Title: Electro-Mechanical Impedance Method for Crack Detection in Thin Wall Structures.
Authors:
Andrei N. Zagrai1 Victor Giurgiutiu2
Paper to be presented at: 3rd International Workshop of Structural Health Monitoring, Sept. 12-14, 2001, Stanford University.
ABSTRACT1 The Electro-Mechanical (E/M) impedance method has achieved acceptance among NDE methods due to its relative simplicity, applicability to complex structures, and low cost of active piezoelectric elements. Thus, allow one to identify the structural dynamics directly by obtaining its E/M impedance or admittance signatures. Previous work reported on this method contained little theoretical work but described many practical applications. The scope of presented research was to extend the positive results obtained for 1-D structure onto 2-D thinwall structures with circular symmetry. Theoretical analysis was analytically performed on simple geometries (circular plates) subjected to particular set of boundary conditions. The experimental results shown that E/M impedance (or admittance) spectrum accurately identify the natural frequency spectrum of the specimens manufactured from the aircraft-grade thin-gage material. Changes in the E/M impedance spectrum due to presence of crack were investigated. The dependence of the frequency spectrum on crack location was studied. The results show changing in E/M impedance spectrum due to damage presence. Additional recommendations are given to improve applications of the E/M impedance method for aging aircraft-type structures to detect incipient cracks and corrosion damage. 1.
INTRODUCTION
In recent years, the damage detection with E/M impedance method has gained increased attention. The method uses small-size piezoelectric active sensors intimately bonded to an existing structure, or embedded into a new composite construction. Experimental demonstrations have shown that the real part of the high-frequency impedance spectrum is directly affected by the presence of damage or defects in the monitored structure (Figure 1). The pioneering work on utilization of E/M impedance method for structural health monitoring was presented by Liang et al. (1994) who performed the coupled 1
Graduate Research Assistant, Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208,
[email protected], 803-777-0619, Fax 803-777-0106 2 Associate Professor, Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208,
[email protected], 803-777-8018, Fax 803-777-0106.
E/M analysis of adaptive systems driven by a surface-attached piezoelectric wafer. However, no modeling of the structural substrate was included, and no prediction of structural impedance for a multi-DOF structure was presented. This work was continued and extended by Sun et al. (1994) who used the half-power bandwidth method to accurately determine the natural frequency values. However, no theoretical modeling of the E/M impedance/admittance response for comparison with experimental data was attempted. 600
Impedance (ReZ)
500
PZT wafer transducer Defect or damage (a)
Cracked
300
Pristine
200 100 0
Structure
100 (b)
Figure 1
400
200
300 400 500 Frequency (kHz)
600
PZT wafer transducer acting as active sensor to monitor structural damage: (a) mounting of the PZT wafer transducer on a damaged structure; (b) the change in E/M impedance due to the presence of a crack.
Subsequently, several authors reported the use of the E/M impedance method for structural health monitoring, whereby the admittance or impedance frequency spectra of pristine and damaged structures were compared (Chaudhry et al (1994,1995), Ayres et al (1996), Giurgiutiu (1998)). The method has been shown to be especially effective at ultrasonic frequencies, which properly capture the changes in local dynamics due to incipient structural damage. (Such changes are too small to affect the global dynamics and hence cannot be readily detected by conventional low-frequency vibration methods). In this paper, brief theoretical analysis for 2-D aluminum circular plates structures is presented. Theoretical analysis was performed for particular boundary conditions to model the experimental set-up. The analytical model is then validated with experimental results. Systematic experiments conducted on statistical samples of incrementally damaged specimens were used to fully understand and calibrate the investigative method. 2.
E/M IMPEDANCE DAMAGE DETECTION STRATEGY
Based on the ability of the E/M method to detect local damage, we can formulate the damage detection strategy. The real part of the E/M impedance (Re Z) reflects the pointwise mechanical impedance of the structure, and the E/M impedance spectrum is equivalent to the pointwise frequency response of the structure. As damage (cracks, corrosion, disbonds) develop in the structure, the pointwise impedance in the damage vicinity changes. Piezoelectric active sensors placed at critical structural locations will be able to detect these near-field changes. In addition, due to the sensing localization property of this method, far-field influences will not be registered in the E/M impedance spectrum. The integrity of the sensor itself, which may also modify the E/M impedance spectrum, is
independently confirmed using the imaginary part of E/M impedance(Im Z), which is highly sensitive to sensor disbond, but much less sensitive than the real part to structural resonances (Giurgiutiu and Zagrai, 2000). To illustrate this damage detection strategy, consider an array of 4 active sensors as presented in Figure 2.
3
3
2
Structural Crack xC, yC a, θ 4
(a)
Corrosion Damage
2
xC, yC a, b, θ 4
1
1
(b)
Figure 2
Damage detection strategy using an array of 4 piezoelectric active sensors and E/M impedance method: (a) detection of structural cracks; (b) detection of corrosion damage. The circles represent the sensing radius of each active sensor.
Each active sensor has its own sensing area resulting from the application of the localization concept. This sensing area is characterized by a sensing radius and the corresponding sensing circle. Inside the sensing area, the sensor detection capability decreases with the distance away from the sensor. A damage feature that is placed in the near field of the sensor is expected to create a disturbance in the sensor response that is larger than a damage feature placed in the far field. Effective area coverage is ensured when the sensing circles overlap. Structural cracks, corrosion, and disbonds/delaminations present in the sensor sensing area can be effectively detected (Figure 2). The limitations of the E/M impedance method reside in its sensing localization which diminishes its ability to detect farfield damage. 3.
MODELING OF A PZT ACTIVE SENSOR INSTALLED ON A STRUCTURAL SUBSTRATE
The goal of the following analysis is to develop expression, which represents the E/M impedance spectrum as measured at the sensors terminals, and accounts for geometry, boundary conditions and, thus, correspondent pointwise dynamic structural stiffness kstr. In order to define kstr we have to consider flexural vibrations of circular plates under steady state excitation produced by piezoelectric active sensor. 3.1
MODAL EXCITATION. The theoretical foundation for transverse vibrations of isotropic circular plates was first published by Airey at 1911 and extended by Colwell (1936). Since then, many authors in various braches of science and engineering formulate and solve vibration problems for specific cases. An outstanding overview of a subject was presented by Leissa, (1969). The theoretical background and numerical results were given for large variety of boundary conditions and plate shapes. However, for complete developments reader should address other references. (Wah, (1962);
Kunukkasseril and Swamidas, (1974); Soedel, (1993); Rao (1999)). The generic treatment of transversely vibrating circular plates usually consider following equation: ∂Qr ∂2w ∂M r M r − M θ Qr + r ⋅ = ρ hr ⋅ 2 (1) + − Qr = 0 (2) r ∂r ∂t ∂r If any external forces or moments are applied, ra they should be accounted accordingly in equations (1) and (2). Consider situation when pzt piezoelectric active sensor, which has a disk A B z = h/2 shape, is installed in the center of a circular plate r so that the centers of a plate and a disk coincides. Plate (Figure 3.) Piezoelectric sensor produces both, force and Figure 3 Schematics on elongation of piezoelectric sensor bounded moment acting upon the plate structure. The on the plate. flexural excitation is produced by a line bending F h moment of magnitude M a = PZT , which 2π r a 2 space-wise distribution is expressed using the Heaviside function, H(ra – r), defined as H(ra – r) = 1 for r < ra, and H(ra – r) = 0 for ra ≤ r. Thus, moment Equation (2) will contain additional term: M re (r , t ) = M ra (r ) ⋅ eiωt , M ra (r , t ) = M a ⋅ H ( ra − r ) (3) Substituting M r = M r* + M re into (2) and expressing moments in terms of displacement gives, ∂ 2 w(r ,θ , t ) ∂ 2 M re (r , t ) 2 ∂M re (r , t ) D∇ 4 w(r ,θ , t ) + ρ h ⋅ = + (4) ∂t 2 ∂r 2 r ∂r The solution of Equation (4) includes series expansion of mode-shape functions expressed as 2 w(r ,θ , t ) = ∑∑ Gmn ⋅ Ymn (r ,θ ) ⋅ eiωt . (5) D ⋅∇ 4Ymn (r ,θ ) = ω mn ⋅ ρ h ⋅ Ymn (r ,θ ) (6) m n where ωmn2 in Equation (6) corresponds to the natural frequencies of circular plate. Using Equations (4), (5), (6), and applying normalization condition on (4) the modal excitation factor yields 2M a / ⋅ Ymn (ra ) − ra ⋅ Ymn (7) Gmn = (ra ) 2 2 2 ρ hR ⋅ ω − ω
(
3.2
mn
)
ELONGATION OF THE PIEZOELECTRIC SENSOR AND DYNAMIC STRUCTURAL STIFFNESS. The displacement of piezoelectric sensor with zero initial elongation is h described by u(r, z) = - z wa/ (r) = − ⋅ wa/ (r ) , where wa(r) is the bending 2 displacements of the neutral axis (Bickford, 1998). Referring to the Figure 3, the difference between B and A points yields h / (8) uPZT ( r , t ) = u B (ra , t ) − u A (0, t ) = − ∑∑ Gmn ⋅ Ymn (ra ) ⋅ eiωt 2 m n
Substitution Equation (8) into (9) gives the following result for the axial displacement of piezoelectric active sensor. −M a / / u PZT (r , t ) = ⋅ ∑∑ Ymn (ra ) − ra ⋅ Ymn (ra ) ⋅Ymn (ra ) ⋅ eiωt (9) 2 2 2 ρ R ⋅ ω mn − ω m n
(
)
Dividing Equation (9) by FPZT = FˆPZT ⋅ eiωt yields the structural frequency response function (FRF) to the Single Input Single Output (SISO) excitation applied by the PZT active sensor. This situation is similar to conventional modal testing (Harris, 1996, Section 21) with the proviso that the PZT wafers are unobtrusive and permanently attached to the structure. The dynamic structural stiffness is the inverse of frequency response function, i.e., if we rewrite moment in terms of force F Thus, expressing the moment in terms of force and noting that k str = PZT , the u PZT following equation for dynamic structural stiffness can be obtained. / /2 Ypq (ra ) ⋅ Ymn (ra ) − ra ⋅ Ypq (ra ) 4π ra R 2 ρ k str (ω ) = − ⋅ ∑∑ 2 h (ω mn −ω2) m n
−1
(10)
3.3
THE PZT ACTIVE SENSOR IMPEDANCE PZT wafer active sensors used in this analysis are of the circular shape. The axially symmetric vibrations of such sensors are well developed and widely presented in literature. (Onoe et al, 1967; Pugachev, 1984; IEEE Std, 1987) The solution of the differential equation is expressed in terms of Bessel functions (IEEE Std. 176 1987) and the displacement ξ(r) at radial position r is given by ξ (r ) = AJ1 (ω r / v p ) ⋅ e jω t (11) E where J1 is the first order Bessel function while ϑ p = 1/ ρ s11 ⋅ (1 − ν a2 ) is the wave speed for axially symmetric radial motion. Upon solution for elastically constrained boundary conditions, one can express the electrical admittance as k p2 (1 + ν a ) Yel (ω ) = jω Ca ⋅ 1 − k p2 1 + ⋅ (12) 2 ϕ J ϕ J ϕ − − ν − κ ( ) / ( ) (1 ) k 1 − 0 1 a a a p
(
)
2 E T is the where ϕ a = ω ra / ϑ p , ra is the radius of a disk, k p = 2d31 / s11 ⋅ (1 − ν a )ε 33
planar coupling factor, and κ = kstr / k PZT is a dynamic stiffness ratio. (Giurgiutiu et al, 2001) 4.
EXPERIMENTAL AND ANALYTICAL DIMENSIONAL STRUCTURES
RESULTS
FOR
TWO-
Since plates are common members of aircraft structures a series of experiments on thin-gage aluminum plates were conducted to understand how application of E/M impedance method could be transitioned from 1-D beam structures into 2-D plate structures. Twenty-five plate specimens (100-mm diameter, 0.8-mm thick) were constructed from aircraft-grade aluminum stock. Each plate was instrumented with one 7-mm circular PZT active sensor placed at its center (Figure 4a). In order to verify theoretical model, initial data set was taken for 5 of these
identical specimens using the HP 4194A Impedance Analyzer. No damage was introduced at this point, hence, data set can represent plates in pristine condition.
Crack (EDM slit)
PZT active sensor
PZT active sensor
Aluminum plate (a)
(b)
Figure 4
(a) Thin-gage aluminum plate specimens with centrally located piezoelectric sensors: 100mm circular plates, thickness – 0.8mm. (b) Photograph of circular plate specimen showing a 7-mm active sensor of the sensor and a simulated crack (EDM slit)
During the experiments, the specimens were supported on commercially available packing foam to simulate free boundary conditions. Plate resonance frequencies were identified from the E/M impedance real part spectra. Figure 5 presents identification of the first 3 natural frequencies for one plate taken from the initial data set. This case can be considered as a typical one since it was previously shown (Giurgiutiu and Zagrai, 2000ASME) that natural frequencies fall in a narrow frequency band, with very good mean and standard deviation values. The results are given in Table 1, “Exp.” column. Re(Z), Ohms. Log scale
1 .10
4
Table 1
Theoretical and experimental results for first 3 natural frequencies of circular plate #2.
Experimental
1 .10
3
Calculated 100
0
2
4
6
Theory kHz
Exp. kHz
∆%
0.720
0.722
-0.28 %
3.056 6.969
3.01 6.946
1.51 % 0.33 %
8
Frequency, KHz Figure 5
Experimental and calculated spectra of E/M impedance of the circular plate #2
In parallel, theoretical impedance spectrum was predicted using the method described in previous section. It is known that the mode-shape function introduced in Equation (6) depends on boundary conditions and expressed in terms of Bessel functions. For the case of free boundary condition at the plates radius i.e., Mr (a) = 0 and Vr (a) = 0, mode-shapes are governed by the following expression (Itaro and Crandall, 1979). λa λa Ymn (r ) = Am 0 ⋅ J 0 am 0 r + Cm 0 ⋅ I 0 am 0 r , n = 0 (13) where Am0 and Cm0 are amplitude and mode-shape parameters determined through normalization process and by solving eigenvalue problem. It should be noted that
(
)
(
)
Equation (13) obtained by assuming that the displacement of a plate is angle independent and, therefore, only zero order Bessel functions are involved in the solution. Substituting this into expression for dynamic structural stiffness (10) and combining with Equation (12) gives the relationship for sensor’s admittance at the sensor’s terminals. After some adjustments, this equation was used to predict the response of pristine circular plate specimens. The comparison between calculated and experimental results is presented on Figure 5 and in Table 1. 5.
SYSTEMATIC CIRCULAR-PLATE EXPERIMENTS
Systematic representation of one experimental set is shown at Figure 6. Each of five sets contains one pristine circular plate and plates with cracks placed at increasing distance from the sensor. In our study, a 10-mm circumferential slit EDM was used to simulate an in-service crack. (Figure 4b) During the experiments, the specimens were supported on packing foam to simulate free conditions as in previous case. Group 1 PZT wafer
Group 2 Crack
PZT wafer
Group 3 PZT wafer
Crack
Group 4 PZT wafer
Crack
Crack
Group 5 7-mm dia. PZT wafer
10mm
Figure 6
40mm
25mm
10mm
3mm
100-mm dia. 0.8-mm thin circular plate
Systematic study of circular plates with simulated cracks (slits) at increasing distance from the E/M impedance sensor.
The experiments were conducted over three frequency bands: 10-40 kHz; 10150 kHz, and 300-450 kHz. The data was process by displaying the real part of the E/M impedance spectrum, and determining a damage metric to quantify the difference between two spectra. Several damage metrics were tried: root mean square (RMS) deviation; mean absolute percentage deviation; covariance change; correlation coefficient (R2) deviation. Figure 7 shows data in the 10-40 kHz band. The superposed spectra of groups 1 and 5 specimens (extreme situations) are shown in Figure 7a, while those of groups 4 and 5 (almost similar situations) are shown in Figure 7b. Figure 7a indicates that the presence of the crack in the close proximity of the sensor drastically modifies the point-wise frequency response function, and hence the real part of the E/M impedance spectrum. Resonant frequency shifts and the appearance of new resonances are noticed. In contrast, the presence of the crack in the far field only marginally modifies the frequency spectrum (Figure 7b). Pristine
1000 100 10
Figure 7
1000 100 10
10
(a)
Pristine Damaged
10000
Damaged
Re Z, Ohms
Re Z, Ohms
10000
15
20 25 30 Frequency, kHz
35
40
10
(b)
15
20 25 30 Frequency, kHz
35
40
E/M impedance results in the 10—40 kHz band: (a) superposed groups 1 & 5 spectra; (b) superposed groups 4 & 5 spectra.
Pristine
100 10 1
1000 100 10 1
300
350 400 Frequency, kHz
(a) Figure 8
Pristine Damaged
10000
Damaged
1000
Re Z, Ohms
Re Z, Ohms
10000
450
300
350 400 Frequency, kHz
(b)
450
E/M impedance results in the 300—450 kHz band: (a) superposed groups 1 & 5 spectra; (b) superposed groups 4 & 5 spectra.
100.0% 83.3%
120 1-Cor. Coeff., %
74.61
80
(1-Cor. Coeff.) 3, %
10-40 kHz band
99.53
83.74
53.12 40
(a)
53.5%
26.6% 25.0% 0.0% 3
10
25 Crack distance, mm
40
51.2%
50.0%
0.01%
5 0 3
300-450 kHz band
75.0%
50
10
25
40
50
Crack distance, mm
(b)
Figure 9 Damage metric variations with the distance between the crack and the sensor: (a) the 10—40 kHz band; (b) the 300—450 kHz band 2
Figure 9a presents the plot of the correlation coefficient deviation, (1-R ). The (1R2) damage metric tends to decrease as the crack moves away from the sensor. However, in this frequency band, the decrease tendency is not uniform. When the higher frequency band 300-450 kHz is examined, the damage metric decreases much more uniformly with crack distance (Figure 9b). We conclude that: a. The crack presence dramatically modifies the pointwise frequency response function, and hence the real part of the E/M impedance spectrum b. This modification decreases as the distance between the sensor and the crack increases However, in order to obtain consistent results, the proper frequency band (usually in high kHz) and the appropriate damage metric must be used. Further work is needed on systematically investigating the most appropriate damage metric to be used for successful processing of the frequency spectra. 6.
DISCUSSION AND CONCLUDING REMARKS
The application of E/M impedance method for damage detection in circular plates was considered in this paper. The quasi-static approach previously presented by Liang et al. (1994) was extended and derived mathematical expressions accounted for the dynamic response of both the sensor and the structure. Positive results obtained previously for one-dimension structure are extended into twodimensions by considering axial symmetry of a problem. Moreover, this paper for the first time presents brief derivation of the expression for E/M admittance of PZT sensor disk installed on the circular plate. The analytical model, developed for 2-D structures, accounts for flexural vibrations only and predicts the E/M impedance response, as it would be measured at the piezoelectric active sensor’s terminals. Experiments were conducted on circular plate specimens in support of the theoretical investigation. The piezoelectric strain sensor of circular shape was
attached in the middle of each specimen. Since strains sensors are sensitive to changing in curvature, the vibration modes, which have maximum curvature at the middle of a plate, should be most successfully determined. In this case, the experimental set up gives us results for zero nodal diameters, n = 0, condition. This assumption perfectly match with experimental and calculated results presented in Table 1. According to the Table 1, the percentage of error between calculated and experimental values is around 1% or less. From another hand, any misalignment of a sensor relative to the center of a plate will give additional frequencies in the spectrum due to presence of other harmonics described with higher order Bessel functions. These harmonics, however, have little effect on frequency spectrum when the sensor is properly aligned. In this case, the solution should be presented in terms of Bessel functions of zero order, which significantly contributes to the spectrum, as it is shown on Figure 5. Therefore, we can conclude that good matching was obtained for calculated and experimental E/M impedance signatures. Five sets of thin plate specimens with five plates in each were designed to study sensor’s sensitivity to the presence of structural damage. It was found that the crack presence dramatically modifies the E/M impedance spectrum and this modification decreases as the distance between the sensor and the crack increases. The consistent results were obtained at high frequency range. Several damage metrics were investigated: root mean square (RMS) deviation; mean absolute percentage deviation; covariance change; correlation coefficient (R2) deviation. For this selection, we found the correlation coefficient (1-R2) to be the most successful damage metric. However, this metric is not without disadvantages and further investigation on the most appropriate damage metric to be used for successful processing of the frequency spectra is needed. The work reported in this paper has demonstrated the ability of permanently attached PZT active sensors to perform structural identification of thin circular plates through the E/M impedance spectrum. It was shown that crack, modeled as a slit, significantly modifies the E/M impedance spectrum. The present method can be a useful and reliable tool for automatic on-line structural identification in the ultrasonic frequencies range. In comparison with traditional modal analysis methods used for structural identification, the E/M impedance method does not require separate instrumentation for excitation (e.g., impact hammer) and recording of structural response (e.g., accelerometers). Therefore, it presents the following advantages: small size of the permanently attached or embedded piezoelectric sensors, ultrasonic frequency range application, and ability to be used for on-line and in service structural health monitoring. ACKNOWLEDGMENTS The financial support of Department of Energy through the Sandia National Laboratories, contract doc. # BF 0133 is thankfully acknowledged. Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.
7.
REFERENCES
Airey, J., (1911) “The Vibration of Circular Plates and their Relation to Bessel Functions”, Proc. Phys. Soc., London, Vol. 23, 1911, pp.225-232. Ayres, T., Chaudhry Z., and Rogers C., (1996) "Localized Health Monitoring of Civil Infrastructure via Piezoelectric Actuator/Sensor Patches," Proceedings, SPIE’s 1996 Symposium on Smart Structures and Integrated Systems, SPIE Vol. 2719, pp. 123-131. Bickford, W. B. (1998) “Advanced Mechanics of Materials”, Addison Wesley Longman, Inc. Chaudhry, Z., Sun, F. P, and Rogers C. A., (1994) "Health Monitoring of Space Structures Using Impedance Measurements," Fifth International Conference on Adaptive Structures, Sendai, Japan, 5-7 December, 1994; pp. 584-591. Chaudhry, Z., Joseph, T., Sun, F., and Rogers, C., (1995). "Local-Area Health Monitoring of Aircraft via Piezoelectric Actuator/Sensor Patches," Proceedings, SPIE North American Conference on Smart Structures and Materials, San Diego, CA, 26 Feb. - 3 March, 1995; Vol. 2443, pp. 268-276. Colwell, R.C., (1936) “The Vacuum Tube Oscillator for Membranes and Plates”, Journal of Acoustical Society of America, Vol. 7, 1936, pp.228-230. Giurgiutiu, V., Reynolds, A., and Rogers, C. A., (1998), “Experimental Investigation of E/M Impedance Health Monitoring of Spot-Welded Structural Joints” submitted for publication to the Journal of Intelligent Material Systems and Structures, July 1998. Giurgiutiu, V and Zagrai, A. (2000) “Damage Detection in Simulated Aging-aircraft Panels Using the Electro-mechanical Impedance Technique”, Adaptive Structures and Material Systems Symposium, ASME Winter Annual Meeting, Nov. 5-10, 2000, Orlando, Florida Giurgiutiu, V and Zagrai, A.N. (2001) “Characterization of Piezoelectric Wafer Active Sensors”, Submitted to: Journal of Intelligent Material Systems and Structures, February 2001 Giurgiutiu, V., Zagrai, A., (2001a), “Embedded Self-Sensing Piezoelectric Active Sensors for OnLine Structural Identification”, Submitted to: Transactions of ASME, Journal of Vibration and Acoustics, January 2001. Harris, C., M., (1996) “Shock and Vibration Handbook”, McGraw-Hill, USA, 1996. IEEE Std. 176 (1987), IEEE Standard on Piezoelectricity, The Institute of Electrical and Electronics Engineers, Inc., 1987 Itao, K., Crandall, S.H. (1979) “Natural Modes and Natural Frequencies of Uniform, Circular, FreeEdge Plates”, Journal of Applied Mechanics, Vol. 46, 1979, pp. 448-453. Kunukkasseril, V.X., Swamidas, A.S.J. (1974) “Vibration of Continuous Circular Plates”, International Journal of Solids Structures, Vol. 10, 1974, pp. 603-619. Liang, C., Sun, F. P., and Rogers C. A., (1994) “Coupled Electro-Mechanical Analysis of Adaptive Material System-Determination of the Actuator Power Consumption and System energy Transfer”, Journal of Intelligent Material Systems and Structures, Vol. 5, January 1994, pp. 1220 Liessa, A. (1969) “Vibration of Plates”, Published for the Acoustical Society of America through the American Institute of Physics, Reprinted in 1993 Onoe, M., Jumonji, H. (1967) “Useful formulas for piezoelectric ceramic resonators and their application to Measurement of parameters”, IRE, Number 4, Part 2, 1967 Pugachev, S. I, Ganapol’sky, V. V., Kasatkin, B. A., Legusha, F. F., Prud’ko, N. I. (1984) Handbook “Piezoceramic Transducers”, Sudostroenie, St.- Petersburg (In Russian) Rao, J.S. (1999) “Dynamics of Plates”. Marcel Dekker, Inc., Narosa Publishing House, 1999 Soedel, W. (1993) “Vibrations of Plates and Shells”, Marcel Dekker, Inc., 1993 Sun, F. P., Liang C., and Rogers, C. A., (1994) “Experimental Modal Testing Using Piezoceramic Patches as Collocated Sensors-Actuators”, Proceeding of the 1994 SEM Spring Conference & Exhibits, Baltimore, MI, June 6-8, 1994. Wah, T., (1962) “Vibration of Circular Plates”, Journal of Acoustical Society of America, Vol. 34, 1962, pp.275-281.
