- Email: [email protected]
An exact analysis of forced thickness-twist vibrations of multi-layered piezoelectric plates
Acta Mechanica Solida Sinica, Vol. 20, No. 3, September, 2007 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-007-0725-x
ISSN 0894-9166
AN EXACT ANALYSIS OF FORCED THICKNESS-TWIST VIBRATIONS OF MULTI-LAYERED PIEZOELECTRIC PLATES Hu Hongping1
Chen Ziguang2
Yang Jiashi1,3
Hu Yuantai1 1
1
( Department of Mechanics, School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China) (2 Institute of Mechanics and Sensing Technology, School of Civil Engineering and Architecture, Central South University, Changsha 410082, China) (3 Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588, USA)
Received 20 November 2006; revision received 26 June 2007
ABSTRACT This paper deals with the thickness-twist vibration of a multi-layered rectangular piezoelectric plate of crystals of 6 mm symmetry or polarized ceramics. An exact solution is obtained from the three-dimensional equations of linear piezoelectricity. The solution is useful to the understanding and design of composite piezoelectric devices. A piezoelectric resonator, a piezoelectric transformer, and a piezoelectric generator are analyzed as examples.
KEY WORDS piezoelectricity, plate, resonator, transformer, generator
I. INTRODUCTION Thickness-twist vibrations of crystal and ceramic plates are often used as the operating modes of piezoelectric resonators and acoustic wave sensors, e.g., Refs.[1–6]. Polarized ceramic plates have been used for a long time. Recently, thin film resonators and sensors of crystals with 6 mm symmetry including AlN and ZnO have become the cynosure of interest[7, 8] . AlN and ZnO plates with the six-fold axis normal and in-plane to the plates, or with a tilted six-fold axis are all being developed[9] . When the six-fold axis is in-plane, a 6 mm crystal plate can support thickness-twist modes with electromechanical coupling. Since the material tensors of polarized ceramics that are transversely isotropic have the same structures as those of crystals of 6 mm symmetry[10] , their electromechanical coupling behavior is the same as that of 6 mm crystals. Ceramic plates with in-plane poling can also be driven into thickness-twist vibrations by a thickness-electric field. Piezoelectric devices are also often multilayered[11–13] , which causes considerable complications in modeling. Known exact theoretical results concerning thicknesstwist waves are mostly for infinite plates[5] or semi-infinite wedges[6] . For multilayered infinite plates, exact results of general plate waves were obtained in Ref.[14]. For a real device of a finite size, the operating modes are modes of finite bodies. Owing to anisotropy and piezoelectric coupling, known exact modes of finite piezoelectric bodies are relatively few. Usually, two-dimensional approximate plate equations are used for theoretical analysis of finite plates[15, 16] . Some exact results on finite, laminated piezoelectric plates were obtained for the analysis of smart structures[17–22] where the choice of materials, their orientation and the plate deformation modes are
Corresponding author. Email: [email protected] Project supported by Chinese Postdoctoral Science Foundation (No. 20070410944).
· 212 ·
ACTA MECHANICA SOLIDA SINICA
2007
different from applications in resonator, sensor and other resonant piezoelectric devices. For smart structures usually stress and strain are calculated by static analysis for strength considerations rather than analysis of acoustic wave devices where the frequency and/or wave speed as signals, or output electric voltage or power are of main concern. Recently, exact thickness-twist modes in a single-layered rectangular plate of polarized ceramics or piezoelectric crystals of 6 mm symmetry have been obtained in Ref.[23]. The result of Ref.[23] offers the possibility of obtaining exact solutions in a finite multi-layered plate of 6 mm crystals, which is the purpose of this paper. The structure and the mathematical problem are defined in §II. An exact analysis is performed in §III where a general solution is obtained. A piezoelectric composite resonator, a transformer and a generator are analyzed in §IV respectively as examples. Some conclusions are drawn in §V.
II. GOVERNING EQUATIONS Consider an N -layered piezoelectric plate of total thickness 2h with the x2 axis normal to the plate (see Fig. 1). The x3 axis is determined from the x1 and x2 axes by the right-hand rule. The two plate major surfaces and the N − 1 interfaces are sequentially determined by x2 = −h = h0 , h1 , . . . , hN −1 , and hN = h. Each layer is of a different 6 mm crystal with the x3 axis as the six-fold axis or the poling Fig. 1 Cross-section of a multilayered plate of 6 mm crystals. axis of ceramics. This includes a nonpiezoelectric dielectric layer as a special case when the piezoelectric constant is set to zero. The plate is unbounded in the x3 direction. Figure 1 shows a cross-section. On the two major surfaces of the plate at x2 = ±h are electroded under given shear stress T23 (x1 , t) and potential φ(x1 , t). According to the theory of piezoelectricity the electric potential is a constant on an electrode (at most a function of time). Multiple electrodes are needed to realize the x1 dependence of the electric potential. The two minor surfaces at x1 = ±a are unelectroded and traction-free. Thickness-twist modes are described by[5, 24] u1 = u2 = 0,
u3 = u(x1 , x2 , t),
φ = φ(x1 , x2 , t)
(1)
where ui is the displacement in the xi direction and φ is the electric potential. We introduce a function ψ through[24, 25] eu (2) φ=ψ+ ε where e = e15 and ε = ε11 are the relevant piezoelectric and dielectric constants. Then the governing equations for u and ψ are[24, 25] c¯∇2 u = ρ¨ u, ∇2 ψ = 0 (3) where ∇2 = ∂12 + ∂22 is the two-dimensional Laplacian, c¯ = c44 + e2 /ε, and c44 is the relevant elastic constant. The nonzero stress and electric displacement components are[24, 25] T23 = c¯u,2 + eψ,2 ,
T31 = c¯u,1 + eψ,1 ,
D1 = −εψ,1 ,
D2 = −εψ,2
(4)
where an index after a comma denotes partial differentiation with respect to the coordinate associated with the index. At the top and bottom surfaces the boundary conditions are T23 (x1 ) = T ± (x1 ) exp(iωt), ±
V (x1 ) = V (x1 ) exp(iωt),
x2 = ±h x2 = ±h
(5)
which are sufficient to describe the applications we have in mind. At an interface, we require the continuity of u, T23 , φ and D2 . At the two ends, the boundary conditions are T13 = 0,
D1 = 0,
x1 = ±a
(6)
Vol. 20, No. 3
Hu Hongping et al.: An Exact Analysis of Forced Thickness-Twist Vibrations
· 213 ·
III. AN EXACT SOLUTION Consider the i-th layer, with i = 1, 2, 3, . . . , N . For time-harmonic motions we write u, ψ, φ, T23 and D2 as (u, ψ, φ, T23 , D2 ) = Re{(U, Ψ, Φ, T, D) exp(iωt)} (7) From the modes in Ref.[23] or separation of variables we construct the following solution for the i-th layer: ∞ (i) (i) (i) (i) (i) (i) (i) (i) U = A(0) cos η(0) x2 + B(0) sin η(0) x2 + A(m) cos η(m) x2 + B(m) sin η(m) x2 X(m) (x1 ) m=1
∞ (i) (i) (i) (i) C(m) cosh(ξ(m) x2 ) + D(m) sinh(ξ(m) x2 ) X(m) (x1 ) Ψ = C(0) + D(0) x2 + m=1
(8) where and are undetermined constants, the superscript (i) is for the i-th layer, the subscript (m) is for summation within a layer, and m sin(ξ(m) x1 ), m = 1, 3, 5, · · · π, X(m) (x1 ) = ξ(m) = cos(ξ(m) x1 ), m = 2, 4, 6, · · · 2a (9) 2 ρ(i) ω 2 m 2 (i) π , m = 0, 1, 2, 3, · · · η(m) = − 2a c¯(i) Equation (8) satisfies Eq.(3) and the boundary conditions at x1 = ±a. To apply the interface continuity conditions and the boundary conditions at x2 = ±h, we calculate the following: ∞ (i) e(i) e e(i) (i) (i) (i) (i) (i) (i) (i) (i) A cos η x Φ = (i) A(0) cos η(0) x2 + (i) B(0) sin η(0) x2 + C(0) + D(0) x2 + 2 (m) ε ε ε(i) (m) m=1 e(i) (i) (i) (i) (i) + (i) B(m) sin η(m) x2 + C(m) cosh(ξ(m) x2 ) + D(m) sinh(ξ(m) x2 ) X(m) (x1 ) (10) ε ∞ (i) (i) (i) D = −ε(i) D(0) + −ε(i) C(m) ξ(m) sinh(ξ(m) x2 ) − ε(i) ξ(m) D(m) cosh(ξ(m) x2 ) X(m) (x1 ) (11) (i) A(m) ,
(i) B(m) ,
(i) C(m)
(i) D(m)
m=1
(i)
T = −¯ c
(i) (i) A(0) η(0)
(i) (i) +¯ c(i) B(m) η(m)
sin
(i) η(0) x2
(i)
+ c¯
(i) (i) B(0) η(0)
cos
(i) η(0) x2
+e
(i)
(i) D(0)
+
∞ m=1
(i) (i) (i) −¯ c(i) A(m) η(m) sin η(m) x2
(i) (i) (i) cos η(m) x2 +e(i) C(m) ξ(m) sinh(ξ(m) x2 ) + e(i) D(m) ξ(m) cosh(ξ(m) x2 ) X(m) (x1 ) (12)
At the interface between the i-th and (i + 1)-th layer at x2 = hi (where i = 1, 2, 3, . . . , N − 1), we apply the following continuity conditions + U (h− i ) = U (hi )
(13)
+ Φ(h− i ) = Φ(hi ) + D(h− i ) = D(hi ) + T (h− i ) = T (hi )
Equations (13)-(16) imply that ⎧ (i) ⎫ ⎧ (i+1) ⎫ ⎪ ⎪ A(0) ⎪ A(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (i) ⎪ ⎬ ⎬ ⎨ B (i+1) ⎪ (i) (0) (0) = [T i = 1, 2, 3, · · · , N − 1 ] (i) (0) ⎪ (i+1) ⎪ , ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (i) ⎪ ⎭ ⎭ ⎩ (i+1) ⎪ D(0) D(0) ⎧ (i) ⎫ ⎧ (i+1) ⎫ ⎪ ⎪ A(m) ⎪ A(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (i) ⎪ ⎬ ⎬ ⎨ B (i+1) ⎪ (m) (m) = [T (i) ] , i = 1, 2, 3, · · · , N − 1, (i) (i+1) ⎪ ⎪ C(m) ⎪ C(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (i) ⎪ ⎭ ⎭ ⎩ (i+1) ⎪ D(m) D(m)
(14) (15) (16)
(17)
m = 1, 2, 3, · · ·
(18)
· 214 ·
ACTA MECHANICA SOLIDA SINICA
2007
(i)
where [T(0) ] and [T (i) ] are transfer matrices whose expressions are straightforward from Eqs.(13)-(16) and are too lengthy to be presented here. With Eqs.(17) and (18) we can express all undetermined coefficients in terms of those of the N -th layer. In particular, between the first and the last layers, we have ⎧ (1) ⎫ ⎧ (N ) ⎫ ⎪ ⎪ A(0) ⎪ A(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (1) ⎪ ⎬ ⎬ ⎨ B (N ) ⎪ (1) (2) (N −1) (0) (0) = [T (19) ][T ] · · · [T ] (1) (N ) (0) (0) (0) ⎪ ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (1) ⎪ ⎭ ⎭ ⎩ (N ) ⎪ D(0) D(0) ⎧ (1) ⎫ ⎧ (N ) ⎫ ⎪ ⎪ ⎪ A(m) ⎪ ⎪ ⎪ ⎪ A(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (1) ⎪ ⎬ ⎬ ⎨ B (N ) ⎪ (m) (m) = [T (1) ][T (2) ] · · · [T (N −1) ] (20) (1) (N ) ⎪ ⎪ C(m) ⎪ C(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (1) ⎪ ⎭ ⎭ ⎩ (N ) ⎪ D(m) D(m) We also write the boundary conditions in Eq.(5) in series form: ± T ± (x1 ) = T(0) +
∞ m=1
± T(m) X(m) (x1 ),
± V ± (x1 ) = V(0) +
∞ m=1
± V(m) X(m) (x1 )
(21)
± ± ± ± , T(m) , V(0) and V(m) are known expansion coefficients of the applied load. Then the boundary where T(0) conditions imply that, for m = 0 :
(1) (1) (1) (1) (1) (1) (1) − c¯(1) A(0) η(0) sin η(0) h + c¯(1) B(0) η(0) cos η(0) h + e(1) D(0) = T(0) (N ) (N ) (N ) (N ) (N ) (N ) (N ) + −¯ c(N ) A(0) η(0) sin η(0) h + c¯(N ) B(0) η(0) cos η(0) h + e(N ) D(0) = T(0)
(22) (23)
and for m = 1, 2, 3, . . . : (1) (1) (1) (1) (1) (1) c¯(1) A(m) η(m) sin η(m) h + c¯(1) B(m) η(m) cos η(m) h (1)
(1)
− −e(1) C(m) ξ(m) sinh(ξ(m) h) + e(1) D(m) ξ(m) cosh(ξ(m) h) = T(m) (N ) (N ) (N ) (N ) (N ) (N ) −¯ c(N ) A(m) η(m) sin η(m) h + c¯(N ) B(m) η(m) cos η(m) h (N )
(24)
(N )
+ +e(N ) C(m) ξ(m) sinh(ξ(m) h) + e(N ) D(m) ξ(m) cosh(ξ(m) h) = T(m)
(25)
as well as for m = 0:
e(1) e(1) (1) (1) (1) (1) (1) (1) − A cos η h − B sin η h + C(0) − D(0) h = V(0) (0) (0) (0) (0) (1) (1) ε ε e(N ) e(N ) (N ) (N ) (N ) (N ) (N ) (N ) + A(0) cos η(0) h + (N ) B(0) sin η(0) h + C(0) + D(0) h = V(0) (N ) ε ε
(26) (27)
and for m = 1, 2, 3, . . . : e(1) e(1) (1) (1) (1) (1) (1) (1) − A(m) cos η(m) h − (1) B(m) sin η(m) h + C(m) cosh(ξ(m) h) − D(m) sinh(ξ(m) h) = V(m) (28) (1) ε ε e(N ) e(N ) (N ) (N ) (N ) (N ) (N ) (N ) + A cos η h + B sin η h + C(m) cosh(ξ(m) h) + D(m) sinh(ξ(m) h) = V(m) (29) (m) (m) ε(N ) (m) ε(N ) (m) (1)
(1)
(1)
(1)
(N )
(N )
(1)
(1)
Equations (19), (22), (23), (26) and (27) are eight equations for A(0) , B(0) , C(0) , D(0) , A(0) , B(0) ,
(N )
(N )
(1)
C(0) and D(0) . Equations (20), (24), (25), (28) and (29) are eight equations for A(m) , B(m) , C(m) , (1)
(N )
(N )
(N )
(N )
D(m) , A(m) , B(m) , C(m) and D(m) for m = 1, 2, 3, . . ..
Vol. 20, No. 3
Hu Hongping et al.: An Exact Analysis of Forced Thickness-Twist Vibrations
· 215 ·
IV. EXAMPLES To show the effectiveness of the solution obtained in the preceding section we analyze a few examples of piezoelectric devices with respect to their most basic characteristics. 4.1. A Piezoelectric Resonator First consider a two-layered piezoelectric plate driven electrically as a resonator shown in Fig. 2 under the following driving voltage: T23 (x1 ) = 0,
V (x1 ) = V0± exp(iωt),
x2 = ±h
(30)
Fig. 2. A two-layered plate as a resonator driven electrically. Fig. 3. Resonances in a two-layered resonator.
where V0± are constants. There is a pair of electrodes which are shown by the thick lines in Fig.2. The two ceramics have significantly different material properties. Some damping is introduced by allowing the elastic material constant c44 to assume complex values, which can represent viscous damping in the material. In our calculations c44 is replaced by c44 (1 + iQ−1 ) where Q is a large and real number. For polarized ceramics the value of Q is of the order of 102 to 103 . We fix Q = 100 and choose h = 1 mm, and a = 10 mm. The equations in the preceding section are solved numerically on a computer. In Fig.3 we plot the displacement at the center of the upper surface of the plate versus the driving frequency which is normalized by c¯ π ω0 = (31) 2h ρ ω0 is close to the lowest thickness-twist resonant frequency. The figure shows the locations of the resonant frequencies. In this example, because of the uniform driving voltage the plate is in fact vibrating like an infinite plate without x1 dependence. The modes corresponding to different resonant frequencies have different wave numbers in the x2 direction. These modes may also be called thickness-shear modes in the x3 direction. The resonant frequencies of these modes are not exactly the integral multiples of the fundamental resonant frequency. They will be called major resonances below. Note that for a onelayered plate modes symmetric about the middle plane, e.g., the second resonance cannot be excited by a thickness electric field. A two-layered plate does not have symmetry about the middle plane and its modes cannot be separated into symmetric and antisymmetric modes and all modes can be excited. The displacement distribution along the plate thickness (mode shape) near the first resonance is shown in Fig. 4 Mode shapes near the first resonance. Fig.4, which is the fundamental thickness mode.
· 216 ·
ACTA MECHANICA SOLIDA SINICA
2007
4.2. A Piezoelectric Transformer Piezoelectric materials can be used to make transformers with two pairs of electrodes, one for input and the other for output. Next consider a two-layered plate as a piezoelectric transformer shown in Fig.5 with the following electrical load: T23 (x1 ) =0, x2 = ±h ±V01 exp(iωt), V (x1 ) = ∓V02 exp(iωt),
x2 = ±h, x2 = ±h,
−a < x1 < 0 0 < x1 < a
(32)
where V01 and V02 are constants. V01 is a known input voltage. V02 is an unknown output voltage. To determine V02 the situation at the output electrodes needs to be specified. We consider the simple case of open output electrodes with I2 = Q˙ 2 = iωQ2 = iω
a 0
D2 |x2 =−h dx1 = 0
(33)
where Q2 and I2 are the charge and current on the lower output electrode. In Fig.6 we plot the transforming ratio versus the driving frequency. The figure shows that near the major resonant frequencies the transforming ratio assumes maximum. Therefore a transformer is a resonant device with a significant output only at certain frequencies (filter). After the third major resonance the output is little because for higher-order thickness-twist modes there are quite a few nodal points along the plate thickness and voltage generated by strains along the plate thickness through piezoelectric coupling mostly cancels itself. Near the major resonances there are also minor resonances which are due to the presence of many modes with essentially the same wave number along the plate thickness but are with different dependence on x1 .
Fig. 5. A two-layered plate as a transformer driven electrically.
Fig. 6. Transforming ratio versus driving frequency.
4.3. A Piezoelectric Generator Piezoelectric materials can also be used as generators for converting mechanical energy to electrical energy. Finally we consider a two-layered plate as a generator shown in Fig.7 with the following mechanical load: T exp(iωt), x2 = ±h, −a < x1 < 0 T23 (x1 ) = −T exp(iωt), x2 = ±h, 0 < x1 < a (34) ±V exp(iωt), x2 = ±h, −a < x1 < 0 V (x1 ) = ∓V exp(iωt), x2 = ±h, 0 < x1 < a where T and V are constants. T is known but V is not. Again two pairs of electrodes are needed. We will still assume open output electrodes. Output voltage versus driving frequency is shown in Fig.8. The maximum output can be picked up at resonances. Note that the second major resonance is essentially not excited.
Vol. 20, No. 3
Hu Hongping et al.: An Exact Analysis of Forced Thickness-Twist Vibrations
Fig. 7. A two-layered plate as a generator driven mechanically.
· 217 ·
Fig. 8. Output voltage versus driving frequency.
V. CONCLUSION An exact piezoelectric solution from the three-dimensional equations of linear piezoelectricity is obtained for the thickness-twist vibration of a finite, rectangular plate of 6 mm crystals or polarized ceramics. The solution is useful in analyzing resonant piezoelectric devices. For multi-layered plates the symmetry about the middle plane is usually lost and the modes cannot be separated as symmetric or anti-symmetric about the middle plane. Thickness-twist modes with essentially the same wave number in the thickness direction and different small in-plane wave numbers have frequencies close to each other. Therefore the resonant behavior of thickness-twist devices is relatively complicated.
References [1] Mindlin,R.D., Thickness-twist vibrations of an infinite, monoclinic, crystal plate. International Journal of Solids and Structures, 1965, 1: 141-145. [2] Mindlin,R.D., Bechmann’s number for harmonic overtones of thickness/twist vibrations of rotated Y-cut quartz plates. The Journal of the Acoustical Sociaty of America, 1967, 41: 969-973. [3] Pearman,G.T., Thickness-twist vibrations in beveled AT-cut quartz plates. The Journal of the Acoustical Sociaty of America, 1968, 45: 928-934. [4] Bleustein,J., Thickness-twist and face-shear vibrations of a contoured crystal plate. International Journal of Solids and Structures, 1966, 2: 351-360. [5] Bleustein,J.L., Some simple modes of wave propagation in an infinite piezoelectric plate. The Journal of the Acoustical Sociaty of America, 1969, 45: 614-620. [6] Yang,J.S., Shear horizontal vibrations of a piezoelectric/ferroelectric wedge. Acta Mechanica, 2004, 173: 13-17. [7] Pang,W., Zhang,H. and Kim,E.S., Micromachined acoustic wave resonator isolated from substrate. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2005, 52: 1239-1246. [8] Link,M., Schreiter,M., Weber,J., Primig,R., Pitzer,D. and Cabl,R., Solidly mounted ZnO shear mode film bulk acoustic wave resonators for sensing applications in liquids. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2006, 53: 492-496. [9] Martin,F., Jan,M.E., ReyMermet,S., Belgacem,B., Su,D., Cantoni,M. and Muralt,P., Shear mode coupling and tilted grain growth of AlN thin films in BAW resonators. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2006, 53: 1339-1343. [10] Tiersten,H.F., Linear Piezoelectric Plate Vibrations, Plenum, New York, 1969. [11] Mansfeld,G.D., New types of BAW composite resonators and their properties//Proc. IEEE Ultrasoncis Symp., 1995, 917-920. [12] Mansfeld,G.D., Modern state and perspective of bulk acoustic wave composite resonators//Proc. Joint EFTF-IEEE International Frequency Control Symposium, 1999, 843-846. [13] Makkonen,T., Holappa,A., Ella,J. and Salomaa,M.M., Finite element simulation of thin-film composite BAW resonators. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2001, 48: 12411258. [14] Stewart,J.T. and Yong,Y.K., Exact analysis of the propagation of acoustic waves in multilayered anisotropic piezoelectric plates. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1994, 41: 375-390.
· 218 ·
ACTA MECHANICA SOLIDA SINICA
2007
[15] Zhang,Z. and Yong,Y.K., Numerical analysis of thickness shear thin film piezoelectric resonators using a laminated plate theory. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1995, 42: 734-746. [16] Yao,L.Q., Lu,L., Wang,Z.H., Zhu,W.G. and Dai,Y., Exact solution of multilayered piezoelectric diaphragms. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2003, 50: 1262-1271. [17] Yang,J.S., Batra,R.C. and Liang,X.Q., The cylindrical bending vibration of a laminated elastic plate due to piezoelectric actuators. Smart Materials and Structures, 1994, 3: 485-493. [18] Batra,R.C., Liang,X.Q. and Yang,J.S., Shape control of a vibrating simply supported rectangular plate by using piezoelectric actuators. AIAA Journal, 1996, 34: 116-122. [19] Batra,R.C., Liang,X.Q. and Yang,J.S., The vibration of a simply supported rectangular elastic plate due to piezoelectric actuators. International Journal of Solids and Structures, 1996, 33: 1597-1618. [20] Vel,S.S. and Batra,R.C., Three-dimensional analytical solution for hybrid multilayered piezoelectric plates. ASME Journal of Applied Mechanics, 2000, 67: 558-567. [21] Vel,S.S. and Batra,R.C., Exact solution for rectangular sandwich plates with embedded piezoelectric shear resonators. AIAA Journal, 2001, 39: 1363-1373. [22] Baillargeon,B.P. and Vel,S.S., Exact solution for the vibration and active damping of composite plates with piezoelectric shear actuators. Journal of Sound and Vibration, 2005, 282: 781-804. [23] Yang,J.S. and Guo,S.H., Thickness-twist modes in a rectangular piezoelectric resonator of hexagonal crystals. Applied Physics Letters, 2006, 88: 153506. [24] Yang,J.S., An Introduction to the Theory of Piezoelectricity, Springer, New York, 2005. [25] Bleustein,J.L., A new surface wave in piezoelectric materials. Applied Physics Letters, 1968, 13: 412-413.
ISSN 0894-9166
AN EXACT ANALYSIS OF FORCED THICKNESS-TWIST VIBRATIONS OF MULTI-LAYERED PIEZOELECTRIC PLATES Hu Hongping1
Chen Ziguang2
Yang Jiashi1,3
Hu Yuantai1 1
1
( Department of Mechanics, School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China) (2 Institute of Mechanics and Sensing Technology, School of Civil Engineering and Architecture, Central South University, Changsha 410082, China) (3 Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588, USA)
Received 20 November 2006; revision received 26 June 2007
ABSTRACT This paper deals with the thickness-twist vibration of a multi-layered rectangular piezoelectric plate of crystals of 6 mm symmetry or polarized ceramics. An exact solution is obtained from the three-dimensional equations of linear piezoelectricity. The solution is useful to the understanding and design of composite piezoelectric devices. A piezoelectric resonator, a piezoelectric transformer, and a piezoelectric generator are analyzed as examples.
KEY WORDS piezoelectricity, plate, resonator, transformer, generator
I. INTRODUCTION Thickness-twist vibrations of crystal and ceramic plates are often used as the operating modes of piezoelectric resonators and acoustic wave sensors, e.g., Refs.[1–6]. Polarized ceramic plates have been used for a long time. Recently, thin film resonators and sensors of crystals with 6 mm symmetry including AlN and ZnO have become the cynosure of interest[7, 8] . AlN and ZnO plates with the six-fold axis normal and in-plane to the plates, or with a tilted six-fold axis are all being developed[9] . When the six-fold axis is in-plane, a 6 mm crystal plate can support thickness-twist modes with electromechanical coupling. Since the material tensors of polarized ceramics that are transversely isotropic have the same structures as those of crystals of 6 mm symmetry[10] , their electromechanical coupling behavior is the same as that of 6 mm crystals. Ceramic plates with in-plane poling can also be driven into thickness-twist vibrations by a thickness-electric field. Piezoelectric devices are also often multilayered[11–13] , which causes considerable complications in modeling. Known exact theoretical results concerning thicknesstwist waves are mostly for infinite plates[5] or semi-infinite wedges[6] . For multilayered infinite plates, exact results of general plate waves were obtained in Ref.[14]. For a real device of a finite size, the operating modes are modes of finite bodies. Owing to anisotropy and piezoelectric coupling, known exact modes of finite piezoelectric bodies are relatively few. Usually, two-dimensional approximate plate equations are used for theoretical analysis of finite plates[15, 16] . Some exact results on finite, laminated piezoelectric plates were obtained for the analysis of smart structures[17–22] where the choice of materials, their orientation and the plate deformation modes are
Corresponding author. Email: [email protected] Project supported by Chinese Postdoctoral Science Foundation (No. 20070410944).
· 212 ·
ACTA MECHANICA SOLIDA SINICA
2007
different from applications in resonator, sensor and other resonant piezoelectric devices. For smart structures usually stress and strain are calculated by static analysis for strength considerations rather than analysis of acoustic wave devices where the frequency and/or wave speed as signals, or output electric voltage or power are of main concern. Recently, exact thickness-twist modes in a single-layered rectangular plate of polarized ceramics or piezoelectric crystals of 6 mm symmetry have been obtained in Ref.[23]. The result of Ref.[23] offers the possibility of obtaining exact solutions in a finite multi-layered plate of 6 mm crystals, which is the purpose of this paper. The structure and the mathematical problem are defined in §II. An exact analysis is performed in §III where a general solution is obtained. A piezoelectric composite resonator, a transformer and a generator are analyzed in §IV respectively as examples. Some conclusions are drawn in §V.
II. GOVERNING EQUATIONS Consider an N -layered piezoelectric plate of total thickness 2h with the x2 axis normal to the plate (see Fig. 1). The x3 axis is determined from the x1 and x2 axes by the right-hand rule. The two plate major surfaces and the N − 1 interfaces are sequentially determined by x2 = −h = h0 , h1 , . . . , hN −1 , and hN = h. Each layer is of a different 6 mm crystal with the x3 axis as the six-fold axis or the poling Fig. 1 Cross-section of a multilayered plate of 6 mm crystals. axis of ceramics. This includes a nonpiezoelectric dielectric layer as a special case when the piezoelectric constant is set to zero. The plate is unbounded in the x3 direction. Figure 1 shows a cross-section. On the two major surfaces of the plate at x2 = ±h are electroded under given shear stress T23 (x1 , t) and potential φ(x1 , t). According to the theory of piezoelectricity the electric potential is a constant on an electrode (at most a function of time). Multiple electrodes are needed to realize the x1 dependence of the electric potential. The two minor surfaces at x1 = ±a are unelectroded and traction-free. Thickness-twist modes are described by[5, 24] u1 = u2 = 0,
u3 = u(x1 , x2 , t),
φ = φ(x1 , x2 , t)
(1)
where ui is the displacement in the xi direction and φ is the electric potential. We introduce a function ψ through[24, 25] eu (2) φ=ψ+ ε where e = e15 and ε = ε11 are the relevant piezoelectric and dielectric constants. Then the governing equations for u and ψ are[24, 25] c¯∇2 u = ρ¨ u, ∇2 ψ = 0 (3) where ∇2 = ∂12 + ∂22 is the two-dimensional Laplacian, c¯ = c44 + e2 /ε, and c44 is the relevant elastic constant. The nonzero stress and electric displacement components are[24, 25] T23 = c¯u,2 + eψ,2 ,
T31 = c¯u,1 + eψ,1 ,
D1 = −εψ,1 ,
D2 = −εψ,2
(4)
where an index after a comma denotes partial differentiation with respect to the coordinate associated with the index. At the top and bottom surfaces the boundary conditions are T23 (x1 ) = T ± (x1 ) exp(iωt), ±
V (x1 ) = V (x1 ) exp(iωt),
x2 = ±h x2 = ±h
(5)
which are sufficient to describe the applications we have in mind. At an interface, we require the continuity of u, T23 , φ and D2 . At the two ends, the boundary conditions are T13 = 0,
D1 = 0,
x1 = ±a
(6)
Vol. 20, No. 3
Hu Hongping et al.: An Exact Analysis of Forced Thickness-Twist Vibrations
· 213 ·
III. AN EXACT SOLUTION Consider the i-th layer, with i = 1, 2, 3, . . . , N . For time-harmonic motions we write u, ψ, φ, T23 and D2 as (u, ψ, φ, T23 , D2 ) = Re{(U, Ψ, Φ, T, D) exp(iωt)} (7) From the modes in Ref.[23] or separation of variables we construct the following solution for the i-th layer: ∞ (i) (i) (i) (i) (i) (i) (i) (i) U = A(0) cos η(0) x2 + B(0) sin η(0) x2 + A(m) cos η(m) x2 + B(m) sin η(m) x2 X(m) (x1 ) m=1
∞ (i) (i) (i) (i) C(m) cosh(ξ(m) x2 ) + D(m) sinh(ξ(m) x2 ) X(m) (x1 ) Ψ = C(0) + D(0) x2 + m=1
(8) where and are undetermined constants, the superscript (i) is for the i-th layer, the subscript (m) is for summation within a layer, and m sin(ξ(m) x1 ), m = 1, 3, 5, · · · π, X(m) (x1 ) = ξ(m) = cos(ξ(m) x1 ), m = 2, 4, 6, · · · 2a (9) 2 ρ(i) ω 2 m 2 (i) π , m = 0, 1, 2, 3, · · · η(m) = − 2a c¯(i) Equation (8) satisfies Eq.(3) and the boundary conditions at x1 = ±a. To apply the interface continuity conditions and the boundary conditions at x2 = ±h, we calculate the following: ∞ (i) e(i) e e(i) (i) (i) (i) (i) (i) (i) (i) (i) A cos η x Φ = (i) A(0) cos η(0) x2 + (i) B(0) sin η(0) x2 + C(0) + D(0) x2 + 2 (m) ε ε ε(i) (m) m=1 e(i) (i) (i) (i) (i) + (i) B(m) sin η(m) x2 + C(m) cosh(ξ(m) x2 ) + D(m) sinh(ξ(m) x2 ) X(m) (x1 ) (10) ε ∞ (i) (i) (i) D = −ε(i) D(0) + −ε(i) C(m) ξ(m) sinh(ξ(m) x2 ) − ε(i) ξ(m) D(m) cosh(ξ(m) x2 ) X(m) (x1 ) (11) (i) A(m) ,
(i) B(m) ,
(i) C(m)
(i) D(m)
m=1
(i)
T = −¯ c
(i) (i) A(0) η(0)
(i) (i) +¯ c(i) B(m) η(m)
sin
(i) η(0) x2
(i)
+ c¯
(i) (i) B(0) η(0)
cos
(i) η(0) x2
+e
(i)
(i) D(0)
+
∞ m=1
(i) (i) (i) −¯ c(i) A(m) η(m) sin η(m) x2
(i) (i) (i) cos η(m) x2 +e(i) C(m) ξ(m) sinh(ξ(m) x2 ) + e(i) D(m) ξ(m) cosh(ξ(m) x2 ) X(m) (x1 ) (12)
At the interface between the i-th and (i + 1)-th layer at x2 = hi (where i = 1, 2, 3, . . . , N − 1), we apply the following continuity conditions + U (h− i ) = U (hi )
(13)
+ Φ(h− i ) = Φ(hi ) + D(h− i ) = D(hi ) + T (h− i ) = T (hi )
Equations (13)-(16) imply that ⎧ (i) ⎫ ⎧ (i+1) ⎫ ⎪ ⎪ A(0) ⎪ A(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (i) ⎪ ⎬ ⎬ ⎨ B (i+1) ⎪ (i) (0) (0) = [T i = 1, 2, 3, · · · , N − 1 ] (i) (0) ⎪ (i+1) ⎪ , ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (i) ⎪ ⎭ ⎭ ⎩ (i+1) ⎪ D(0) D(0) ⎧ (i) ⎫ ⎧ (i+1) ⎫ ⎪ ⎪ A(m) ⎪ A(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (i) ⎪ ⎬ ⎬ ⎨ B (i+1) ⎪ (m) (m) = [T (i) ] , i = 1, 2, 3, · · · , N − 1, (i) (i+1) ⎪ ⎪ C(m) ⎪ C(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (i) ⎪ ⎭ ⎭ ⎩ (i+1) ⎪ D(m) D(m)
(14) (15) (16)
(17)
m = 1, 2, 3, · · ·
(18)
· 214 ·
ACTA MECHANICA SOLIDA SINICA
2007
(i)
where [T(0) ] and [T (i) ] are transfer matrices whose expressions are straightforward from Eqs.(13)-(16) and are too lengthy to be presented here. With Eqs.(17) and (18) we can express all undetermined coefficients in terms of those of the N -th layer. In particular, between the first and the last layers, we have ⎧ (1) ⎫ ⎧ (N ) ⎫ ⎪ ⎪ A(0) ⎪ A(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (1) ⎪ ⎬ ⎬ ⎨ B (N ) ⎪ (1) (2) (N −1) (0) (0) = [T (19) ][T ] · · · [T ] (1) (N ) (0) (0) (0) ⎪ ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ C(0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (1) ⎪ ⎭ ⎭ ⎩ (N ) ⎪ D(0) D(0) ⎧ (1) ⎫ ⎧ (N ) ⎫ ⎪ ⎪ ⎪ A(m) ⎪ ⎪ ⎪ ⎪ A(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B (1) ⎪ ⎬ ⎬ ⎨ B (N ) ⎪ (m) (m) = [T (1) ][T (2) ] · · · [T (N −1) ] (20) (1) (N ) ⎪ ⎪ C(m) ⎪ C(m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (1) ⎪ ⎭ ⎭ ⎩ (N ) ⎪ D(m) D(m) We also write the boundary conditions in Eq.(5) in series form: ± T ± (x1 ) = T(0) +
∞ m=1
± T(m) X(m) (x1 ),
± V ± (x1 ) = V(0) +
∞ m=1
± V(m) X(m) (x1 )
(21)
± ± ± ± , T(m) , V(0) and V(m) are known expansion coefficients of the applied load. Then the boundary where T(0) conditions imply that, for m = 0 :
(1) (1) (1) (1) (1) (1) (1) − c¯(1) A(0) η(0) sin η(0) h + c¯(1) B(0) η(0) cos η(0) h + e(1) D(0) = T(0) (N ) (N ) (N ) (N ) (N ) (N ) (N ) + −¯ c(N ) A(0) η(0) sin η(0) h + c¯(N ) B(0) η(0) cos η(0) h + e(N ) D(0) = T(0)
(22) (23)
and for m = 1, 2, 3, . . . : (1) (1) (1) (1) (1) (1) c¯(1) A(m) η(m) sin η(m) h + c¯(1) B(m) η(m) cos η(m) h (1)
(1)
− −e(1) C(m) ξ(m) sinh(ξ(m) h) + e(1) D(m) ξ(m) cosh(ξ(m) h) = T(m) (N ) (N ) (N ) (N ) (N ) (N ) −¯ c(N ) A(m) η(m) sin η(m) h + c¯(N ) B(m) η(m) cos η(m) h (N )
(24)
(N )
+ +e(N ) C(m) ξ(m) sinh(ξ(m) h) + e(N ) D(m) ξ(m) cosh(ξ(m) h) = T(m)
(25)
as well as for m = 0:
e(1) e(1) (1) (1) (1) (1) (1) (1) − A cos η h − B sin η h + C(0) − D(0) h = V(0) (0) (0) (0) (0) (1) (1) ε ε e(N ) e(N ) (N ) (N ) (N ) (N ) (N ) (N ) + A(0) cos η(0) h + (N ) B(0) sin η(0) h + C(0) + D(0) h = V(0) (N ) ε ε
(26) (27)
and for m = 1, 2, 3, . . . : e(1) e(1) (1) (1) (1) (1) (1) (1) − A(m) cos η(m) h − (1) B(m) sin η(m) h + C(m) cosh(ξ(m) h) − D(m) sinh(ξ(m) h) = V(m) (28) (1) ε ε e(N ) e(N ) (N ) (N ) (N ) (N ) (N ) (N ) + A cos η h + B sin η h + C(m) cosh(ξ(m) h) + D(m) sinh(ξ(m) h) = V(m) (29) (m) (m) ε(N ) (m) ε(N ) (m) (1)
(1)
(1)
(1)
(N )
(N )
(1)
(1)
Equations (19), (22), (23), (26) and (27) are eight equations for A(0) , B(0) , C(0) , D(0) , A(0) , B(0) ,
(N )
(N )
(1)
C(0) and D(0) . Equations (20), (24), (25), (28) and (29) are eight equations for A(m) , B(m) , C(m) , (1)
(N )
(N )
(N )
(N )
D(m) , A(m) , B(m) , C(m) and D(m) for m = 1, 2, 3, . . ..
Vol. 20, No. 3
Hu Hongping et al.: An Exact Analysis of Forced Thickness-Twist Vibrations
· 215 ·
IV. EXAMPLES To show the effectiveness of the solution obtained in the preceding section we analyze a few examples of piezoelectric devices with respect to their most basic characteristics. 4.1. A Piezoelectric Resonator First consider a two-layered piezoelectric plate driven electrically as a resonator shown in Fig. 2 under the following driving voltage: T23 (x1 ) = 0,
V (x1 ) = V0± exp(iωt),
x2 = ±h
(30)
Fig. 2. A two-layered plate as a resonator driven electrically. Fig. 3. Resonances in a two-layered resonator.
where V0± are constants. There is a pair of electrodes which are shown by the thick lines in Fig.2. The two ceramics have significantly different material properties. Some damping is introduced by allowing the elastic material constant c44 to assume complex values, which can represent viscous damping in the material. In our calculations c44 is replaced by c44 (1 + iQ−1 ) where Q is a large and real number. For polarized ceramics the value of Q is of the order of 102 to 103 . We fix Q = 100 and choose h = 1 mm, and a = 10 mm. The equations in the preceding section are solved numerically on a computer. In Fig.3 we plot the displacement at the center of the upper surface of the plate versus the driving frequency which is normalized by c¯ π ω0 = (31) 2h ρ ω0 is close to the lowest thickness-twist resonant frequency. The figure shows the locations of the resonant frequencies. In this example, because of the uniform driving voltage the plate is in fact vibrating like an infinite plate without x1 dependence. The modes corresponding to different resonant frequencies have different wave numbers in the x2 direction. These modes may also be called thickness-shear modes in the x3 direction. The resonant frequencies of these modes are not exactly the integral multiples of the fundamental resonant frequency. They will be called major resonances below. Note that for a onelayered plate modes symmetric about the middle plane, e.g., the second resonance cannot be excited by a thickness electric field. A two-layered plate does not have symmetry about the middle plane and its modes cannot be separated into symmetric and antisymmetric modes and all modes can be excited. The displacement distribution along the plate thickness (mode shape) near the first resonance is shown in Fig. 4 Mode shapes near the first resonance. Fig.4, which is the fundamental thickness mode.
· 216 ·
ACTA MECHANICA SOLIDA SINICA
2007
4.2. A Piezoelectric Transformer Piezoelectric materials can be used to make transformers with two pairs of electrodes, one for input and the other for output. Next consider a two-layered plate as a piezoelectric transformer shown in Fig.5 with the following electrical load: T23 (x1 ) =0, x2 = ±h ±V01 exp(iωt), V (x1 ) = ∓V02 exp(iωt),
x2 = ±h, x2 = ±h,
−a < x1 < 0 0 < x1 < a
(32)
where V01 and V02 are constants. V01 is a known input voltage. V02 is an unknown output voltage. To determine V02 the situation at the output electrodes needs to be specified. We consider the simple case of open output electrodes with I2 = Q˙ 2 = iωQ2 = iω
a 0
D2 |x2 =−h dx1 = 0
(33)
where Q2 and I2 are the charge and current on the lower output electrode. In Fig.6 we plot the transforming ratio versus the driving frequency. The figure shows that near the major resonant frequencies the transforming ratio assumes maximum. Therefore a transformer is a resonant device with a significant output only at certain frequencies (filter). After the third major resonance the output is little because for higher-order thickness-twist modes there are quite a few nodal points along the plate thickness and voltage generated by strains along the plate thickness through piezoelectric coupling mostly cancels itself. Near the major resonances there are also minor resonances which are due to the presence of many modes with essentially the same wave number along the plate thickness but are with different dependence on x1 .
Fig. 5. A two-layered plate as a transformer driven electrically.
Fig. 6. Transforming ratio versus driving frequency.
4.3. A Piezoelectric Generator Piezoelectric materials can also be used as generators for converting mechanical energy to electrical energy. Finally we consider a two-layered plate as a generator shown in Fig.7 with the following mechanical load: T exp(iωt), x2 = ±h, −a < x1 < 0 T23 (x1 ) = −T exp(iωt), x2 = ±h, 0 < x1 < a (34) ±V exp(iωt), x2 = ±h, −a < x1 < 0 V (x1 ) = ∓V exp(iωt), x2 = ±h, 0 < x1 < a where T and V are constants. T is known but V is not. Again two pairs of electrodes are needed. We will still assume open output electrodes. Output voltage versus driving frequency is shown in Fig.8. The maximum output can be picked up at resonances. Note that the second major resonance is essentially not excited.
Vol. 20, No. 3
Hu Hongping et al.: An Exact Analysis of Forced Thickness-Twist Vibrations
Fig. 7. A two-layered plate as a generator driven mechanically.
· 217 ·
Fig. 8. Output voltage versus driving frequency.
V. CONCLUSION An exact piezoelectric solution from the three-dimensional equations of linear piezoelectricity is obtained for the thickness-twist vibration of a finite, rectangular plate of 6 mm crystals or polarized ceramics. The solution is useful in analyzing resonant piezoelectric devices. For multi-layered plates the symmetry about the middle plane is usually lost and the modes cannot be separated as symmetric or anti-symmetric about the middle plane. Thickness-twist modes with essentially the same wave number in the thickness direction and different small in-plane wave numbers have frequencies close to each other. Therefore the resonant behavior of thickness-twist devices is relatively complicated.
References [1] Mindlin,R.D., Thickness-twist vibrations of an infinite, monoclinic, crystal plate. International Journal of Solids and Structures, 1965, 1: 141-145. [2] Mindlin,R.D., Bechmann’s number for harmonic overtones of thickness/twist vibrations of rotated Y-cut quartz plates. The Journal of the Acoustical Sociaty of America, 1967, 41: 969-973. [3] Pearman,G.T., Thickness-twist vibrations in beveled AT-cut quartz plates. The Journal of the Acoustical Sociaty of America, 1968, 45: 928-934. [4] Bleustein,J., Thickness-twist and face-shear vibrations of a contoured crystal plate. International Journal of Solids and Structures, 1966, 2: 351-360. [5] Bleustein,J.L., Some simple modes of wave propagation in an infinite piezoelectric plate. The Journal of the Acoustical Sociaty of America, 1969, 45: 614-620. [6] Yang,J.S., Shear horizontal vibrations of a piezoelectric/ferroelectric wedge. Acta Mechanica, 2004, 173: 13-17. [7] Pang,W., Zhang,H. and Kim,E.S., Micromachined acoustic wave resonator isolated from substrate. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2005, 52: 1239-1246. [8] Link,M., Schreiter,M., Weber,J., Primig,R., Pitzer,D. and Cabl,R., Solidly mounted ZnO shear mode film bulk acoustic wave resonators for sensing applications in liquids. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2006, 53: 492-496. [9] Martin,F., Jan,M.E., ReyMermet,S., Belgacem,B., Su,D., Cantoni,M. and Muralt,P., Shear mode coupling and tilted grain growth of AlN thin films in BAW resonators. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2006, 53: 1339-1343. [10] Tiersten,H.F., Linear Piezoelectric Plate Vibrations, Plenum, New York, 1969. [11] Mansfeld,G.D., New types of BAW composite resonators and their properties//Proc. IEEE Ultrasoncis Symp., 1995, 917-920. [12] Mansfeld,G.D., Modern state and perspective of bulk acoustic wave composite resonators//Proc. Joint EFTF-IEEE International Frequency Control Symposium, 1999, 843-846. [13] Makkonen,T., Holappa,A., Ella,J. and Salomaa,M.M., Finite element simulation of thin-film composite BAW resonators. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2001, 48: 12411258. [14] Stewart,J.T. and Yong,Y.K., Exact analysis of the propagation of acoustic waves in multilayered anisotropic piezoelectric plates. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1994, 41: 375-390.
· 218 ·
ACTA MECHANICA SOLIDA SINICA
2007
[15] Zhang,Z. and Yong,Y.K., Numerical analysis of thickness shear thin film piezoelectric resonators using a laminated plate theory. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1995, 42: 734-746. [16] Yao,L.Q., Lu,L., Wang,Z.H., Zhu,W.G. and Dai,Y., Exact solution of multilayered piezoelectric diaphragms. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2003, 50: 1262-1271. [17] Yang,J.S., Batra,R.C. and Liang,X.Q., The cylindrical bending vibration of a laminated elastic plate due to piezoelectric actuators. Smart Materials and Structures, 1994, 3: 485-493. [18] Batra,R.C., Liang,X.Q. and Yang,J.S., Shape control of a vibrating simply supported rectangular plate by using piezoelectric actuators. AIAA Journal, 1996, 34: 116-122. [19] Batra,R.C., Liang,X.Q. and Yang,J.S., The vibration of a simply supported rectangular elastic plate due to piezoelectric actuators. International Journal of Solids and Structures, 1996, 33: 1597-1618. [20] Vel,S.S. and Batra,R.C., Three-dimensional analytical solution for hybrid multilayered piezoelectric plates. ASME Journal of Applied Mechanics, 2000, 67: 558-567. [21] Vel,S.S. and Batra,R.C., Exact solution for rectangular sandwich plates with embedded piezoelectric shear resonators. AIAA Journal, 2001, 39: 1363-1373. [22] Baillargeon,B.P. and Vel,S.S., Exact solution for the vibration and active damping of composite plates with piezoelectric shear actuators. Journal of Sound and Vibration, 2005, 282: 781-804. [23] Yang,J.S. and Guo,S.H., Thickness-twist modes in a rectangular piezoelectric resonator of hexagonal crystals. Applied Physics Letters, 2006, 88: 153506. [24] Yang,J.S., An Introduction to the Theory of Piezoelectricity, Springer, New York, 2005. [25] Bleustein,J.L., A new surface wave in piezoelectric materials. Applied Physics Letters, 1968, 13: 412-413.