Correspondence
Vibration Characteristics of Composite
Piezoceramic Plates at Resonant Frequencies:
Experiments and Numerical Calculations
Chi-Hung Huang and Chien-Ching Ma
Abstract—The experimental measurement of the reso-nant frequencies for the piezoceramic material is generally performed by impedance analysis. In this paper, we em-ploy an optical interferometry method called the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) to investigate the vibration characteristics of piezo-ceramic/aluminum laminated plates. The AF-ESPI is a powerful tool for the full-field, noncontact, and real-time measurement method of surface displacement for vibrat-ing bodies. As compared with the conventional film record-ing and optical reconstruction procedures used for holo-graphic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Because the clear fringe patterns measured by the AF-ESPI method will be shown only at resonant frequencies, both the resonant frequencies and corresponding vibration mode shapes are obtained experimentally at the same time. Excellent quality of the interferometric fringe patterns for both the in-plane and out-of-plane vibration mode shapes are demonstrated. Two different configurations of piezoce-ramic/aluminum laminated plates, which exhibit different vibration characteristics because of the polarization direc-tion, are investigated in detail. From experimental results, we find that some of the out-of-plane vibration modes (Type A) with lower resonant frequencies cannot be measured by the impedance analysis; however, all of the vibration modes of piezoceramic/aluminum laminated plates can be obtained by the AF-ESPI method. Finally, the numerical finite element calculations are also performed, and the re-sults are compared with the experimental measurements. Excellent agreements of the resonant frequencies and mode shapes are obtained for both results.
I. Introduction
W
hen acoherent laser beam illuminates a rough surface, reflected wavelets from each point of the rough surface will form a random pattern of laser speckles and fill the entire space. This is known as the speckle effect. The speckle effect is no longer looked upon as only noise but also as an information carrier, which can be used in many applications. To digitize the speckles and to process them directly is the present research of interest, because it would eliminate the use of photographic film. ESPI, which was first proposed by Butters and Leendertz [1] to investigate the out-of-plane vibration of disks, is a full-field, noncontact, and real-time measurement method of defor-mation for structures subjected to various kinds of loadings.Manuscript received June 15, 2000; accepted February 6, 2001. The authors gratefully acknowledge the financial support of this research by the National Science Council (Republic of China) under Grant NSC 89-2212-E-002-016.
C.-H. Huang is with the Department of Mechanical Engineering, Ching Yun Institute of Technology, Chung-Li, Taiwan 320, Republic of China.
C.-C. Ma is with the Department of Mechanical Engineering, Na-tional Taiwan University, Taipei, Taiwan 106, Republic of China (e-mail: [email protected]).
As compared with the conventional film recording and optical reconstruction procedures used for holographic interferometry [2], the interferometric fringe patterns of ESPI can be displayed directly in a video monitor. ESPI was developed by combining the techniques of holographic and speckle interferometry by employing an image hologram configuration and the method of double-exposure holography.
The most common light source for ESPI is a continuous wave (CW) laser because of the lower price and optical power require-ment. When the CW source is employed, time-averaged inter-ferometric fringes are produced for the harmonically vibrating object, which offers a good observation of the vibration mode shape. The disadvantage of this time-averaged method is that the interferometric fringes represent the amplitude, but not the phase, of the vibration. To overcome this limitation, the phase-modulation method, using the reference beam phase-modulation tech-nique, was developed by Løkberg and Hogmoen [3] to determine the relative phase of displacement. Shellabear and Tyrer [4] used ESPI to make three-dimensional vibration measurements. Three different illumination geometries were constructed, and the orthogonal components of vibration amplitude and mode shape were determined. For the purpose of reducing the noise coming from environment, Creath and Slettemoen [5] devel-oped the subtraction method to reduce this noise. The sub-traction method differs from the time-averaged method in that the reference frame is first recorded before vibration and con-tinuously subtracted from the incoming frames after vibration. Doval et al. [6] proposed the additive stroboscopic TV holog-raphy for out-of-plane vibration analysis, which exhibited the enhanced contrast with constant visibility fringes and dynamic phase shifting. In 1996, an AF-ESPI method was proposed by Wang et al. [7] for the out-of-plane vibration measurement. In the AF-ESPI method, the reference frame is recorded in a vibrating state and subtracted from the incoming frame. Con-sequently, it combines the advantages of the time-averaged and subtraction methods, i.e., good visibility and noise reduction.
Since Pierre and Jacques Curie discovered the piezoelectric effect in 1880, there had been a large amount of research and applications addressed in later literatures (Berlincourt et al. [8]; Zelenka [9]). Piezoelectric transducers are widely used in elec-tromechanical sensors, actuators, non-destructive testing de-vices as well as electro-optic modulator, etc. Piezoelectric ef-fect is applied to many engineering applications because it ex-presses the connection between the electrical and mechanical fields. Piezoelectricity describes the phenomenon in which the material generates electric charge when subjected to stress and, conversely, generates strain when the electric field is applied. Although the vibration characteristics of piezoelectric materi-als can be determined by three-dimensional equations of the linear elasticity, the Maxwell equation, and piezoelectric con-stitutive equations (Tiersten [10]), it is difficult to obtain ana-lytical solutions even for a simple geometry. On account of the complication mentioned previously, the variational approxima-tion method and analytical method with certain simplificaapproxima-tion are usually employed to study the vibration problems of piezo-electric transducers. Adelman and Stavsky [11] employed the plate-type theory to investigate the flexural-extensional behav-ior of composite piezoelectric circular plates. To determine the
optimal thickness ratio and influence of the electrode thickness, metal-piezoelectric unimorphs and PZT-5H bimorphs with sim-ply supported conditions are chosen as the numerical examples. Ricketts [12] worked out the natural frequency expression for flexural vibrations of the completely free composite piezoelec-tric polymer plate. The vibration modes were expressed as the type p/q (p, q = 0, 1, 2, . . . ) where p and q are the number of nodal lines in the x1and x2directions. Rudnitskii and Evseichik
[13], [14] studied the vibration of piezoelectric ceramic trans-ducers of bimorph (metal-piezoelectric) types with the same radius and with different radii by the Kirchhoff-Love kinematic hypotheses. The mathematical model is simplified on the as-sumption that the Poisson’s ratios of composed materials are the same. Lee and Jiang [15] used the state space approach to analyze three-dimensional piezoelectric lamina based on the lin-ear theory of piezoelectricity. By determining the transfer ma-trix of a single layer, the analytical approach can be extended for multilayered lamina with interlayer contact and boundary conditions. Rogacheva et al. [16] used the generalized electroe-lasticity with Kirchhoff-Love hypothesis to investigate the dy-namic characteristics for a three-layered piezoelectric/elastic laminate cantilever beam. By different arrangements of applied voltages, axial expansion/contraction, transverse bending, and their coupled motion can be stimulated individually. Chang and Tung [17] employed the similar methodology as mentioned pre-viously to derive the vibration characteristics of two-layered clamped rectangular piezoelectric laminated plates. The nu-merical calculations of frequency parameters were presented for the design of sensors and actuators in microelectromechanical system (MEMS) applications.
The investigation of the dynamic characteristics for piezo-electric materials is a problem of great practical interest. How-ever, very few experimental results, especially for the full-field measurement of mode shapes, are available in the literature. In this paper, we report experimental results of vibration char-acteristics for piezoceramic/aluminum laminated plates based on the AF-ESPI and an impedance analysis. The advantage of using the AF-ESPI method is that both resonant frequen-cies and the corresponding mode shapes can be obtained at the same time. Both in-plane and out-of-plane vibrations are studied and discussed. The fringe patterns shown in the experi-mental results are correspondent to the vibrating mode shapes. Because of the polarization of piezoceramics with respect to elastic layer, two different types of stacking sequences for the three-layered laminated plates are investigated. According to experimental results obtained in this study, it is interesting to note that several modes with lower resonant frequencies for out-of-plane vibrations cannot be measured by impedance anal-ysis. In addition to the AF-ESPI method, numerical computa-tions based on a finite element package are also presented, and good agreements of resonant frequencies and mode shapes are found in comparison with experimental results. For the purpose of self-explanation, several formulas reported in [18] are given here.
II. Theory of AF-ESPI Method and Piezoelectricity
A. Out-of-Plane Vibration
When the specimen vibrates periodically, we record the first image as a reference. The light intensity of this reference
im-age detected by a charge-coupled device (CCD) camera can be expressed as I1= 1 τ
τ 0 IA+ IB+ 2 √ IAIBcos φ +2π λ(1 + cos θ)A cos ωt
dt
(1)
where
IA= the object light intensity,
IB= the reference light intensity,
τ = the CCD refresh time,
φ = the phase difference between object and reference light, λ = the wavelength of laser,
θ = the angle between object light and observation direction, A = the vibration amplitude, and
ω = the angular frequency. Let Γ = 2π
λ (1 + cos θ) and τ = 2mπ
ω ; m is an integer; then, (1) can be worked out as
I1= IA+ IB+ 2
√
IAIB(cos φ)J0(Γ A) (2)
where J0 is a zero-order Bessel function of the first kind.
After image processing and rectifying, the intensity of the first image can be expressed as
I1 = IA+ IB+ 2
√
IAIB
(cos φ)J0(Γ A). (3)As the vibration of the specimen goes on, we assume that the vibration amplitude has changed from A to A+∆A because of the electronic noise or instability of the apparatus. The light intensity of the second image can be represented as
I2= 1 τ
τ 0 IA+ IB+ 2 √ IAIBcos
φ + Γ (A + ∆A) cos ωtdt.
(4)
Expanding (4) using Taylor series expansion, keeping the first two terms and neglecting the higher order terms, we can rewrite (4) as follows I2= IA+ IB+ 2 √ IAIB(cos φ)
1−1 4Γ 2 (∆A)2
J0(Γ A). (5)
By image processing and rectifying, I2 can be similarly
ex-pressed as I2= IA+ IB+ 2 √ IAIB
(cos φ)1−1 4Γ 2 (∆A)2
J0(Γ A) . (6)
When these two images (the first and second images) are subtracted by the image processing system, i.e., subtract (3) from (6), and rectified, the resulting image intensity can be expressed as I = I2− I1 = √ IAIB 2
(cos φ)Γ 2 (∆A)2J0(Γ A). (7)B. In-Plane Vibration
Similar to the out-of-plane vibration case, the first and sec-ond image intensity, i.e., I1 and I2, for in-plane vibration are
expressed as I1= IA+ IB+ 2 √ IAIB
(cos φ)J0(ΓA)and (8) I2= IA+ IB+ 2 √ IAIB (cos φ)1−1 4Γ 2(∆A)2
J0(ΓA) (9) where
IA= IB= the object light intensity,
A= the vibration amplitude of in-plane vibration, Γ= 2π
λ (2 sin θ
), and
θ= half of the angle between two illumination lights. Subtracting (9) from (8) and rectifying by the image pro-cessing system, we can obtain the resulting image intensity as
I = I2− I1 = √ IAIB 2
(cos φ)Γ
2(∆A)2 J0(ΓA). (10)From (7) and (10), it is interesting to note that the fringe pattern for both the out-of-plane and in-plane vibrating mo-tions obtained by AF-ESPI method are controlled by a zero-order Bessel function J0. Ma and Huang [18] provided a
de-tailed discussion of the AF-ESPI method. They also investi-gated the three-dimensional vibration of piezoelectric rectangu-lar parallelepipeds by using the AF-ESPI method. Combining the out-of-plane and in-plane optical setups by the AF-ESPI method, we can construct complete vibration characteristics of the piezoceramic/aluminum laminated plate, including res-onant frequencies and mode shapes at the same time. This is different from the conventional impedance analysis, which has been used widely in determining only the resonant frequency for piezoelectric material.
C. Piezoelectricity
The vibration of piezoelectric material is electroelastic in nature, and it is necessary to include the coupled electrical field with the elastic behavior. In other words, the equation of linear elasticity is coupled to the charge equation of electrostatics by means of the piezoelectric constants. The system of governing equations needed to determine the vibration characteristics of a piezoelectric material was presented in detail by Tiersten [10]. The linear piezoelectric constitute equations are
τij= cEijklskl− ekijEkand
Di= eiklskl+ εsikEk
(11)
where τij, Di, skl, and Ekrepresent the stress, electric
displace-ment, strain, and electric field, respectively, and cE
ijkl, ekij, and
εsik are the elastic, piezoelectric, and dielectric constants,
re-spectively.
Because of the symmetry, the compressed matrix notation is introduced in place of the tensor notation in general. This
matrix notation consists of replacing ij or kl by p or q, where i, j, k, and l take the values 1, 2, 3 and p, q take the values 1 through 6. By virtue of the transformation, we can make the identifications
cEijkl≡ c E
pq, eikl≡ eiq, and τij≡ Tp, (12)
and the constitutive equations (11) can be rewritten as Tp= cEpqSq− ekpEkand
Di= eiqSq+ εsikEk
(13)
where skl = Sq when k = l, q = 1, 2, 3 and 2skl = Sq when
k= l and q = 4, 5, 6.
Because the polarized piezoelectric ceramics have the same symmetry of a hexagonal crystal in class C6v = 6 mm, which
can be modeled as a transversely isotropic material. The elastic, piezoelectric, and dielectric constants, respectively, are repre-sented in matrix forms as
cEij=
cE 11 cE12cE13 0 0 0 cE12 cE11cE13 0 0 0 cE13 cE13cE33 0 0 0 0 0 0 cE44 0 0 0 0 0 0 cE 44 0 0 0 0 0 0 c E 11− c E 12 2
, (14) eip= 0 0 0 0 e 150 0 0 0 e15 0 0 e31 e31e33 0 0 0 , and (15) εsij= εs 11 0 0 0 εs11 0 0 0 εs33 . (16)III. Experimental Results and Numerical Analysis
The piezoceramic/aluminum laminated plates as shown in Fig. 1 are three-layered rectangular plates. The piezoceramics are in upper and lower layers, which are completely coated with silver electrodes at two surfaces and are adhered symmetrically with respect to the aluminum layer. The arrow with symbol
P denotes the polarization direction of the piezoceramic layer.
Owing to the polarization with respect to the middle layer, two types of piezoceramic/aluminum laminated configurations are considered and are named Specimen 1 and Specimen 2, as shown in Fig. 1(a and b, respectively). The material of piezoce-ramic and elastic layers are PIC-151(Germany Physik Instru-ments Company) and 6061T6 aluminum alloy, respectively, and their material properties are listed in Table I.
The schematic layout of self-arranged, time-averaged AF-ESPI optical systems, as shown in Fig. 2 and Fig. 3, is used to perform the out-of-plane and in-plane experimental measure-ments for resonant frequencies and corresponding mode shapes. A 30-mW He-Ne laser with wavelength λ = 632.8 nm is used as the coherent light source. We use a CCD camera (Pulnix Com-pany) and a P360F (Dipix Technologies, Inc.) frame grabber with a digital signal processor onboard to record and process the images. As shown in Fig. 2 for the out-of-plane measure-ment, the laser beam is divided into two parts, the object and reference beam, by a beamsplitter. The object beam travels to the specimen and then reflects to the CCD camera. The refer-ence beam goes directly to the CCD camera via a mirror and
Fig. 1. Geometric dimensions and polarization directions of Specimen 1 (a) and Specimen 2 (b).
TABLE I
Material Properties of the PIC-151 Ceramics and 6061T6 Aluminum Alloy. PIC-151 cE 11(1010N/m2) 10.76 cE 33 10.04 cE 12 6.312 cE 13 6.385 cE 44 1.962 cE 66= (cE11− cE12)
2 2.224 e31(N/Vm) −9.6 e33 15.1 e15 12.0 εs 11 ε0 1110 εs 33 ε0 852 ρ(kg/m3) 7760 ε0= 8.85× 10−12F/m 6061T6 E(109N/m2) 70 ν 0.33 ρ(kg/m3) 2700a reference plate. Note that the optical path length and the light intensities of these two beams are maintained equal in the experimental setup. For the in-plane measurement system, as shown in Fig. 3, two laser beams with the same optical path and light intensity are symmetrically incident to the specimen and then reflect to the CCD camera. The CCD camera con-verts the intensity distribution of the interference pattern of the object into a corresponding video signal at 30 frames/s. The signal is electronically processed and converted into an image on the video monitor. The interpretation of the fringe image is similar to the reading of a contour map. To achieve the sinusoidal output, a digitally controlled function genera-tor HP33120A (Hewlett Packard) connected to a 4005 power amplifier (NF Corporation) is used.
The experimental procedure of the AF-ESPI technique for both the in-plane and the out-of-plane vibrations is performed as follows. First, a reference image is taken after the speci-men vibrates; then, the second image is taken. The reference image is subtracted by the image processing system. If the vi-brating frequency is not the resonant frequency, only random distributed speckles are displayed, and no fringe patterns will be shown. However, if the vibrating frequency is in the neigh-borhood of the resonant frequency, stationary distinct fringe patterns will be observed in the monitor. Then, the function generator is carefully and slowly turned; the number of fringes will increase, and fringe pattern will become clearer as the reso-nant frequency is approached. From the aforementioned exper-imental procedure, the resonant frequencies and the correspon-dent mode shapes can be determined at the same time using the AF-ESPI method.
Based on the experimental results obtained in this study, we find that the resonant frequencies and correspondent mode shapes of Specimen 1 can be classified into two types, named Type A and B modes. Comparing with Type B modes, the res-onant frequencies of Type A modes are much lower than those of Type B modes, and the correspondent mode shapes can be obtained only by the out-of-plane measurement. So, we can conclude that the Type A modes are the out-of-plane vibration modes (or bending modes) and the Type B modes are the in-plane vibration modes (or extensional modes). In addition to the experimental measurement, numerical calculations of reso-nant frequencies as well as mode shapes are also investigated by the commercially available software ABAQUS finite element package [19]. The 20-node three-dimensional solid piezoelectric element (C3D20E) in ABAQUS is selected to analyze the piezo-ceramic layer, and the 20-node three-dimensional solid contin-uum element (C3D20R) is selected for the aluminum layer. Fig. 4 and Fig. 5 show the experimental and numerical results for the first four mode shapes of Type A and the first two mode shapes of Type B, respectively. We indicate the phase of displacements in finite element results as solid or dashed line. The solid lines are in the opposite direction to the dashed lines. The transition from solid lines to dashed lines corresponds to a zero displacement line or a nodal line. The zero-order fringe, which are the brightest fringes on the experimental results, represents the nodal lines of the piezoceramic/aluminum lami-nated plate at resonant frequencies. The rest of the fringes are contours of constant displacement, which can be quantitatively calculated by J0(Γ A) = 0 (or J0(ΓA) = 0) according to (7)
[or (10)] for out-of-plane (or in-plane) measurement. Because Γ = 2π(1 + cos θ)/λ and Γ = 4π sin θ/λ, the sensitivity of the out-of-plane measurement will increase as θ decreases, and the sensitivity of the in-plane measurement will increase as θ
Fig. 2. Schematic diagram of AF-ESPI setup for out-of-plane measurement.
Fig. 5. Mode shapes of Specimen 1 obtained by AF-ESPI and the finite element method (Type B).
increases. Therefore, we choose θ = 10◦and θ= 60◦for the ex-perimental setup. The mode shapes obtained by exex-perimental results can be checked by the nodal lines and fringe patterns with the numerical finite element calculations, and excellent agreements are found.
Because the electrical impedance of the piezoceramic ma-terial drops to a local minimum when it vibrates at a reso-nant frequency, the resoreso-nant frequency can also be determined by the impedance analysis. Herein, it is carried out by using HP4194A impedance/gain-phase analyzer (Hewlett Packard), and the impedance curve for the piezoceramic plate measured from HP4194A is shown in Fig. 6. The local minimums ap-pearing in the impedance curves are the corresponding reso-nant frequencies at resonance. Unexpectedly, we find that only the resonant frequencies of the Type B modes are indicated in Fig. 6, i.e., those of the Type A modes cannot be obtained by the impedance analysis. This phenomenon can be explained by means of the characteristics of piezoelectricity. When the piezo-electric plate vibrates at a resonant frequency, the charge will greatly be induced on the electrode surfaces owing to the vibra-tion deformavibra-tion, named the direct piezoelectric effect, and the impedance will drop to a local minimum value. This is the rea-son that the rerea-sonant frequencies of piezoceramic plates can be determined by using the impedance analyzer. However, if the summation of the induced charge distributed over the electrode surfaces is zero, we won’t be able to find the large variation of impedance at the resonant frequency. Type A modes are just the situation mentioned previously, and we cannot obtain the
Fig. 6. Impedance variation curve of Specimen 1.
resonant frequencies for these modes from the impedance curve shown in Fig. 6. Table II shows the first few resonant frequen-cies of the piezoceramic/aluminum laminated plate obtained by using the AF-ESPI, impedance analysis, and the finite ele-ment method. The discrepancy of resonant frequencies between AF-ESPI and impedance analysis for Type B modes is smaller than that between AF-ESPI and the finite element method. However, the difference between the experimental data and the finite element method may result from the measurement of the
TABLE II
Results of Resonant Frequencies Obtained from AF-ESPI, Impedance Analysis, and the Finite Element Method (FEM) for Specimen 1.
Type A Type B
AF-ESPI Impedance analysis FEM AF-ESPI Impedance analysis FEM
Mode (Hz) (Hz) (Hz) (Hz) (Hz) (Hz)
1 7910 — 7602 77900 77913 75718
2 11830 — 11313 110850 110850 108022
3 19780 — 19154
4 34470 — 33675
TABLE III
Results of Resonant Frequencies Obtained from AF-ESPI, Impedance Analysis and the Finite Element Method (FEM)
for Specimen 2.
AF-ESPI Impedance analysis FEM
Mode (Hz) (Hz) (Hz) 1 14950 14950 14352 2 34350 34113 33498 3 64550 64400 62984 4 81830 81450 80607 5 120270 119225 119896 6 132100 131925 132285
Fig. 8. Impedance variation curve of Specimen 2.
material properties and the faults of specimens that are gener-ated by the manufacturing or adhesion process.
Fig. 7 shows the experimental and numerical results for the first six mode shapes of Specimen 2. We find that mode shapes of Specimen 2 are all out-of-plane vibration modes for an isotropic square plate, which result from the anti-symmetric polarization directions with respect to the middle plane. A com-plete discussion of out-of-plane vibration mode shapes for dif-ferent boundary conditions of the isotropic plate was presented by Huang and Ma [20]. From observing the experimental re-sults, the characteristics of mode shapes for Specimen 2 is that the nodal lines do not pass the center of Specimen 2. This phe-nomenon is completely different from that obtained in Speci-men 1 in which the nodal lines of the mode shapes in Type A (Fig. 4) all pass the center of Specimen 1. It is also inter-esting to note that the mode shapes shown in Fig. 7 are vi-bration modes that are symmetric about coordinate axes and diagonals. Furthermore, the induced in-plane motions by the out-of-plane vibration modes caused by the Poisson’s effect are large enough so that the fringe patterns (see Fig. 7) can also be observed in the in-plane experimental measurements. Ta-ble III shows the resonant frequencies of Specimen 2 obtained by AF-ESPI, impedance analysis, and finite element methods. The impedance variation curve is shown in Fig. 8, and we find that the impedance variations of modes 2, 4, and 5 are much smaller than those of the other modes. This may result from the fact that these modes are stimulated, owing to the geo-metric configuration, and will disappear for the circular plate.
Consequently, the mode shapes of Modes 1, 3, and 6 are sim-ilar the nodal circles as the circular plates. In other words, if we gradually change the configuration from a square plate to a circular plate, then Modes 1, 3, and 6, shown in Fig. 7, will reduce to one, two, and three nodal circles, respectively.
IV. Conclusion
Optical techniques have been shown to have certain advan-tages for vibration analysis, and ESPI has been applied to many vibration problems. The advantages of the optical ESPI method include noncontact, full-field measurement; real-time observa-tion; submicrometer sensitivity; and digital image processing, among others. As compared with the film recording and optical reconstruction procedures used for holographic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system.
It is known that the vibration characteristics of piezoelec-tric materials are important in many engineering applications. Most of the works for vibration analysis of piezoceramic/elastic laminated plates published in the literature are analytical and numerical results. There are only very few experimental re-sults available for the full-field configuration of mode shapes for vibrating plates. In this study, a self-arranged AF-ESPI optical setup with good fringe visibility and noise reduction has been employed to investigate the resonant frequencies and the corresponding mode shapes of free vibrating piezoce-ramic/aluminum laminated plates. The resonant frequencies of piezoceramic/aluminum laminated plates are also determined by impedance analysis. Herein, we classify the vibration modes into two types: the out-of-plane (bending) and in-plane (ex-tensional) modes. Based on the experimental results obtained by the AF-ESPI method, it is noted that the three-layered laminated plates exhibit different vibration characteristics for different stacking sequence. The resonant frequencies of Type A bending modes for Specimen 1 cannot be measured by the impedance analysis, and only the extensional modes are indi-cated in the impedance curve. However, all of the resonant fre-quencies for bending modes of Specimen 2 can be determined by the impedance analysis. In addition, the out-of-plane mode shapes of the first four modes of Specimen 1 and six modes of Specimen 2 are the same as those of isotropic plates. Numeri-cal Numeri-calculations of resonant frequencies and mode shapes based on a finite element package are also performed, and excellent agreements for the mode shapes are obtained when compared with results obtained by AF-ESPI. The results shown in this study demonstrate that the AF-ESPI method is applicable to many situations in vibration analysis for piezoceramic lami-nated plates as long as the amplitude for the in-plane or out-of plane vibration modes reaches the sensitivity out-of AF-ESPI method.
It is concluded that the resonant frequencies for all of the in-plane and part of the out-of-plane vibration modes can be determined by the impedance analysis. However some out-of-plane vibration modes cannot be determined from a similar procedure. Hence, a careful investigation for the vibration mode shapes should be performed to identify the measured resonant frequencies from the impedance analysis.
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