ABSTRACT 1. INTRODUCTION

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1 Analytical modeling of PWA in-plane and out-of-plane electromechanical impedance spectroscopy (MI) Tuncay Kamas, Bin Lin, Victor Giurgiutiu epartment of Mechanical ngineering, University of outh Carolina, Columbia, C kamas@ .sc.edu, linbin@cec.sc.edu, victorg@sc.edu ABTRACT This paper discusses theoretical analysis of electro-mechanical impedance spectroscopy (MI) of piezoelectric wafer active sensor (PWA). Both free and constrained PWA MI models are developed for in-plane (lengthwise) and outof plane (thickness wise) mode. The paper starts with the general piezoelectric constitutive equations that express the linear relation between stress, strain, electric field and electric displacement. This is followed by the PWA MI models with two assumptions: 1) constant electric displacement in thickness direction ( ) for out-of-plane mode; 2) constant electric field in thickness direction ( ) for in-plane mode. The effects of these assumptions on the free PWA in-plane and out-of-plane MI models are studied and compared. The effects of internal damping of PWA are considered in the analytical MI models. The analytical MI models are verified by Coupled Field Finite lement Method (CF-FM) simulations and by experimental measurements. The extent of the agreement between the analytical and experimental MI results is discussed. The paper ends with summary, conclusions, and suggestions for future work. Keywords: thickness mode, electro-mechanical impedance spectroscopy, piezoelectric constitutive equations, piezoelectric wafer active sensors 1. INTROUCTION Piezoelectric wafer active sensor (PWA) 1,2 is light-weighted, inexpensive, unobtrusive, minimally intrusive sensor requiring low-power. PWA is made of piezoelectric ceramic with electric field polarization,, across the electrodes deposited on both surfaces. It has recently been extensively employed in many applications for structural health monitoring (HM) and non-destructive evaluations (N),4.lectro-mechanical impedance spectroscopy (MI) method has been widely used to determine the dynamic characteristics of a free PWA and bonded PWA for in-situ ultrasonics 5 such as in the work presented by un and Liang 6,7. They utilized the MI method for high frequency local modal sensing. Many rigorous researches on the thickness (out-of-plane) mode theory have been conducted for piezoelectric crystal and ceramic resonators. Tiersten 8 presented a pioneering work to develop the analytical solution for the thickness vibration of an anisotropic piezoelectric plate. He used the resonator theory with traction-free T = 0 boundary conditions at surfaces of a plate. Thickness vibration in an infinite piezoelectric plate was explored 8 based on lossless ideal linear theory. He assumed a medium that is perfectly elastic and perfectly insulating to electric current so that the coupling of mechanical field and electric field is omitted. Meeker 9 adopted Tiersten s basic equations to develop general impedance equations with arbitrary boundary conditions 10,11 He used a matrix method to analyze the parallel and perpendicular electrical field excitation of piezoelectric plates in thickness direction. The resonant and anti-resonant frequencies and the coupling factors were determined by solving transcendental equations. Yamada and Niizeki 10,11 extended the thickness mode solution for both thickness and lateral excitation. Mason 12 further developed the equivalent electrical circuit theory to predict the impedance of the simple thickness mode piezoelectric transducer. These previous analytical solutions have been focused on piezoelectric crystal transducers. The analytical in-plane impedance for piezoelectric ceramic transducers such as PWA has been developed by Zagrai and Giurgiutiu 5,1.One and two dimensional in-plane /M impedance models for free PWA and constrained PWA were derived to model the dynamics of PWA and substrate structure in terms of MI. They assumed the constant electric field,, to derive the in-plane MI. However, the analytical study for thickness mode of PWA-MI has ensors and mart tructures Technologies for Civil, Mechanical, and Aerospace ystems 201, edited by Jerome Peter Lynch, Chung-Bang Yun, Kon-Well Wang, Proc. of PI Vol. 8692, PI CCC code: X/1/$18 doi: / Proc. of PI Vol

2 not been fully performed yet. The present work aims to extend the MI model of a rectangular free PWA at high frequencies (up to 15MHz). We adopted the constant electric displacement assumption used in the literature 9,14 and solved the piezoelectric constitutive equations for the thickness mode. Coupled-field finite element method (CF-FM) was used to model and simulate free and constrained PWA-MI. In addition, a set of experiments was conducted using free-rectangular PWA and bonded PWA on an aluminum bar. The analytical and numerical models are validated with the experimental results. The comparison between theoretical prediction, simulation, and experimental data are illustrated and discussed. 2. ANALYTICAL /M IMPANC MOLING OF PWA The modeling of a free PWA is useful for (a) understanding the electromechanical coupling between the mechanical vibration response and the complex electrical response of the sensor; and (b) sensor screening and quality control prior to installation on the monitored structure. lectromechanical resonances reflect the coupling between the mechanical and electrical variables, they happen under electric excitation, which produces electromechanical response (i.e., both a mechanical vibration and a change in the electric admittance and impedance). When a PWA is excited harmonically with a constant voltage at a given frequency, electrical resonance is associated with the situation in which a device is drawing very large currents. At Z = 1/ Y goes to zero. As the admittance resonance, the admittance ( Y ) becomes very large whereas the impedance ( ) becomes very large, the current drawn under constant-voltage excitation also becomes very large because I = YV. In piezoelectric devices, the mechanical response at electrical resonance also becomes very large. This happens because the electromechanical coupling of the piezoelectric materials transfers energy from the electrical input into the mechanical response IN-PLAN MO OF PWA RONATOR WITH CONTANT AUMPTION This section addresses the behavior of both in-plane MI models for free PWA considering axial and flexural vibrations. Zagrai and Giurgiutiu 15,16 applied the constant assumption for 1- in-plane impedance equation (2) by utilizing the piezoelectric constitutive equations(1) under stress-free boundary conditions. The constitutive equations possess the stress and the electric field as the independent variables and linearly relate the mechanical and electrical properties of piezoelectric materials. Zagrai 17 also discussed the thickness mode of the electro-mechanical impedance response of active sensor at resonance frequencies. However, he has not explicitly derived the thickness mode PWA- MI in his research. = s T + d d T kl kl k k T j = jkl kl + ε jk k (1) where is the strain tensor, Tkl is the stress tensor, j is the electrical displacement (charge per unit area), s kl is the T mechanical compliance matrix at zero electric field, ε jk is the dielectric constant at zero stress, d jkl is the induced strain coefficient (mechanical strain per unit electric field). The following assumptions for PWA were used for the in-plane MI model 18. The PWA of length l, width b, and thickness h, undergoing piezoelectric expansion induced by the thickness polarization electric field,. The electric field is generated by harmonic voltage () ˆ i t Vt = Ve ω between the top and bottom surface electrodes. is assumed to be uniform over the piezoelectric wafer. Thus, its derivative with respect to x is zero i.e. / x = 0. The voltage excitation is harmonic so that the electric field ˆ i t = e ω and the mechanical response in terms of particle displacement are also harmonic, i.e. uxt (,) = uxe ˆ() iωt where ux ˆ( ) is the x dependent complex amplitude that incorporates any phase difference between the excitation and response. Giurgiutiu and Zagrai obtained the following frequency dependent impedance equation that can be used to predict the frequency response of PWA excited at anti-resonance frequencies. The electro-mechanical impedance follows the electrical impedance function, 1/iω C0 where C 0 is the capacitance of the sensor. To this purpose, we note that the term 1 ϕ is a function of frequency and wave speed, i.e. ϕ = / 2 ωl c and the electro-mechanical coupling is denoted by 2 κ 1 term. Proc. of PI Vol

3 1 2 1 Z = 1 κ 1 1 iωc0 ϕcotϕ 1 (2) i O 104 t ::: 10 ( Ce Frequency (khz) Frequency (khz) Figure 1 imulated frequency response of admittance and impedance of a PWA (including internal damping effects of δ = η = 0.01 ) Frequency plots of admittance ( Y 1/ Z) = and impedance shows the graphical determination of the resonance and antiresonance frequencies. Figure 1 presents the numerical simulation of admittance and impedance response for a piezoelectric active sensor ( l = 7mm, b = 1.68mm, h = 0.2mm, APC-850 piezo-ceramic) THICKN MO OF PWA RONATOR WITH CONTANT AUMPTION In this section, the behavior of a free PWA in thickness mode will be addressed by implying the constant electric displacement as opposed to the implication of constant electric field in in-plane MI model. ince the axial vibration in thickness direction is assumed to be non-perturbed by other lateral modes because the lateral sizes are h<< b<< l i.e. the thickness motions are assumed to be much greater as compared to the thickness size of PWA ( ) decoupled from width and length motions. Thus, from Gauss law, due to no free charge inside the transducer, the electrical charge density, i.e. electric displacement, can be assumed to be constant throughout the polarization direction and the gradient of the electric displacement becomes zero as shown in q.(). iv = + + = 0 x1 x2 x () Redwood 19 explained the basic assumptions from which the mechanical resonant frequencies of a piezoelectric plate resonator is determined. He assumed that the differentials of the electrical displacement with respect to the lateral directions are zero. Therefore, the mechanical resonant frequency was determined entirely by its thickness dimension. The plane waves propagate undistorted in thickness direction and are reflected at the lateral surfaces without losing their plane wave propagation nature 1 2 = = 0 x x 1 2 (4) Therefore, / x = 0 (5) Proc. of PI Vol

4 x u T. - dt. I'y n, f PZT-active-sensor-Q length l, width b, thickness h Length i,thickness t :width b Figure 2 chematic of thickness mode of a piezoelectric wafer active sensor and infinitesimal axial element The constitutive equations shown in q. (6) possess the strain and the electric displacement as the independent variables. a) b) T = c h h = + β (6) The thickness mode MI model with the constant assumption has been determined using the corresponding piezoelectric constants in qs. (7) as presented in the literature 14,20. d = ε g = e s T g = β d = h s T e = ε h = d c h = β e = g c (7) Mechanical Response of free PWA in thickness mode In this subsection, the mechanical response of the free PWA in terms of the displacement and the strain in thickness mode is derived. The wave equation u c u 2 2 = 2 2 t ρ x (8) and the general solution of the wave equation for the particle displacement are implied. u = C sinγ x + C cosγ x (9) 1 t 2 t The wave speed in direction of x axis, c / = c ρ, is introduced. C1 and C 2 are determined from the traction-free boundary conditions. T ( x = ± h/2) = 0 (10) To this purpose, q. (6)a is substituted into q. (10) using the strain-displacement relation ( = u / x); ( ) γt ( ( γt ) ( γt )) T x =± h/2 = c C cos 1/2 h m C sin 1/2 h h = 0 (11) 1 2 Proc. of PI Vol

5 whereγ t = ω / c is the wave-number in thickness mode i.e. the ratio of the angular frequency to the phase velocity of the wave in thickness direction. Imposing the boundary conditions shown in q. (10) and summing up the stresses on both surfaces in q. (11),one obtains h C = 1 γ cos tc ϕ (12) t C 2 = 0 since sinϕ t is assumed to be non-trivial term where ϕ t = 1/2 γ h t.the mechanical response of PWA in terms of the particle displacement u and the strain can be expressed as t γtc cosϕt h uˆ ( x ) = sinγ x (1) h = uˆ = cosγ x (14) t c cosϕ lectrical response of free PWA in thickness mode Consider a 1- PWA under electric excitation as illustrated in Figure 2. Recall q. (6)b representing the thickness polarization electric field. The electrical impedance Z = V / I and t V = dx (15) 0 ubstituteq. (14) into q. (6)b to obtain h = x + (16) 2 s cosγt β c cosϕt 2 2 s 2 Recalling the piezoelectric constant relations in q. (7),one can derive these relations h = e / ε using the first s s relation in q(7); β = 1/ ε using the combination of the third and fourth of q(7); and e = d / s using the first. 2 s Finally one can come up with the expression, ( ) 2 h = βd / s, and plug it into q(16) noting that c = 1/ s and introduce the electro-mechanical coupling coefficient, κ, defined as κ = e / cε Upon rearrangement of the q.(16) 2 cosγ t x = β 1 κ cosϕt (17) and upon substitution of q. (17) into q. (15)a, we obtain h 2 h κβ 2 1 V = βdx cosγt xdx = βh 1 κ cosϕ cot 0 t ϕ 0 t ϕt (18) Proc. of PI Vol

6 Recall the electrical current and charge relation, i.e. I = Q& = Q/ t = iωq,as well as the electrical charge and electrical displacement relation, Q = dxdx 1 2. Upon substitution of the electrical charge into the current equation, we obtain A the current as a function of the electrical displacement, i.e. d I = da= i bl dt ω (19) A ubstitute q. (19) into the impedance, Z = V / I, to get βh 2 1 Z = 1 κ iωbl ϕt cotϕt (20) Recall the capacitance of the PWA, C0 = bl / hβ, hence; V Z = = 1 κ I iωc0 ϕt cotϕt (21) Note that the impedance is purely imaginary and consists of the capacitive impedance, 1/iω C0, modified by the effect of piezoelectric coupling between mechanical and electrical fields. This effect is apparent in the term containing the electro-mechanical coupling coefficient, κ that is defined for thickness mode MI. 2. FFCT OF INTRNAL AMPING The internal damping can be modeled analytically by complex compliance and dielectric constant ( 1 η) ( 1 η) s = s i c = c i ε = ε (1 iδ) (22) The internal damping is implied in free (the boundaries are unbounded) and in order to determine the thickness mode impedance assuming η andδ are 2%. The impedance becomes complex expression Z = 1 κ 1 1 iωc0 ϕcotϕ 1 (2) Z = 1 κ iωc0 ϕt cotϕt (24) where 2 2 T κ1 = d1 / s11ε for constant assumption and 2 2 κ = e / cε for constant assumptions the complex coupling factor, C0 = (1 iδ ) C0, and ϕ = ϕ 1 η. The material properties are given in Table 1. t t i Proc. of PI Vol

7 Table 1Properties of APC 850 piezoelectric ceramic ( Property APC 850 ρ ( kg / m ) d m V ( / ) d m V ( / ) 1 g Vm N (1 0 / ) g Vm N (1 0 / ) 1 s m N ( / ) 11 s m N ( / ) ε / ε T 0 κ p κ κ Poisson s ratio COUPL-FIL FINIT LMNT ANALYI The first part of this section presents the CF-FA model of a free PWA in order to illustrate the MI calculation.then, CF-FA modeling will be presented for a PWA affixed on an aluminum structure Free PWA MI Model A free PWA has been modeled without the presence of the host structure in order to have understanding of the multiphysics based modeling approach and its extent of agreement with both analytical and experimental analysis. ANY multi-physics software with the implicit solver was used to obtain MI computation in frequency domain. To perform the coupled stress and electric field analysis of PWA transducers, coupled field piezoelectric elements were used. These coupled field finite elements consist of both mechanical and electrical fields. The elements that represent piezoelectric effects in our analysis are the 2- coupled field solid elements i.e. PLAN 1 that have four nodes with displacement degrees of freedom (OF) along with electric voltage as another OF. The reaction forces FX, FY correspond to the UX, UY displacement OF, respectively. The electrical charge Q is the electrical reaction corresponding to the voltage OF. The charge Q is then used to calculate the admittance and impedance data. The admittance Y is calculated as I / V, where I is the current in ampere and V is the applied potential voltage in volts. The current comes from the charge accumulated on the PWA surface electrodes and is calculated as I = iω Q with ω being the operating frequency, i is the complex number, and Qi is the summed nodal charge. A free square i shaped PWA of dimension PWA as follows mm was modeled. The APC-850 material properties were assigned to the C p = GPa (25) Proc. of PI Vol

8 ε p = F/m (26) 0 0 e p = C/m (27) where C p is the stiffness matrix, ε p is the dielectric matrix, and the PWA material is assumed to be ρ = 7700 kg/m. e p is the piezoelectric matrix. The density of 4.2. Constrained PWA model on an aluminum strip In CF-FA approach, the mechanical coupling between the structure and the sensor is implemented by specifying boundary conditions of the sensor, while the electromechanical coupling is modeled by multi-physics equations for the piezoelectricc material. The first coupling allows the mechanical response sensed by the piezoelectric element to be reflected in its impedancee signature. The aluminum beam was modeled as a homogeneous isotropic material with assumed density ρ = 2780 kg/m and elastic modulus = 72.4 GPa. PWA - - F F- Figure Interactionn between PWA and structure The coupled field FM matrix element can be expressed as follows [ M ] [ 0] [ 0] [ 0] {} u&& [ ] [ ] { } C 0 u& + { V&& } [ 0] [ 0] { V& } [ K ] + K Z T K K Z d { u } { V } { F} { L} = (28) where [ M ], [ C ], and [ K] are the structural mass, damping, and stiffness matrices, respectively; {} u and { V} are the vectors of nodal displacement and electric potential, respectively, r with dot abovee variables denoting time derivative; { F } is the force vector; { L} is the vector of nodal, surface, and body charges; { K Z } is the piezoelectric coupling matrix; and { K d } is the dielectric conductivity. This CF-FA method is very convenient for evaluating the impedance signatures as it is measured by the impedance 2 2 analyzer. The aluminum beam dimensions are mm and the PWA mm. For a plane strain analysis, only a longitudinal section of the specimen and PWA were analyzed; hencee 2- meshed CF-FA model was generated which reduced considerably the computational time. The 2 plane element PLAN42 is used for the aluminum beam; this element has 4 nodes and 2 OF at each node. The 2 plane element PLAN1 is used to model the PWA using the coupled field formulation presented in q.(28). Then, the impedance spectrum up to 15 MHz was calculated. Proc. of PI Vol

9 5. PWA MI XPRIMNTAL MAURMNT The measurement of the intrinsic MI was done with an HP 4194A impedance phase-gain analyzer Figure 4.For free- PWA MI test, a test fixture was used for measuring the intrinsic MI of the PWA. We used a metallic plate with a lead connected at one corner. The PWA was centered on the bolt head and held in place with the probe tip. Thus, the PWA could vibrate freely. The PWA sample was tested by scanning a predetermined frequency range in high frequency band (up to 15MHz) and recording the complex impedance spectrum. A LabView data acquisition program was used to control the impedance analyzer and sweep the frequency range. PWA Fixture HP 4194A Impedance Analyzer Figure 4 xperimental setup for measuring the impedance characteristics of the PZT active sensors with HP 4194A Impedance Phase-Gain Analyzer Free PWA MI results 6. RULT AN ICUION We considered a free PWA under a traction-free boundary condition to conduct a one-dimensional theoretical /M impedance analysis. The results in terms of impedance are compared with CF-FA as well as experimental analysis as shown in Figure 5 and Figure 6.uring the visualization of the frequency sweep, the peaks appearing in the frequency range up to 2500 khz are associated with the anti-resonances of the in-plane modes. These peaks are progressively smaller, with the fundamental resonance being the strongest. This is consistent with the fact that higher modes need more energy to get excited. Under constant energy excitation, higher modes would have lower amplitudes. At around 11 MHz, a new solitary strong peak appears. This is associated with the fundamental resonance of the thickness mode.the first and second in-plane impedance spectra in the experimental results were found to be significantly different from the theoretical predictions. The significant differences are indicative of the 2- stiffening effect, typical of in-plane vibrations of low-aspect ratio piezo-plates. These 2- stiffening effects could not be captured by the 1- theory at higher modes. However, the 2- stiffening effect diminishes, and the agreement between theory and experiment improves at the thickness mode spectra because the thickness mode is hardly affected by the in-plane shape of the PWA. Proc. of PI Vol

10 a-) b-) Figure 5 Comparison of analytical analysis results with coupled field finite element analysis results for a-) in-plane MI and b-) out-of-plane MI of 7mmx0.2mm PWA a-) b-) Figure 6 Comparison of analytical results with experimental results for a-) in-plane MI and b-) out-of-plane MI of 7mmx0.2mm PWA pectra from anaytical, numerical simulation and experiment of the free-pwa are obtained for in-plane MI in relatively low frequency range and for out-of-plane MI in high frequency range. Globally good matching is observed, however some discrepancies between the analytical PWA-MI and CF-FA PWA-MI are visible especially for the thickness mode peak in Figure 5b. mall differences at high frequencies are expected between the analytical and the numerical responses due to the simplifying assumptions made in the one-dimensional analytical analysis. Nonetheless the comparison illustrates a reasonable agreement in high frequency in-plane modes of PWA. It is also noticable that Proc. of PI Vol

11 in high frequency range, the impedance results agreereasonably well in comparing the analytical thickness mode MI model of PWA with the corresponding CF-FA and experimental results Constrained PWA MI results The analysis of a constrained PWA is essential for the analysis of the PWA as a structural modal sensor. When affixed to a structure, the PWA is constrained by the structure and its dynamic behavior is essentially modified. The PWA /M impedance will closely follow the dynamics of the structure and the PWA becomes a sensor of the dynamical modal behavior of the structure. To understand the MI method for constrained PWA, active sensor experiments and coupled field finite element analyses were conducted on a thin metallic bar of a 0.8 mm thick 2024 aluminum alloy (217mm x 25.4mm) The bar specimen was instrumented with a PWA positioned at central. Figure 7 shows the log-scale plot of the real part of the electro-mechanical impedance computed with the coupled-field FM analysis. The experimental MI results are superposed with the FA simulation results in one plot. The impedance peak amplitudes at high frequencies are relatively small in experimental results with the fundamental resonance being the strongest since the modes at high frequencies need more energy to get excited. Under constant energy excitation, higher modes would have lower amplitudes. xperimental CF-FA Figure 7Comparison of two-dimensional coupled field finite element impedance analysis results with experimental analysis results for 7mm x 0.2mm PWA mounted onto an aluminum strip in size of 217mm x 0.8mm The anti-resonance frequencies from both experimental and CF-FA results and the extent of their agreement are shown in Table 2 Comparison of real part anti-resonance frequency results from CF-FA and experiment for PWA bonded on an aluminum bar.table 2 for the two peaks appearing in high frequency range between 2-6 MHz. The second peaks match better having 9% difference in acceptable range. However, the first simulated impedance peak is shifted slightly more with respect to the experimental measurement. The difference between the first anti-resonance peaks is 1.7%. This may be explained by the fact that the finite element size and the time step of the harmonic analysis may not be sufficiently fine for such high frequencies and thus can yield corrupted results. The effect of element size and time step at high frequencies should always be explored when doing FM analysis. Nevertheless, the CF-FA simulation of constrained PWA-MI overall gives a good matching with the trend of the experimental MI results. Proc. of PI Vol

12 Table 2 Comparison of real part anti-resonance frequency results from CF-FA and experiment for PWA bonded on an aluminum bar. Anlysis Thickness mode anti-resonance frequencies [MHZ] for constrained PWA 1 2 CF-FA xperiment % ifference CONCLUION In this paper, the in-plane /M impedance methods for both free PWA and constrained PWA were discussed. The inplane MI approach previously presented by Zagrai 17 and Giurgiutiu 18 was extended by developing the theoretical outof-plane (thickness-wise) MI model for free PWA. A one-dimensional Cartesian coordinate analysis for axial vibrations was presented. The thickness mode vibration of free-pwa-mi solution was developed to assure the dynamic characteristics and the quality of PWA itself in high frequency range before it is installed on a structure. The coupled field finite element analysis for free PWA was conducted to predict the /M impedance response throughout the broadband frequency range (100 Hz 15 MHz), as it would be measured at PWA terminals. In addition, the thickness mode coupled field FA model of constrained PWA-MI was developed to investigate the coupling between the dynamics of the structural substrate and the dynamics of PWA. A set of experiments was also conducted to assess the theoretical investigation. Free PWA and constrained PWA in square form as in the theoretical models are used to measure the electro-mechanical impedance spectra. The comparisons illustrated a reasonable agreement in both in-plane and out-of-plane MI analysis. A novel feature of the present work is the one-dimensional analytical modeling and coupled field multi-physics finite element modeling of a PWA under elastic boundary conditions and the prediction of its broad band /M impedance thickness modes of the PWA.PWA has traction-free boundaries and as constrained on a metallic bar. To obtain more consistent results from thickness mode vibration analysis of MI of PWA, the dimensions and assumptions made in analytical and numerical modeling should be closer to the experimental investigation. 1- MI analytical approximation is convenient to understand the impedance spectra in high frequency range however the sensitivity of the analytical model needs to be improved by further work. The effects of element size and time step at high frequencies require more investigation when doing FM analysis. An analytical model for thickness mode of constrained PWA-MI is under development. The analytical model for the bonded PWA MI in thickness mode can predict thickness-wise defects in the monitored substrate material using the modal expansion analysis of the coupled PWA structure. 8. ACKNOWLGMNT upport from National cience Foundation Grant # CM ; Air Force Office of cientific Research #FA , r. avid targel, Program Manager; are thankfully acknowledged. Proc. of PI Vol

13 9. RFRNC [1] Lin, B. & Giurgiutiu, V., "Modeling of Power and nergy Transduction of mbedded Piezoelectric Wafer Active ensors for tructural Health Monitoring" PI mart tructure and Materials + Nondestructive valuation and Health Monitoring 2010, ensors and mart tructures Technologies for Civil, Mechanical, and Aerospace ystems, 7981, 76472P 76472P 12 (2010). [2] Giurgiutiu, V., Bao, J. & Zhao, W., "Active ensor Wave Propagation Health Monitoring of Beam and Plate tructures" Proc of PI s 8th International ymposium on mart tructures and Materials, (2001). [] Giurgiutiu, V. & Zagrai, A., "amage etection in imulated Aging-Aircraft Panels Using The lectro-mechanical Impedance Technique" Adaptive tructures and Material ystems ymposium, AM Winter Annual Meeting (2000). [4] Giurgiutiu, V., Zagrai, a. & Jing Bao, J., "Piezoelectric Wafer mbedded Active ensors for Aging Aircraft tructural Health Monitoring," tructural Health Monitoring, 1, (2002). [5] Zagrai, A. N. & Giurgiutiu, V., "lectro-mechanical Impedance Method for amage Identification in Circular Plates," Journal of Intelligent Material ystems and tructures, 12, 40 (2001). [6] Liang, C., un, F. P. & Rogers, C. A., "Coupled lectro-mechanical Analysis of Adaptive Material ystems -- etermination of the Actuator Power Consumption and ystem nergy Transfer" Journal of Intelligent Material ystems and tructures 5, (1994). [7] un, F. P., Liang, C. & Rogers, C. A. "tructural modal analysis using collocated piezoelectric actuator/sensors: an electromechanical approach" Proc. PI 2190, mart tructures and Materials 1994: mart tructures and Intelligent ystems, 28 (1994). [8] Tiersten, H. F., "Thickness Vibrations of Piezoelectric Plates" J. Acoustic ociety of America, 5, 5 58 (196). [9] Meeker, T. R. "Thickness mode piezoelectric transducers" Ultrasonics, 10, 26 6 (1972). [10] Yamada, T. & Niizeki, N., "Admittance of piezoelectric plates vibrating under the perpendicular field excitation," Proceedings of the I, (1970). [11] Yamada, T., "Formulation of Admittance for Parallel Field xcitation of Piezoelectric Plates," Journal of Applied Physics, 41, 604 (1970). [12] Mason, W. P., "lectromechanical Transducers and Wave Filters,". Van Nostrand Company, Inc. (1948). [1] Zagrai, A. & Giurgiutiu, V., "lectro-mechanical Impedance Method for Crack etection in Thin Plates," Journal of Intelligent Material ystems and tructures, 12, (2001). [14],"I Ultrasonics I tandard on Piezoelectricity," (1987). [15] Giurgiutiu, V. & Rogers, C., "Modal xpansion Modeling of the lectro-mechanical (/M) Impedance Response of 1- tructures," uropean COT F Conference, 6 9 (2000) [16] Giurgiutiu, V., "Piezoelectric wafer active sensors (PWA) for structural health monitoring and embedded ultrasonics",2 5 [17] Zagrai, A. N., "Piezoelectric Wafer Active ensor lectro-mechanical Impedance tructural Health Monitoring,", (2002) [18] Giurgiutiu, V., "Piezoelectric Wafer Active ensors tructural Health Monitoring with Piezoelectric Wafer Active ensors,", (2008). [19] Redwood, M., "Transient performance of a piezoelectric transducer," Acoustical ociety of America,, (1961). [20] Berlincourt,. A., Curran,. R. & Jaffe, H., "Piezoelectric and Piezomagnetic Materials and Their Function in Transducers. Physical Acoustics and The Properties of olids", (1958). Proc. of PI Vol