechanical impedance. Both concepts are extensively described in the following sections.

2.2.1 Mechanical impedance

The mechanical impedance of a structure is a quantitative measure of the ability of that structure to resist a harmonic force. The mechanical impedance can be defined as the complex ratio of the exciting force to the resulting sinusoidal velocity of the same frequency. When both are measured at the same point, the ratio is denoted the driving point impedance. The ratio is designated as the transfer impedance if the force and velocity are measured each

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 9 in a different point [23]. In this work, mechanical impedance will mean the driving point mechanical impedance unless specifically stated otherwise. The mechanical impedance (Zm)

can be expressed as

Zm=

F

˙u (2.9)

The harmonic force can be represented by a rotating phasor on the complex plane. This phasor is rotating anti-clockwise at an angular frequency ω, which is the angular frequency of the harmonic force. Further is the magnitude of the phasor, the peak amplitude F0 of the

harmonic force. The mechanical impedance can be expressed using phasor notation as

Zm = F ˙u = F0· ejωt ˙u0· ej(ωt−φ) = F0 ˙u0 ejφ (2.10)

With this definition, the mechanical impedance of a pure mass m, spring with spring constant k and damper with damping constant c can be derived.

Mass Newton’s second law of motion states that the acceleration ¨u of an ideal mass m is proportional to the applied force F . Imposing a harmonic force onto the mass, the mechanical impedance of an ideal mass is defined using equation 2.10:

ZM =

F0· ejω

F0· ejω/jM ω

= jM ω (2.11)

Spring Combining Hooke’s law with equation 2.10, the mechanical impedance of an ideal spring can be derived as:

Zk=

F0· ejω

jω· F0· ejω/k

= −jk

ω (2.12)

Damper The mechanical impedance of an ideal damper is calculated using the proportional relation between the damping force and the speed. Using this relation and equation 2.10, the mechanical impedance of a damper is:

Zc=

F0· ejω

jF0· ejω/c

= c (2.13)

A mechanical system is regularly modelled as a combination of these three basic components. Consider a parallel combination of ’n’ components, the equivalent mechanical impedance is calculated as followed: Zeq,p= n X i=1 Zi (2.14)

Analogously, the following expression can be obtained for a series combination of ’n’ compo-nents: 1 Zeq,s = n X i=1 1 Zi (2.15)

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 10 Highly simplified, any structure can be built up with a discrete number of these basic me-chanical components [5, 24].

Piezoelectric transducer

As the mechanical impedance is evaluated with a to the host bonded PZT patch, the measured impedance will be affected by the piezoelectric transducer itself. In addition to the structure impedance Zm, the PZT patch also has a non-negligible mechanical impedance Za. The

mechanical impedance of the PZT is determined in short circuited condition shown in figure 2.5. With l, w and h respectively the length, width and height of the PZT and ¯EE represents the complex Young’s modulus of the PZT patch under constant electric field.

PZT patch 1 2 3 F w l h short circuit

Figure 2.5: Determination of mechanical impedance of a PZT patch

Hence, the piezoelectric effect is eliminated and a pure mechanical response is invoked. With the force F applied to the PZT patch, the short-circuited mechanical impedance Zabecomes

Za= F_|x=l ˙u_|x=l = whT1|x=l ˙u_|x=l = wh ¯E E_S 1|x=l jωu_|x=l (2.16) For a circular patch as the PWAS, the short-circuited impedance is given by

Za= F_|r=a ˙u|r=a = 2πaTrr ˙u|r=a (2.17) with a half of the patch diameter.

2.2.2 PZT-structure interaction

For an electromechanical impedance measurement, a PZT patch is used for the actuation of the the structure and for sensing the response to the implied ultrasonic vibrations. Over the years, different interaction models have been developed to characterise this complex duality. Reviewing existing interaction models is therefore vital for gaining in depth knowledge about the electromechanical impedance [5, 25].

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 11 Single degree of freedom model

In a first attempt by Liang et al. [25] to model the impedance, the patch is condensed to a thin bar undergoing axial vibrations caused by the applied electric field. The bonding layer is replaced by a simple connection at the outer ends of the bar, shown in figure 2.6a. Furthermore, the PZT host-structure system in figure 2.6a can be modelled as a mechanical impedance coupled to the axial vibrating thin bar, as illustrated in figure 2.6b. The PZT patch vibrates in direction 1 as result of the uniform alternating electric fieldE3 applied in

direction 3:

∂1E3 = ∂2E3= 0 (2.18)

The patch has a half-length l, width w and thickness h. The host structure is assumed as one dimensional element with sectional properties lumped along the neutral axis 1. Consequently, vibrations of the PZT in direction 2 can be ignored. By assuming that the involved frequencies are lower than the first resonant frequency for thickness, the PZT loading in direction 3 is ignored. The vibrating PZT patch is negligible in terms of mass and stiffness to the host structure. Therefore, the structure has uniform dynamic stiffness over the bonded area and the two end points of the patch encounter equal mechanical impedance Zm, from the

host structure as shown in figure 2.6b. Assuming this symmetry, the PZT patch has no displacement at the mid-point (x = 0) [26].

Host structure PZT patch

1 2

3

(a) Thin bar model of PZT patch

Zm l PZT patch Host structure 1 2 3 Ɛ3 ~ _w h _Z m

(b) Interaction model of PZT patch and host structure

Figure 2.6: Simplified model for PZT-structure interaction

Implementing these assumptions in the constitutive relations given by 2.1 and 2.2, the fol-lowing expressions can be derived

D3= ¯T33E3+ d3T1 (2.19)

S1 =

T1

¯

EE + d31E3 (2.20)

where D3 is the electric displacement over the PZT patch in direction 3, S1 the strain in

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 12 represents the complex Young’s modulus of the PZT patch under constant electric field and ¯

T

33 the complex permittivity in direction 3 of the PZT material under constant stress. Both

complex properties can be rewritten as ¯

EE = EE(1 + ηj) (2.21)

¯ T

33= T33(1 + δj) (2.22)

Here, η and δ denote respectively the mechanical and dielectric loss factor of the piezoelectric material. The electromechanical impedance of the thin bar representation of the PZT patch can be analytically derived. A small element with length dx, situated at a distance x from the center of the patch is considered for the derivation, shown in figure 2.7.

1 (x) x l dx h Center of patch T1 T₁+-δx∂T1 ∂x

Figure 2.7: An infinitesimal element of PZT patch under dynamic equilibrium

The infinitesimal element under dynamic equilibrium has a mass dM and can be calculated as

dM = ρwhdx (2.23)

In equation 2.23, ρ denotes the density of the PZT material, w the width and h the thickness of the patch as shown in figure 2.6a. Take u(x) as the displacement at any point in the PZT. Applying the D’Alembert’s principle on the infinitesimal element yields

T1+ ∂T1 ∂x δx − T1 wh = dM∂ 2_u ∂t2 (2.24)

By combining equations 2.23 and 2.24 the following expression is obtained ∂T1

∂x = ρ ∂2_u

∂t2 (2.25)

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 13

T1 = ¯EES1= ¯EE

∂u

∂x (2.26)

Substituting equation 2.26 into 2.25 yields ¯ EE∂ 2_u ∂x = ρ ∂2_u ∂t2 (2.27)

The one-dimensional wave equation can be identified in this partial differential equation and solved using the method of separation of variables. The solution yields

u = (C1sin(κx) + C2cos(κx)) ejωt (2.28)

Where κ, the wave number, related to the angular frequency of excitation ω, the density ρ and the complex Young’s modulus of elasticity is defined as

κ = ω r _ρ

¯

EE (2.29)

Applying the boundary condition of no displacement (u = 0) at the mid-point (x = 0) implies that C2 = 0 in equation 2.27. The mechanical impedance Zmof the structure can be expressed

as

F_|x=l =−Zm˙u|x=l (2.30)

Note the negative sign in equation 2.30, which implies that a positive displacement u gives rise to a force in the opposite direction. The displacement is harmonic, which results in

˙u = jωu (2.31)

Substituting equations 2.28 and 2.30 in 2.30 and using the fact C2= 0, yields

F_|x=l =−C1jωZmsin(κl)ejωt (2.32)

Additionally, the strain in the PZT patch becomes S1(x) =

∂u

∂x = C1κ cos(κx)e

jωt _(2.33)

Combining equations 2.32 and 2.33 with 2.20 for x = l and using T1 = F|x=l wh (2.34) E3= V h (2.35)

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 14 C1κ cos(κx)ejωt= −C1 jωZmsin(κl)ejωt wh ¯YE + d31 h V (2.36)

Using equation 2.28 for u and 2.33 for S1, the expression for the mechanical impedance of the

PZT becomes

Za=

whκ ¯EE

jω tan(κl) (2.37)

Combining equations 2.36 and 2.37 and solving for C1 gives

C1= ZaV0d31 hκ cos(κl) (Zm+ Za) (2.38) Here, V0 is defined as V = V0ejωt (2.39)

The stress T1 in the PZT patch under general electric field conditions is given by equation

2.20 and is further resolved by using equations 2.33 and 2.39

T1 = Zacos(κx) cos(κl) (Zm+ Za) V0E¯Ed31ejωt h (2.40)

Substituting 2.40 in equation 2.19, gives for the electric charge density on the surfaces of the PZT patch D3= Zad231E¯Ecos(κx) (Zm+ Za) cos(κl) + (¯T 33− d231E¯E) V0 he jωt _(2.41)

The current is obtained by integrating the rate of charge of the electric charge D3 over a

surface of the PZT and is,

I = Z h 2 −h₂ Z x=l −l dD3 dt dxdy = Z h 2 −h₂ Z l −l jωD3dxdy (2.42)

Substituting equation 2.41 yields

I = 2V0wlωje jωt h Zad231E¯2 (Zm+ Za) tan(κl) κl + ¯ T 33− d231E¯T (2.43) So, the admittance of the PZT patch is given by

Y = 1 Ze = I V = 2ωj wl h ¯ T₃₃_{− d}2₃₁E¯T+ Za Zm+ Za d2₃₁E¯T tan(κl) κl (2.44) This electromechanical coupling between the mechanical impedance Zm of the host structure

and the electrical admittance Y of the PZT is used for damage detection with the EMI technique.

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 15 2-D model

For structures where 2-D coupling is significant, Liang’s 1-D approach [25] might introduce serious errors. Zhou et al. [5] extended the single degree of freedom (SDOF) model to a 2-D PZT element coupled to a 2-D host structure shown in figure 2.8.

1 2 3 l w Zxx Zxy Zyx Zyy

Figure 2.8: 2D modelling of the mechanical impedance coupling for the PZT-structure interaction

The structural impedance in matrix notation becomes " F1 F 2 # =₋ " Zxx Zxy Zyx Zyy # " ˙u1 ˙u2 # (2.45)

Whereby Zxxand Zyyrepresents the direct impedances and Zxyand Zyxthe cross impedances.

Using d’Alembert’s principle along the two directions and applying the boundary conditions, the following expression for the admittance is derived

Y = jωwl h " T₃₃−2d 2 31E¯E 1_{− ν} + 2d2 31E¯E 1_{− ν} h sin(κl) l sin(κw) w i N−1 " 1 1 ## (2.46) where κ is the 2-D wave number and N a 2x2 matrix

κ = ω r ρ(1− ν2₎ ¯ EE (2.47) N =  κcos(κl) 1− νw l Zxy Zaxx + Zxx Zaxx κcos(κw)_wl Zyx Zayy − ν Zyy Zayy κcos(κl)w l Zxy Zaxx− ν Zxx Zaxx κcos(κw)1_{− ν} l w Zyx Zayy + Zyy Zayy   (2.48)

with Zaxx and Zayy the two components of the mechanical impedance of the PZT patch in

the two principal directions. The mechanical admittance Y is a function of the structural parameters and is influenced by any structural damage. Although the derivations of Zhou et al. are more accurate than the 1-D approach of Liang, experimental difficulties prohibit the direct application of the structural impedances of the structure. Solving the 2-D impedance approach, equation 2.46 must be solved for 4 complex unknowns Zxx, Zxy, Zyx, Zyy. With

the experimental PZT impedance measurement, only 1 complex quantity is obtained. So the system is indeterminate and can’t be used for experimental determination of the mechanical impedance of a structure [26].

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 16 Effective impedance model

Bhalla and Soh [5] tried to bridge the gap between the 1-D model of Liang et al. and the 2-D model of Zhou et al.. They introduced a new concept of ’effective impedance’. In the first interaction models, the interaction between the PZT and the structure is condensed to an end point of the PZT. Nonetheless, the mechanical interaction is not restricted to the end points, but extends all over the finite surface of the PZT patch. In the effective impedance, the PZT-interaction is represented as the boundary force f per unit length, varying harmoinically over time [27]. This harmonic force results in planar harmonic deformations in the PZT patch. Using the definition of the mechanical impedance (eq. 2.10), the ’effective mechanical impedance’ of a PZT patch is defined as

Za,ef f = F ˙uef f = H f· ˆnds ˙uef f (2.49) With ˆn the unit vector normal to the boundary and F the overall planar or effective force. The effective displacement of the PZT patch is defined as

uef f =

δA p0

(2.50) Here, δA is the total change in surface area of the PZT and p0 denotes the perimeter in

undeformed condition.

Bhalla et al. [27] derived the effective impedance for a squared PZT patch under 2D interac-tion with a host structure by assuming force transmission between PZT patch and structure along the entire boundary of the patch and plane stress condition within the patch. Further is the patch assumed to be square shaped and infinitesimally small compared to the host structure.

Electromechanical admittance signature

The electromechanical impedance technique is based on the measured electromechanical ad-mittance ¯Y (the inverse of the electromechanical impedance) of the PZT patch. The elec-tromechanical admittance is a complex variable and consists of real and imaginary part, respectively the conductance (G) and the susceptance (B).

¯

Y = G + jB (2.51)

The EMI signature is a plot of both variables over a wide frequency range and serves as a characteristic of the structure. In figure 2.9, the conductance and susceptance plots for a PWAS bonded to a undamaged aluminium bar are shown for a frequency range of 30 to 160kHz.

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CHAPTER 2. ELECTROMECHANICAL IMPEDANCE TECHNIQUE 17 40 60 80 100 120 140 160 0 0.01 0.02 0.03 40 60 80 100 120 140 160 -0.05 -0.04 -0.03 -0.02 -0.01 0

Figure 2.9: Conductance and susceptance plots of a PWAS patch bonded to aluminium beam

2.2.3 Damage assessment

The electromechanical impedance technique aims to asses damage using mechanical impedance of the structure under investigation. This is achieved by measuring the electromechanical impedance signature of the to the host structure bonded PZT patch.

Raw-signature method

Sun et al. showed that the appearance of new peaks in the electromechanical signature and lateral and vertical shift of peaks are prominent effects of structural damage [4]. As a result, damage assessment can be done quickly and easily without having to calculate the mechanical impedance from the electromechanical admittance signature. However, damage assessment based on the raw-signature will be less informative than the mechanical impedance method [12].

Beside the single peak approach, where changes in the size and location of the peaks are used as damage metric, different statistical techniques are employed for damage assessment in the electromechanical impedance technique. Frequently used statistical methods for comparing different EMI signatures are the root mean square deviation or RMSD, mean absolute percent deviation or MAPD and the correlation coefficient or CC [28, 29].

The RMSD index is defined as

RM SD = sPN i=1(G1i − G0i)2 PN i=1G0i (2.52) Where G1 _{is the conductance is damage state and G}0 _{is the corresponding pre-damage}