The development of the smart structures and MEMS

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1 95 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 4, july 999 lectro-lastic Characteristics of Asymmetric Rectangular Piezoelectric Laminae Shuo Hung Chang, Member, I, and Yi Chung Tung Abstract The electro-elastic characteristics of clamped rectangular piezoelectric laminated plates were analytically investigated. A fully covered electrode piezoelectric layer was laminated on an elastic layer to form a nonsymmetrical laminated plate. Using the electro-elastic theory with the Kirchhoff-Love hypothesis, formulation of analyses for mechanical, electrical, and electromechanical characteristics of the laminae are presented. Numerical analysis was carried out using the extended Kantorovich method to yield eigenvalues and eigenfunctions. Theoretical predictions of dynamic characteristics were validated by comparing results with finite element analysis data. The calculated natural frequencies are presented in easy-to-use figures that are useful for sensor and actuator design in microelectromechanical systems MMS. electro-mechanical analysis. We started from the electroelastic theory with Kirchhoff-Love hypothesis, and equations for the generalized piezoelectric/elastic laminae were derived. The governing differential equations of twolayered rectangular laminae were formulated for the bending vibrations. Its solutions were obtained by an iterative scheme using the extended Kantorovich method. Finally, analysis using the finite element method FM was conducted to verify the theoretical results. II. lectro-lastic Theory I. Introduction The development of the smart structures and MMS draws much attention to the use of piezoelectric actuators and sensors. Applications of smart structures can be found including control of vibration [], control of shape [], optical scanner [3], and precision positioning mechanisms [4]. Devices fabricated by MMS include square piezoelectric microphones [5], rectangular monograph piezoelectric sonar transducers [6], and piezoelectric flexors [7]. By nature of the MMS lithography process, the asymmetric rectangular piezoelectric/elastic laminae are commonly seen in sensors and actuators of MMS devices. The laminated piezoelectric or piezofilm structures were investigated for static solution [8], constituent equation [9], torsion and bending solution [], extension and bending [], and torsion coupled with bending []. The thin piezofilms were also used for vibration study [], [], [3], and [4]. The in-plane strains of piezoelectric actuators were used to induce bending of rectangular plate [3], shell [5], multilayer shell [], circular plates [6], and disk [7], [8] structures. The study in [3] is for symmetric rectangular piezoelectric/elastic laminated plates. lectro-elastic behavior of the asymmetric rectangular piezoelectric laminated plate was not investigated. This paper presents a theoretical formulation of the rectangular asymmetric piezoelectric laminated plates for Manuscript received June 9, 998; accepted January 5, 999. The authors acknowledge the support of this work from the National Science Council, Taiwan, Republic of China, under Grant no. NSC The authors are with the Department of Mechanical ngineering, National Taiwan University, Taiwan, Republic of China shchang@ccms.ntu.edu.tw. Fig. shows the nonsymmetric rectangular piezoelectric laminae with a dimension of a b that includes a layer of elastic material of thickness h and a piezoelectric layer of thickness h h. The piezoelectric layer is polarized in the thickness direction, and its major surfaces are completely covered with electrodes with negligible thickness. When the electrodes are connected to an electrical alternating voltage of magnitude V and circular frequency ω, the in-plane extension of the piezoelectric layer excites the bending vibration of the laminae. In this section, we use the mutually perpendicular coordinates α and α on the middle surface; the third coordinate α 3 is normal to the middle surface. The complete system of equations includes the following [9]: equilibrium equation: A i T ii α i A j T ij α j k j T ii T jj a k i T ij T i3 ρü i and α 3 T 3 T 3 k T 3 b A α A α k T 3 T 33 α 3 ρü 3 A i α 3, A 3, k i A i A j A i α j and i, j,, i j; 885 3/99$. c 999 I Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

2 chang and tung: electro-elastic characteristics of piezoelectric laminae 95 Hooke s law for the elastic layer: T ii v T ij S ii v S jj v v T 33, 4a v S ij, 4b S 33 [T 33 v T T ], and S i3 v T i3 ; and 4c 4d electro-static equation: D, ψ. 5 Fig.. Top view and side view of the clamped piezoelectric/elastic rectangular laminae. strain-displacement relationship: S ii A i u i α i k i u j, S 3 u 3, α 3 S ij A i A j α j ui A i S i3 u i α 3 A i u 3 α i ; A j A i α i uj A i, and a b c d In the just mentioned equations, A i is the Lamé parameter; T ij, S ij,andu i are the components of stress, strain, and displacement, respectively; s ij, d ij, andε T ij are the elastic compliances at a constant electric field, the piezoelectric constants, and dielectric permittivities at a constant stress of piezoelectric material, respectively; D i and i are the components of electric displacement and electric field, respectively; ψ is the electric potential; and v are the Young s modulus and Poisson s ratio of the elastic layer, respectively; and ρ is the material density. The equation with subscripts i and j contains two equations, one for i and j and another for i and j. The Kirchhoff-Love hypothesis is applied here for a the transverse stress T 33 can be neglected as compared with the principle stresses T and T and b the transverse displacement u 3 is a constant along the plate thickness, and displacements u i vary linearly along the plate thickness, i.e.,: T 33 T,T, u 3 u 3, and u i u i α 3 u i u u 3 i α 3 A i α i 6a 6b 6c superscript n denotes the order of the nth power with respect to the plate thickness. Substituting 6b and 6c into b and d yields: S ii Sii α 3 Sii and S ij Sij α 3 Sij 7a 7b constitutive equations for the piezoelectric layer: S ii s T ii s T jj s 3T 33 d 3 3, S 33 s 3T T s 33T 33 d 33 3, S ij s 66T ij, S i3 s 44T i3 d 5 i, D i d 5 T i3 ε T ii i, and D 3 d 3 T T d 33 T 33 ε T 33 3 ; 3a 3b 3c 3d 3e 3f membrane strains: S u u A A α A α S A u α u S A A α u A A A α A A, 7c, and 7d α u A ;and 7e Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

3 95 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 4, july 999 bending strains: S u u A A α A α, 7f, and 7g S u u A A α A α S A u A u. 7h A α A A α A The electric potential is assumed to vary quadratically along the plate thickness as: ψ ψ α 3 ψ α 3 ψ, 8 and the electrical displacement D 3 is constant with respect to the plate thickness as: D 3 D3. 9 The electrical boundary conditions for the piezoelectric layer with fully covered electrodes are: ψ α3h h n V; ψ α3h h n V. From the electrical boundary conditions and 8 and 9, we have: ψ V h h h n ψ and h h 3 V d 3 h h ε T h h h n T T, 33 3 d 3 ε T T T. 33 Substituting 6b into 3a through 3c yields stress-strain relations for the piezoelectric layer: T s v S v S Bh h h n s v B S S d 3 s v h h V, T s v S v S Bh h h n s v B S S d 3 3a 3b s v h h V, T s v S, 3c T [ B s v B Bv S v B Bv S], 3d T [ s v B v B Bv S BBv S], and 3e T s v S 3f v s d 3 s, B ε T 33 s v, and s 66 s s. 3g Similarly, the stress-strain relations for the elastic layer can be formulated as: T v S v S, 4a T v S v S, 4b T v S, 4c T v S v S, 4d T v S v S, and 4e T v S. 4f The membrane forces N ij and the bending moments M ij are defined by the following integrals: N,N,N M,M,M h h n h n T,S,T,S,T,S dα 3 h h n T,P,T,P,T,P dα 3, and h h n 5a h h n h n T,S,T,S,T,S α 3 dα 3 h h n h h n T,P,T,P,T,P α 3 dα 3 5b subscripts s and p denote the quantities for the elastic layer and piezoelectric layer, respectively. After integration over the plate thickness in 5 and using 3 through 4, the global constitutive relations can be expressed in the matrix form as: N N N M M M A A B B A A B B A 33 B 33 B B D D B B D D B 33 D 33 S S S S S S V V V 4 V 6 V 4 Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

4 chang and tung: electro-elastic characteristics of piezoelectric laminae 953 A h v h h s v, A v h v v h h s v, A 33 h v h h s v, B h h h n v h h h h h n s v, 7a 7b 7c 7d B v h h h n v v h h h h h n s v, 7e B 33 h h h n 4 v h h h h h n 4s v, 7f D h h 3h h n 3h n h h 3 v s v B { 4[h hh h 3h h h n 3h n] 7g B Bv 3Bh h h n v }, D v h h 3h h n 3h n h h 3 v s v B { 4[h hh h 3h h h n 3h n] 7h v B Bv 3Bh h h n v }, D 33 h h 3h h n 3h n 6 v h h [h hh h 3h h h n 3h n] 6s v, 7i d 3 V s v, and 7j V 4 d 3h h h n s v h h h n V. 7k It is noted that the piezoelectric effect only exists in components D ij and V ij. The coordinate h n in Fig. is to define the reference plane. Without the loss of validity, it can be arbitrarily selected such that B. From 7, we deduce the following: h n s vh vh h s v h v h h, 8a A A A 33, 8b D D D 33, and, 8c B 33 B v v h h h n 4 v. 8d For further reduction of the complexity of 6, these papers [5] [8] introduced the relationship v v into 8d, which yields the removal of B 33 and B. However, very often the Poisson s ratio of the piezoelectric materials is quite different from the elastic materials. In this study, we do not follow this approach, and errors induced by this approximation are evaluated in Section V. The equilibrium equations and may be treated the same way as in the classical plate theory. One is to integrate the equations over the thickness coordinate, taking into account the Kirchhoff-Love hypothesis. The other one is to multiply the equations by the thickness coordinate and to integrate over the plate thickness. For the sought for modes of vibration of the thin piezoelectric laminae, we neglected the effects of the rotary inertia []. The deviation is rather tedious and is omitted here. The equations of the piezoelectric/elastic laminae are: N N A α A α A N N N A X, A A α A A α 9a N N A α A α A N N N A X, and A A α A A α 9b A Q 3 Q 3 α A α A A X Q 3 A α Q 3 A α X 3 9c ρü dα 3 R ü, 9d α 3 X ρü dα 3 R ü, 9e α 3 X 3 ρü 3 dα 3 R ü 3, 9f α 3 M Q 3 ρt 3 dα 3 M A α A α α 3 A M M M A R ü A A α A A α, 9g M Q 3 ρt 3 dα 3 M A α A α α 3 A M M M A R ü A A α A A α, 9h R h h n h n ρdα 3 ρ S h ρ P h h, and 9i h h n ρ S R ρα 3 dα 3 h h h n h n ρ P h h h h h n. 9j Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

5 954 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 4, july 999 III. Rectangular Piezoelectric/lastic Laminae For the rectangular piezoelectric/elastic laminated plate considered in Fig., the corresponding coordinate relations are: α x, α y, α 3 z, A A and A A A A. α α α α Substituting these coordinate relations and the straindisplacement relations into 9a through 9c yields the governing equations: N xx x N yy Q xz x N xy R ü x, a N xy x R ü y, b Q yz Q xz M xx x Q yz M yy R ü z, c M xy R ü x, and d M xy x R ü y. e Here, we assume the piezoelectric ceramic subjects to the time harmonic electric voltage at a circular frequency ω. The displacements are also time harmonic with the same frequency: u x W x x, ye iωt, u y W y x, ye iωt, and u z W x, ye iωt W x, W y,andw are the amplitude of the displacements in the x, y, andzcoordinates, respectively. Substituting the displacement quantities into and taking into account 6, 9, and yields the governing differential equations as: A x A 33 R ω W x A A 33 W y x a A A 33 W x x A 33 x A R ω W y, and b R ω Wx x W y 4 [D x 4 4 ] x 4 4 R ω W. c quations a and b represent the in-plane membrane vibrations. According to c, the in-plane membrane vibration is no longer decoupled from the bending vibration, which is due to the two-layered lamina that has no plane of symmetry in the thickness coordinate. In our study, the laminae thickness is far smaller than the other dimensions. Quantity R in 9j has the quadratic order term of the thickness and is much smaller compared with the linear term R in 9i. Therefore, we neglect the R term in c and solve the bending vibration problem. Mechanical boundary conditions for the clamped edges are: W, W, W x atx±afor b y b and 3a W aty±bfor a x a. 3b IV. The xtended Kantorovich Method The extended Kantorovich method [] was successfully used to solve the bending vibration of homogeneous plates [] and the vibration of orthotropic plates [3]. The method has the advantage of providing solutions independent of the initial choices and has rapid convergence to yield accurate eigenvalues and eigenfunctions. Therefore, it was applied here to solve c for the bending vibration of the piezoelectric laminae. The extended Kantorovich method starts with the separation of the displacement as in []: W x, y f xg y 4a g y y b 4b satisfied the boundary conditions in 3b, and the function f x will be found. Using the Galerkin method and substituting 4 into c, the iterative scheme can be established. During the derivations, the solution in each iteration was formulated for symmetric and antisymmetric modes of vibration, respectively. Using the initial iterative equation 4b, the function f x can be derived by solving the fourth-order ordinary partial differential equation obtained in the procedure of Galerkin method. Depending on the symmetric and antisymmetric modes, f x can be expressed as 5a and 5b, respectively []: x f x cosq cosh p a f x sinq sinh x p a coshp cos sinhp sin x q a q x a and 5a 5b coefficient p, q, p,andq satisfy the boundary conditions. Then, after another iteration, the function g y can also be formulated into the similar form of 5. However, the tedious procedure can be found in [] and is omitted here. The resonant frequencies can be found from the derived frequency equations. Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

6 chang and tung: electro-elastic characteristics of piezoelectric laminae 955 V. Numerical Analysis Introducing the frequency parameter F, the frequency factor λ, and the thickness ratio r as: F ωa R D, λ D R, r h h 6a D D h 3 r[r 3rh n 3h n ] 3 v r s v B { 4[ r r 3 rh n 3h n ] B Bv 3B r h n v } 6b and R R h ρ Sr ρ P r, h n h n h, the resonant frequency of the clamped rectangular piezoelectric/elastic laminae is: f Fλ h πa. 7 Taking into account the piezoelectric stress-strain relationship 3a and 3b and neglecting the membrane vibration, the electric displacement D 3 in 3f and 9 can be found as: [ D 3 ε T d ] 3 33 ε T 33 s v 3. 8 The electric current can be found as: I iω D 3 ds e S e a iωε T 33 B b a b Bh h h n d 3 B [ V h h W x ] W dydx, 9 and the antiresonant frequencies can be determined by solving: I. 3 Then, the dynamic electromechanical coupling coefficient MCC K d is found by using the formula [4]: kd f a fr fa 3 f a and f r denote the antiresonant and resonant frequencies, respectively. By substituting 6 into 9, the electric current at different frequencies can be calculated. Consequently, Fig.. Frequency parameter, ωa R /D, versus aspect ratio of a/b. the resonant and antiresonant frequencies of the piezoelectric/elastic laminae can be identified. In this study, the commercial software Mathematica Wolfram Research, Champaign, IL was used to handle the numerical computations. Table I lists the calculated coefficients in 5 in the first four interactions and illustrates the rapid convergence of the iteration scheme. Frequency parameter F is listed for modes of vibration in terms of l, m order. Table II lists the calculated frequency parameters for various aspect ratios a/b for the first six vibration modes. The modes corresponding to,,, 3, and 3, are symmetric, and modes,,,, and, are antisymmetric, with respect to the middle lines of the plate. The results of frequency parameter versus various aspect ratios were also shown in Fig. for easier identification of the transition points of the vibration modes. For example, the transition point for the, 3 and, modes occurs near the aspect ratio of.66, and is near.8 for the, 3 and, modes. The calculated frequency factors versus the thickness ratio for four piezoelectric materials laminated with elastic materials SiO, Al, and Si 3 N 4 are plotted in Figs. 3a through c, respectively. These elastic materials are popularly used in the MMS applications; material properties are listed in Table III. Material properties for piezoelectric materials were taken from [3]. Table II, Fig., and Fig. 3a through c provide an easy-to-use design tool for calculation of the resonant frequencies by using 7. To evaluate the errors of the approximation of v v further, we repeated the numerical computation by assuming the elastic layer has the same Poisson s ratio as the piezoelectric material. The calculated errors based on the frequency factor for v v are shown in Fig. 4a through c. rrors increase for the thicker elastic layers. The errors depend on the composition of the laminae and materials, ranging from to 5.4%. ffect of the piezoelectric material can be evaluated by introducing piezoelectric constant d 3. In this case, Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

7 956 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 4, july 999 TABL I The Convergence of Iteration for Coefficients p n, q n, p n, q n, and frequency parameters F. n 3 4 F, n,n p n q n F, n,n p n q n F, n,n p n q n F, n,n p n q n F, n,n p n q n F, n,n p n q n F, 3 n,n p n q n F, 3 n,n p n q n TABL II Frequency Parameters ωa R /D, for the First Six Vibration Modes at Various Aspect Ratios. Frequency parameter ωa R /D Ratio a/b F, F, F, F, F, 3 F 3, the piezoelectric material dependence frequency factor λ was computed with and without piezoelectric effects for various widths of the laminae. Fig. 5 shows the results of a PbZr.54 Ti.46 O 3 /SiO laminae. The piezoelectric effects increase the frequency factor by a maximum factor of 8%. We also studied the sensitivity of the piezoelectric material properties, including the piezoelectric constant, elastic compliance, dielectric permittivity, and density. ach of these material properties of PbZr.54 Ti.46 O 3 was changed from.5 to.5 times, and resonant frequencies were calculated. The results of change of resonant frequency of the clamped square PbZr.54 Ti.46 O 3 /SiO laminated plate are shown in Fig. 6. It is found that the density and the elastic compliance s are the dominate sensitive factors. VI. Validation of the Model The numerical results described in the previous section were obtained with the approximation that the in-plane vibration was decoupled from the bending vibration. The validity of the approximation approach needs to be evaluated. In this study, the FM was employed to verify the previously mentioned results. A FM software ABAQUS Swanson Analysis Systems, Inc., Houston, PA was used to solve the transverse vibration. Because the thickness of the laminae is much smaller than the other dimensions, the two-dimensional two-layered composite shell elements S8R5 were selected to construct the FM model. The S8R5 shell element in ABAQUS represents the 8-node reduced Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

8 chang and tung: electro-elastic characteristics of piezoelectric laminae 957 a a b b c c Fig. 3. Dependence of the thickness ratio on the frequency factor for four piezoelectric materials laminated with a SiO, b Al, and c Si 3 N 4. Fig. 4. rrors of frequency factor λ induced by v v for four piezoelectric materials laminated with a SiO, b Al, and c Si 3 N 4. The errors were based on the frequency factor λ for v v. Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

9 958 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 4, july 999 TABL III Material Properties for Piezoelectric and lastic Materials. Piezoelectric material Material constants PbZr.54 Ti.46 O 3 PZT4 PZT5 BaTiO 3 S, m /N S, d 3, C/N ε T 33 /ε ρ, 3 kg/m lastic material Material constants SiO Al Si 3 N 4, GPa v ρ, 3 kg/m TABL IV Material Properties for PbZr.54 Ti.46 O 3 Used in ABAQUS. Material constants PbZr.54 Ti.46 O 3 S, m /N.6 S33, 4.8 S, 3.33 S3, 4.97 S44, 45. S66, 9.9 d 33, C/N 5 d 3, 6. d 5, 44 ε T /ε 54 ε T 33 /ε 45 ρ, 3 kg/m Fig. 7. Comparison of resonant frequencies obtained by theory and FM versus the width of the clamped square SiO plate. Fig. 5. Influence of piezoelectric effects on frequency factor for PbZr.54 Ti.46 O 3 laminated with SiO. integration element and has 3 degrees of freedom dof in translation and dof in rotation at each node. A mesh of elements in the major plane and five divisions in thickness was created. We chose the elastic layer of SiO and the piezoelectric material of PbZr.54 Ti.46 O 3 for the laminae. The material properties of SiO in Table III and piezoelectric properties in Table IV were used in ABAQUS. Fig. 6. Change of resonant frequency versus variation of the piezoelectric material properties for clamped square PbZr.54 Ti.46 O 3 /SiO laminae. The validity of the theoretical model was first verified by introducing the piezoelectric layer thickness as i.e., the laminae were simply purely isotropic elastic plates. In this case, we chose a -µm thick square SiO plate. The theoretical and FM results of resonant frequencies were compared. The errors, based on the FM data, versus the width of the square plate for the first five vibration modes are plotted in Fig. 7, and the errors are within.5%. We proceeded with the analysis for a layer thickness of µm for both the PbZr.54 Ti.46 O 3 and SiO at various widths of the square laminae. The calculated resonant frequencies obtained by both theory and FM are listed in Table V. The theory results in a higher resonant frequencies because Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

10 chang and tung: electro-elastic characteristics of piezoelectric laminae 959 TABL V A Comparison of the First Four Resonant Frequencies for a Clamped Square PbZr.54 Ti.46 O 3 /SiO Laminated Plate with ach Layer of Thickness at µm. Width Mode,; Mode,3; µm Item Mode,, Mode, 3, FM Hz Theory Hz rror % FM Hz Theory Hz rror % FM Hz Theory Hz rror % FM Hz Theory Hz rror % FM Hz Theory Hz rror % FM Hz Theory Hz rror % FM Hz Theory Hz rror % FM Hz Theory Hz rror % of the absence of the membrane deformation. The errors within.5% indicate the acceptance of the proposed theory. VII. Conclusions This paper presents a theoretical approach to the study of electromechanical characteristics using twolayered piezoelectric/elastic laminated rectangular plates. The analytical model of the asymmetric rectangular laminae was formulated using the electro-elastic theory and the Kirchhoff-Love hypothesis. The derived governing differential equations were solved using an efficient iterative scheme of the extended Kantorovich method. For the purpose of validation, the developed analytical model was evaluated by assigning the piezoelectric constants vanished. The numerical results were compared with the FM. A good agreement was found. The numerical analysis was conducted to study the laminae using piezoelectric material PbZr.54 Ti.46 O 3 and the elastic materials SiO,Al,andSi 3 N 4. These materials are commonly used in the silicon-based MMS devices. The effects of the variation of the piezoelectric material properties on the resonant frequency were studied and illustrated. The errors within 5.4% induced by simplification of v v were found. The frequency parameter and the frequency factor were numerically calculated. The results are presented in the easy-to-use figures for the design of sensors and actuators in MMS applications. References [] H. S. Tzou and M. Gadre, Theoretical analysis of a multilayered thin shell coupled with piezoelectric shell actuators for distributed vibration controls, J. Sound Vib., vol. 3, no. 3, pp , 989. [] R. C. Batra and X. Q. Liang, The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators, Comput. Structures, vol. 63, no., pp. 3 6, 997. [3] S. H. Chang and Y. C. Tung, A novel design of piezo-driven dual-dimension optical scanning mechanism, Rev. Sci. Instrum., vol. 69, no. 9, pp , Sep [4] S. H. Chang and B. C. Du, A precision piezodriven micropositioner mechanism with large travel range, Rev. Sci. Instrum., vol. 69, no. 4, pp , Apr [5] M. Pedersen, R. Schellin, W. Olthuis, and P. Bergveld, lectroacoustical measurements of silicon microphones on wafer scale, J. Acoust. Soc. Amer., vol., no. 4, pp. 8, 997. [6] J. J. Bernstein, S. L. Finberg, K. Houston, L. C. Niles, H. D. Chen, L.. Cross, K. K. Li, and K. Udayakumer, Micromachined high frequency ferroelectric sonar transducers, I Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, no. 5, pp , Sep [7] J. G. Smits, W. S. Choi, and A. Ballato, Resonances and antiresonances of symmetric and asymmetric cantilevered piezoelectric flexors, I Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, no., pp. 5 58, Mar [8] L. S. Lee and L. Z. Jiang, xact electroelastic analysis of piezoelectric laminae via state space approach, Int. J. Solids Struct., vol. 33, no. 7, pp , 996. [9] J. G. Smits and W. S. Choi, The constituent equations of piezoelectric heterogeneous bimorphs, I Trans. Ultrason., Ferroelect., Freq. Contr., vol. 38, no. 3, pp. 56 7, May 99. [] C. K. Lee and F. C. Moon, Laminated piezopolymer plates for torsion and bending sensors and actuators, J. Acoust. Soc. Amer., vol. 85, no. 6, pp , 989. [] S. H. Chang and C. C. Chou, lectromechanical analysis of an asymmetric piezoelectric/elastic laminate structure: theory and Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.

11 96 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 4, july 999 experiment, I Trans. Ultrason., Ferroelect., Freq. Contr., vol. 46, no., pp , Mar [] R. Chandra and I. Chopra, Structural modeling of composite beams with induced-strain actuators, AIAA Journal, vol. 3, no. 9, pp. 69 7, Sep [3] D. Ricketts, The frequency of flexural vibration of completely free composite piezoelectric polymer plates, J. Acoust. Soc. Amer., vol. 8, no. 3, pp , Sep [4] J. S. Yang, R. C. Batra, and X. Q. Liang, The vibration of a simply supported rectangular elastic plate due to piezoelectric actuators, Int. J. Solids Struct., vol. 33, pp , 996. [5] Z. Chaudhry, F. Lalande, and C. A. Rogers, Modeling of induced strain actuation of shell structures, J. Acoust. Soc. Amer., vol. 97, no. 5, pp , May 995. [6] N. T. Adelman and Y. Stavsky, Flexural-extension behavior of composite piezoelectric circular plates, J. Acoust. Soc. Amer., vol. 67, no. 3, pp. 89 8, Mar. 98. [7] S. I. Rudnitskii, V. M. Sharapov, and N. A. Shul ga, Vibration of a bimorphic disk transducer of the metal-piezoceramic type, Sov. Appl. Mech., vol. 6, no., pp , Oct. 99. [8] Y. B. vseichik, S. I. Rudnitskii, V. M. Sharapov, and N. A. Shul ga, Sensitivity of a metal-piezoceramic bimorph transducer, Sov. Appl. Mech., vol. 6, no., pp. 74 8, Dec. 99. [9] N. N. Rogacheva, Theory of Piezoelectric Shells and Plates. Boca Raton, FL: CRC Press, 994. [] R. D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech., vol. 8, pp. 3 38, 95. [] A. D. Kerr, An extension of the Kantorovich method, Q. Appl. Math., vol. 6, pp. 9 9, 968. [] R. Jones and B. J. Milne, Application of the extended Kantorovich method to the vibration of clamped rectangular plates, J. Sound Vib., vol. 45, pp , 976. [3] M. Dalaei and A. D. Kerr, Natural vibration analysis of clamped rectangular orthotropic plates, J. Sound Vib., vol. 89, no. 3, pp , 996. [4] W. P. Mason, Physical Acoustics Principles and Methods, vol. -Part A, New York, NY: Academic Press, 964. Shuo Hung Chang M 98 received the B.S. in mechanical engineering from National Cheng-Kung University, Taiwan in 974 and the M.S. and Ph.D. degrees in mechanical engineering from the University of Cincinnati, OH in 98 and 985, respectively. He joined IBM Corp., T. J. Watson Research Division, in Yorktown Heights, NY in 984, he worked in the Input/Output Science and Technology Department and was involved in the R&D on advanced computer peripheral devices, such as printer, data storage, and information display. In 99, he joined the faculty of the Department of Mechanical ngineering, National Taiwan University, Taiwan, Republic of China. His research interests have been in fine actuation mechanism and precision machine design. Dr. Chang is a member of the I, ASM, and the Chinese Society of Mechanical ngineers. Yi Chung Tung received the B.S. and M.S. degrees in the Department of Mechanical ngineering, National Taiwan University, Taiwan, Republic of China in 996 and 998, respectively. He is interested in MMS and precision micromechanism design. Authorized licensed use limited to: National Taiwan University. Downloaded on October, 8 at :36 from I Xplore. Restrictions apply.