Theoretical Analysis and Experimental
Measurement for Resonant Vibration of
Piezoceramic Circular Plates
Chi-Hung Huang, Yu-Chih Lin, and Chien-Ching MaAbstract—Based on the electroelastic theory for
piezo-electric plates, the vibration characteristics of piezoceramic disks with free-boundary conditions are investigated in this work by theoretical analysis, numerical simulation, and ex-perimental measurement. The resonance of thin piezoce-ramic disks is classified into three types of vibration modes: transverse, tangential, and radial extensional modes. All of these modes are investigated in detail. Two optical techniques, amplitude-fluctuation electronic speckle pat-tern interferometry (AF-ESPI) and laser Doppler vibrome-ter (LDV), are used to validate the theoretical analysis. Be-cause the clear fringe patterns are shown only at resonant frequencies, both the resonant frequencies and the corre-sponding mode shapes are obtained experimentally at the same time by the proposed AF-ESPI method. Good quality of the interferometric fringe patterns for both the trans-verse and extensional vibration mode shapes are demon-strated. The resonant frequencies of the piezoceramic disk also are measured by the conventional impedance analy-sis. Both theoretical and experimental results indicate that the transverse and tangential vibration modes cannot be measured by the impedance analysis, and only the resonant frequencies of extensional vibration modes can be obtained. Numerical calculations based on the finite element method also are performed, and the results are compared with the theoretical analysis and experimental measurements. It is shown that the finite element method (FEM) calculations and the experimental results agree fairly well for the reso-nant frequencies and mode shapes. The resoreso-nant frequen-cies and mode shapes predicted by theoretical analysis and calculated by finite element method are in good agreement, and the difference of resonant frequencies for both results with the thickness-to-diameter (h/D) ratios, ranging from 0.01 to 0.1, are presented.
I. Introduction
S
incePierre and Jacques Curie discovered the piezo-electric effect in 1880, there has been much further research and applications [1], [2]. Currently, piezoelectric materials are widely used in electromechanical sensors, ac-tuators, nondestructive testing devices, and electro-optic modulators. The piezoelectric effect is applied to many modern engineering applications because it expresses the connection between the electrical and mechanical fields.Manuscript received March 10, 2003; accepted July 23, 2003. C.-H. Huang is with the Department of Mechanical Engineering, Ching Yun University, Chung-Li, Taiwan 320, Republic of China.
Y.-C. Lin and C.-C. Ma are with the Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 106, Re-public of China (e-mail: [email protected]).
Although the vibration characteristics of piezoelectric ma-terials can be determined by the linear piezoelasticity, the Maxwell equation and piezoelectric constitutive equations [3], it is difficult to obtain analytical solutions for even a simple geometry. In general, there are two numerical meth-ods that are usually used to study the vibration problem of piezoelectric materials: one is the variational approxima-tion method and the other is finite element analysis. Eer Nisse [4] applied the calculus of variation to the analysis of piezoelectric disks and compared the results to the ex-perimental results obtained by Shaw [5]. Holland [6] used the Rayleigh-Ritz method to study the extensional modes of rectangular piezoelectric plates and classified them into four distinct symmetry types. Kunkel et al. [7] studied the vibration modes of PZT-5H ceramics disks concerning the thickness-to-diameter (h/D) ratio ranging from 0.1 to 5. Due to the great flexibility and extensive applicability, the finite element method has become the alternative method to the analysis of piezoelectric material in various configu-rations. Ivina [8] studied the symmetric modes of vibration for circular piezoelectric plates to determine the resonant and antiresonant frequencies, radial mode configurations, and optimum geometrical dimensions to maximize the dy-namic electromechanical coupling coefficient. Guo et al. [9] presented the results for PZT-5A piezoelectric disks with h/D of 0.05 and 0.1. In that study there were five types of modes classified according to the mode shape charac-teristics, and the physical interpretation was clarified. The finite element method is a powerful tool, but sometimes an analytical solution is needed to get a deep and clear understanding of the characteristics of resonant vibrations for piezoceramic plates.
In addition to variational and finite element meth-ods, optical techniques usually are used for experimen-tal measurement of resonant frequencies of piezoelectric plates. Shaw [5] used an optical interference technique in which a stroboscopically illuminated multiple beam was applied to measure the surface motion of thick barium ti-tanate disks. However, only normal modes having symme-try with respect to the axis and to the central plane were observed. Minoni and Docchio [10] proposed an optical self-calibrating technique, based on the signal-processing chain, to measure the vibration amplitude of PZTs for different operating frequencies. To evaluate the piezoelec-tric constants, Ohki et al. [11] used the fiber-optic tech-nique to measure the vibration amplitude of piezoceramic circular rod and disk. Chang [12] used the dual-beam
speckle interferometry to measure the in-plane vibration amplitude on the PZT surface. As the measured displace-ment spectrum reaches the local maximum under some driven voltages, the resonant frequencies of in-plane modes were determined. To obtain the vibration mode shapes si-multaneously, the technique with full-field measurement capability should be used. Koyuncu [13] used electronic speckle pattern interferometry (ESPI) with reference beam modulation to study the vibration amplitudes and vibra-tion modes of PZT-4 transducers in air and water. Os-win et al. [14] used ESPI to validate the finite element model of flextensional transducer with an elliptical shape, studying both in-plane and out-of-plane vibrations. Ma and Huang [15], [16] used the amplitude-fluctuation (AF)-ESPI method to investigate the three-dimensional vibra-tion of piezoelectric rectangular parallelepipeds and cylin-ders, and presented both the resonant frequencies and mode shapes.
Because piezoceramic transducers are usually circular, the vibration characteristics of piezoceramic disks are im-portant in transducer design and application. Though al-gorithms based on the finite element method are able to solve this problem, theoretical methods often are preferred because the numerical approaches do not give sufficient insight into the physical interpretation of the resonant vi-bration of piezoceramic plates. If the piezoceramic disk is thin, the out-of-plane (transverse) vibration and the in-plane (tangential and extensional) vibration are uncou-pled. In this study, we will analyze in detail the vibration behavior for transverse, tangential, and extensional vibra-tions, including the resonant frequencies, mode shapes, and electrical currents. The nonaxisymmetric modes are investigated for the transverse vibration, and tangential and extensional vibrations are restricted to axisymmet-ric modes. To validate the theoretical results, two optical techniques—namely, AF-ESPI and laser Doppler vibrome-ter (LDV)—and the electrical impedance measurement are used to experimentally investigate the vibration behavior of piezoceramic disks in resonance. The advantage of using the AF-ESPI method is that not only resonant frequencies but also the corresponding mode shapes for transverse and extensional modes can be obtained. The fringe patterns of AF-ESPI shown in the experimental results correspond to the vibrating mode shapes. According to experimental results obtained in this study, it is shown that only the radial extensional vibration for the piezoceramic disk can be measured by the impedance analysis, and it also is ver-ified by theoretical analysis. In addition to the AF-ESPI measurement, the resonant frequencies of transverse vibra-tion modes also are determined by a LDV system. A com-mercially available finite element software program is used to provide the numerical calculations. Good agreements of the mode shapes and resonant frequencies are obtained for experimental and theoretical results. And, numerical FEM calculations are performed, and the accuracy of theoretical results are presented in this study for different thickness-to-diameter (h/D) ratios are verified by the results of finite element analyses.
Fig. 1. Geometry and coordinate system of the piezoceramic disk.
II. Theoretical Analysis of Dynamic Characteristics for Circular
Piezoceramic Disks
Fig. 1 shows the geometrical configuration of a piezoce-ramic disk with radius R and thickness h; the cylindrical coordinates (r, θ, z) with the origin in the center of the disk are used. The piezoceramic disk is polarized in the thickness direction, and the two opposite faces of the disk are covered with complete electrodes. The system of gov-erning equations and basic hypotheses needed to analyze the vibration characteristics of the piezoceramic disk is presented following Rogacheva [17]. The differential equa-tions of equilibrium in cylindrical coordinates are:
∂σrr ∂r + 1 r ∂σrθ ∂θ + ∂σrz ∂z + 1 r(σrr− σθθ) + ρ ∂2u ∂t2 = 0, (1a) ∂σrθ ∂r + 1 r ∂σθθ ∂θ + ∂σθz ∂z + 2 rσrθ+ ρ ∂2v ∂t2 = 0, (1b) ∂σrz ∂r + 1 r ∂σθz ∂θ + ∂σzz ∂z + 1 rσrz+ ρ ∂2w ∂t2 = 0, (1c) where σrr, σrθ, σθθ, σzz, σrz, σθz are the components of
stress; u, v and w are the displacement field in the r, θ, and z direction, respectively; and ρ is the density. The strain-mechanical displacement relations are:
err= ∂u ∂r, eθθ= u r + 1 r ∂v ∂θ, ezz= ∂w ∂z, erθ= 1 r ∂u ∂θ + ∂v ∂r− v r, erz= ∂u ∂z + ∂w ∂r, eθz= ∂v ∂z+ 1 r ∂w ∂θ, (2)
where err, erθ, K, ezz are the components of strain. The
linear piezoceramic constitutive equations for a piezoce-ramic material with crystal symmetry class C6mmare:
err eθθ ezz eθz erz erθ Dr Dθ Dz = sE11 s E 12 s E 13 0 0 0 0 0 d31 sE12 sE11 sE13 0 0 0 0 0 d31 sE13 sE13 sE33 0 0 0 0 0 d33 0 0 0 sE 44 0 0 0 d15 0 0 0 0 0 sE 44 0 d15 0 0 0 0 0 0 0 sE 66 0 0 0 0 0 0 0 d15 0 εT11 0 0 0 0 0 d15 0 0 0 εT11 0 d31d31d33 0 0 0 0 0 εT33 σrr σθθ σzz σθz σrz σrθ Er Eθ Ez , (3) where sE
11, sE12, K, sE66 are the compliance constants; d15, d31, d33are the piezoelectric constants; εT11and εT33are the
dielectric constants; Dr, Dθ, Dzare the electrical
displace-ment components, and Er, Eθ, Ezare the electrical fields.
The piezoceramic material is isotropic in the plane normal to the z-axis.
The charge equation of electrostatics is given by: ∂Dr ∂r + 1 r ∂Dθ ∂θ + 1 rDr+ ∂Dz ∂z = 0. (4)
The electric field-electric potential relations are:
Er=− ∂ϕ ∂r, Eθ=− 1 r ∂ϕ ∂θ, Ez=− ∂ϕ ∂z, (5)
where ϕ is the electrical potential. The basic hypotheses are:
•Normal stress σzz can be neglected relative to other
stresses, hence σzz = 0.
•The rectilinear element normal to the middle sur-face before deformation remains perpendicular to the strained surface after deformation, and its elongation can be neglected, i.e., erz= eθz = 0.
•Electrical potential varies with the thickness by the square law, i.e., ϕ = ϕ0+ zϕ1+ z2ϕ2; where ϕ0, ϕ1,
and ϕ2 are constants.
•Electrical displacement Dz is a constant with respect
to the thickness.
According to the first and second hypotheses, the elec-troelasticity relation of (3) can be simplified as
σrr= 1 sE 11 1− v2 p (err+ vpeθθ)− d31 sE 11(1− vp) Ez, (6a) σθθ = 1 sE 11 1− v2 p (eθθ+ vperr)− d31 sE 11(1− vp) Ez, (6b) σrθ= 1 sE 66 erθ= erθ 2sE 11(1 + vp) , (6c) Dz= d31(σrr+ σθθ) + εT33Ez, (6d) in which err= ∂ur ∂r + z ∂2w ∂r2, (7a) eθθ= ur r + 1 r ∂vθ ∂θ + z r ∂w ∂r + ∂ ∂θ 1 r ∂w ∂θ , (7b) erθ= 1 r ∂ur ∂θ + z r ∂2w ∂r∂θ + ∂vθ ∂r + z ∂ ∂r 1 r ∂w ∂θ −vθ r − z r2 ∂w ∂θ, (7c)
and the planar Poisson’s ratio vp=− sE12 sE 11
. It is noted that ur = u z=0 and vθ = v z=0 represent the radial and the
tangential displacements of the middle plane.
Because the piezoceramic circular disk is thin, the out-of-plane (transverse) vibration and the in-plane (tangen-tial and extensional) vibration are decoupled. We ana-lyze in detail the dynamic characteristics for transverse, tangential, and extensional vibrations, including the res-onant frequencies, mode shapes, and electrical currents. The nonaxisymmetric modes are investigated for trans-verse vibration, and tangential and extensional vibrations are restricted to axisymmetric modes.
A. Transverse Vibration
Suppose that the piezoceramic disk is driven by an al-ternating current (AC) voltage V eiωt and the transverse
vibration is nonaxisymmetric, the displacement in the z-direction can be assumed to have the following form:
w(r, θ, t) = W (r, θ)eiωt, (8) where ω is the angular frequency. If the time-dependent term eiωtis uniformly suppressed in the analysis, by
sub-stituting (8) into (7a)–(7c), then (6a)–(6c) of the stress components can be rewritten as follows:
σrr= 1 sE 11 1− v2 p z∂ 2W ∂r2 + vp z r ∂W ∂r + 1 r ∂2W ∂θ2 − d31 sE 11(1− vp) Ez, (9a) σθθ= 1 sE11 1− v2 p z r ∂W ∂r + 1 r ∂2W ∂θ2 + vpz ∂2W ∂r2 − d31 sE 11(1− vp) Ez, (9b) σrθ= 1 sE11(1 + vp) z r ∂2W ∂r∂θ− z r2 ∂W ∂θ . (9c)
Herein it is assumed that the stresses as indicated in (9a)–(9c) are induced by the out-of-plane displacement W (r, θ) only. The electrical potential boundary condition on the electrode-covered surfaces is:
ϕ z=±h 2
From (10), (9a) and (9b), the electric-field is found to be: Ez=− 2V h − d31 εT 33 z sE 11(1− vp) 1− k2 p · ∂2W ∂r2 + 1 r ∂W ∂r + 1 r ∂2W ∂θ2 , (11) where kp = 2d2 31 εT
33S11E(1−vp) is the planar electromechanical
coupling coefficient.
Applying the integral operator −h/2h/2 K dz to the equi-librium equations (1a)–(1c) and using (9a)–(9c) and (11), the governing equation of the transverse vibration has the following form: D ∂4W ∂r4 + 2 r ∂3W ∂r3 − 1 r2 ∂2W ∂r2 + 1 r3 ∂W ∂r − 2 r3 ∂3W ∂r∂θ2 + 2 r2 ∂4W ∂r2∂θ2 + 4 r4 ∂2W ∂θ2 + 1 r4 ∂4W ∂θ4 − ρhω2 W = D∇4W− ρhω2W = 0, (12) where ∇4 is a biharmonic operator and the equivalent
bending stiffness is defined by: D= h 3 12· 2− (1 − vp) k2p 2sE11 1− v2 p 1− k2 p . (13)
According to (12), the general solution of nonaxisym-metric transverse vibration for the piezoceramic disk is:
W (r, θ) = C1(n)Jn(β1r) + C (n) 2 In(β1r) cos nθ, n = 0, 1, 2, 3, K, (14a) in which C1(n) and C (n)
2 are constants and
β14= ρhω2 D = 12ρω2 h2 · 2sE 11 1− v2 p 1− k2 p 2− (1 − vp) kp2 . (14b) In (14a), Jnis an nth order Bessel function of the first kind
and Inis an nth order modified Bessel function of the first
kind. For the circumferential-free boundary conditions at r = R, we have: h/2 −h/2zσrrdz = 0, (15a) and h/2 −h/2σrzdz + 1 r ∂ ∂θ h/2 −h/2zσrθdz = 0. (15b)
Two systems of linear equations are obtained from (15a) and (15b), giving (16) (see next page), where ξ = β1R.
To obtain a nontrivial solution for the constants C1(n) and C2(n), the determinant of coefficient matrix must be equal
to zero, and that will yield the characteristic equation of resonant frequencies for nonaxisymmetric transverse vi-brations as shown in (17) (see next page).
It should be noted that the values n in (17) refer to the number of nodal diameters, and the sequence of roots represents the number of nodal circles. From (14b) and (17), the resonant frequencies for nonaxisymmetric transverse vibration of piezoceramic circular plates with circumferentially-free boundary conditions are:
f = ξ 2h 2πR2 2− (1 − vp) kp2 24ρsE 11 1− v2 p 1− k2 p . (18)
Using (14a) and (16), the transverse displacement can be expressed as given in (19) (see next page).
From (6d), the electrical current I can be expressed as: I = ∂ ∂t S Dzds = iω 2π 0 R 0 d31(1 + vp) z sE 11 1− v2 p ∂2W ∂r2 + 1 r ∂W ∂r + 1 r2 ∂2W ∂θ2 + εT33− 2d231 sE 11(1− vp) Ez rdrdθ = iω2πR 2V εT 33 h · k2p− 1 . (20)
It also should be noted that the integration in (20) can be worked out and expressed in a very simple form. It is shown clearly from (20) that the electrical current I does not approach infinity, even if the transverse vibra-tions are excited at resonant frequencies. This implies that the resonant frequencies of transverse vibrations cannot be measured by the impedance variation method, such as an impedance analyzer.
B. Tangential Vibration
Suppose that the tangential vibration is axisymmetric, the tangential displacement has the following form:
vθ(r, t) = Θ(r)eiωt. (21)
Follow a similar procedure, after suppressing the time-dependent term eiωt, the stress components for the tan-gential vibration can be expressed as:
σrr= σθθ=− d31 sE 11(1− vp) Ez, (22a) σrθ= 1 2sE 11(1 + vp) dΘ dr − Θ r , (22b) where Ez=− 2V h . (22c)
C1(n) 1− vp 1 + vp n2Jn(ξ)− ξJn(ξ) − 1 1 + vp + k 2 p 21− k2 p ξ2Jn(ξ) + C2(n) 1− vp 1 + vp n2In(ξ)− ξIn(ξ) + 1 1 + vp + k 2 p 21− k2 p ξ2In(ξ) = 0, (16a) C1(n) (1− vp) n2 1 + vp [Jn(ξ)− ξJn(ξ)]− 1 1 + vp + k 2 p 21− k2 p ξ3Jn(ξ) + C2(n) (1− vp) n2 1 + vp [In(ξ)− ξIn(ξ)] + 1 1 + vp + k 2 p 21− k2 p ξ3In(ξ) = 0 (16b) 1− vp 1 + vp n2Jn(ξ)− ξJn(ξ) − 1 1 + vp + k 2 p 21− k2 p ξ2Jn(ξ) 1− vp 1 + vp n2In(ξ)− ξIn(ξ) + 1 1 + vp + k 2 p 21− k2 p ξ2In(ξ) = (1− vp) n2 1 + vp [Jn(ξ)− ξJn(ξ)]− 1 1 + vp + k 2 p 21− k2 p ξ3Jn(ξ) (1− vp) n2 1 + vp [In(ξ)− ξIn(ξ)] + 1 1 + vp + k 2 p 21− k2 p ξ3In(ξ) . (17) W (r, θ) = C1(n) Jn(β1r)− 1− vp 1 + vp n2Jn(ξ)− ξJn(ξ) − 1 1 + vp + k 2 p 21− k2 p ξ2Jn(ξ) 1− vp 1 + vp n2In(ξ)− ξIn(ξ) + 1 1 + vp + k 2 p 21− k2 p ξ2In(ξ) · In(β1r) cos nθ (19)
Substituting (22) into the equilibrium (1b) and integrat-ing over the thickness of plates, the governintegrat-ing equation of the tangential vibration has the following expression:
d2Θ dr2 + 1 r dΘ dr − 1 r2Θ− 2ρω 2 sE11(1 + vp) Θ = 0. (23) The general solution of tangential vibrations for the piezoceramic disk is a first-kind Bessel function:
Θ(r) = C3J1(β2r), (24a)
in which C3 is a constant and β22= 2ρs
E
11(1 + vp) ω2. (24b)
With the aid of the boundary condition at r = R, h/2
−h/2σrθdz = 0, (25)
we can obtain the characteristic equation of resonant fre-quencies for tangential vibrations as:
ζJ0(ζ)− 2J1(ζ) = 0, ζ = β2R. (26)
It can be shown that, by solving (26) for ζ, the tangen-tial resonant frequencies for piezoceramic disks are:
f = ζ 2πR 1 2ρsE 11(1 + vp) . (27)
The electrical current I for tangential vibrations can be worked out as:
I = ∂ ∂t S Dzds = iω 2π 0 R 0 ! − 2d231 sE 11(1− vp) + εT33 Ez " rdrdθ = iω2πR 2V εT 33 h · kp2− 1 . (28)
Hence, the resonant frequencies of tangential vibra-tions also cannot be measured by the impedance variation method.
C. Extensional Vibration
Suppose that the extensional vibration is axisymmetric, the radial extensional displacement of the middle plane can be assumed to be:
ur(r, t) = U (r)eiωt. (29)
The stress-displacement relations for the extensional vi-bration are given by:
σrr= 1 sE11 1− v2 p dU dr + vp U r + d31 sE 11(1− vp)· 2V h , (30a) σθθ= 1 sE 11 1− v2 p U r + vp dU dr + d31 sE 11(1− vp)· 2V h , (30b) in which (22c) has been used. Substituting (30) into the equilibrium (1a) and integrating over the thickness of plates, we have the well-known governing equation of ex-tensional vibrations: d2U dr2 + 1 r dU dr − 1 r2U− ρω 2 sE11 1− v2p U = 0. (31) The general solution of (31) is:
U (r) = C4J1(β3r), (32a)
where C4 is a constant and β32= ρs E 11 1− vp2 ω2. (32b)
For the boundary condition at r = R, we have: h/2
−h/2σrrdz = 0. (33)
The constant C4 is found to be: C4=
2V d31(1 + vp)
(1− vp) J1(β3R)− β3RJ0(β3R) R
h. (34) The electrical current I for extensional vibrations can be developed as: I = ∂ ∂t S Dzds = iω 2π 0 R 0 d31(1 + vp) sE 11 1− v2 p dU dr + U r +2ε T 33V h k2p− 1 rdrdθ = iω2πR 2V εT 33 h · 1− vp+ (1 + vp) k2 p k2 p− 1 J1(β3R)− β3RJ0(β3R) (1− vp) J1(β3R)− β3RJ0(β3R) . (35)
From (35), the resonant frequencies can be found when-ever the current I approaches infinity. The characteristic equation of resonant frequencies for extensional vibrations is given by:
ηJ0(η) = (1− vp) J1(η), η = β3R. (36a)
Eq. (36a) is the well-known result for two-dimensional analysis of the radial modes. The antiresonant frequen-cies for which the current through the piezoceramic disk vanishes are also important characteristics, and are deter-mined from the roots of the following equation:
1− vp+ (1 + vp) k2 p k2 p− 1 J1(η) = ηJ0(η). (36b) From (32b) and (36), the radial extensional resonant and antiresonant frequencies for piezoceramic disks with free boundary conditions can be expressed as:
f = η 2πR 1 ρsE11 1− v2 p . (37)
III. Theory of the Time-Averaged AF-ESPI Method
For the out-of-plane vibration measurement, the first image is recorded as a reference after the specimen vibrates periodically. As the vibration of the specimen continues, we assume that the vibration amplitude changes from A to A + ∆A because of the electronic noise or instability of the apparatus. When the vibration amplitude variation ∆A is rather small, subtract the first image from the second by the image processing system, and the resulting image intensity can be expressed as
I = √
I0Ir
2
(cosΦ) · 2π∆Aλ (1 + cos Ψ) 2 · J0 2πA λ (1 + cos Ψ) , (38) where I0 is the object light intensity, Ir is the reference
light intensity, Φ is the phase difference between object and reference light, λ is the wavelength of laser, and Ψ is the angle between object light and observation direction.
Similar to the out-of-plane vibration case, the resulting image intensity for the in-plane vibration measurement is:
I = I0 2 (cos Φ)· 2π∆A λ (2 sin Ψ ) 2· J 0 2πA λ (2 sin Ψ ) (39), where I0 is the object light intensity, A is the vibration
amplitude of in-plane vibration, and Ψis half of the angle between two illumination lights.
From (38) and (39), we can see that the fringe patterns for both the out-of-plane and in-plane vibrations obtained
TABLE I
Material Properties of PIC-151.
PIC-151 Ceramics sE 11(10−12m 2/N) 16.83 sE 33 19.0 sE 12 −5.656 sE 13 −7.107 sE 44 50.96 sE 66 44.97 d31(10−10m/V) −2.14 d33 4.23 d15 6.1 εT 11(10−9 F/m) 17.134 εT 33 18.665 ρ (kg/m3) 7800
by the AF-ESPI method are dominated by a zero-order Bessel function J0. The AF-ESPI method for transverse
vibration was first proposed by Wang et al. [18]. Ma and Huang [15] provided a detailed discussion of the AF-ESPI method to investigate the out-of-plane and in-plane vi-brations of piezoelectric rectangular parallelepipeds for a three-dimensional configuration.
IV. Theoretical and Experimental Results
A piezoceramic disk with diameter D = 30 mm and thickness h = 1 mm is selected for experimental investiga-tions; the modal number is PIC-151 (Physik Instrumente, Lederhose, Germany). The polarization is in the thickness direction, and two opposite faces of the specimen are fully covered by silver electrodes. The piezoceramic disk is ex-cited by the application of an AC voltage across electrodes on the two major surfaces and has completely stress-free boundary conditions. The electroelastic properties of the test specimen PIC-151 are listed in Table I.
The schematic layout of time-averaged AF-ESPI opti-cal systems shown in Figs. 2 and 3 are used to perform the out-of-plane and in-plane experimental measurements for resonant frequencies and corresponding mode shapes, re-spectively. A He-Ne laser with 30 mW and the wavelength λ = 632.8 nm is used as the coherent light source. We use a charge-coupled device (CCD) camera (Pulnix TM-7CN, Pulnix America, Inc., Sunnyvale, CA) and a frame grabber (Dipix P360F, Dipix Technologies, Inc., Ottawa, Ontario, Canada) with a digital signal processor onboard to record and process the images. As shown in Fig. 2 for the out-of-plane measurement, the laser beam is divided into two parts, the object and reference beam, by a beamsplitter. The object beam travels to the specimen, then reflects to the CCD camera. The reference beam goes directly to the CCD camera via a mirror and a reference plate. 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, then reflect to the CCD camera. The CCD camera converts the intensity dis-tribution of the interference pattern of the object into a corresponding video signal at 30 frames per second. The signal is electronically processed then converted into an
Fig. 2. Schematic of AF-ESPI setup for out-of-plane measurement.
Fig. 3. Schematic of AF-ESPI setup for in-plane measurement.
image on the video monitor. Interpretation of the fringe image is similar to reading of a contour map. To achieve the sinusoidal output, a function generator (HP-33120A, Hewlett Packard, Palo Alto, CA) connected to a power amplifier (NF Electronic Instruments 4005 Type, NF Cor-poration, Kohokuku, Yokohama, Japan) is used.
The experimental procedure of the AF-ESPI technique is performed as follows. A reference image is taken after the piezoceramic plate is excited into vibration, then the second image is taken, and the reference image is sub-tracted by the image processing system. If the frequency of vibration is not the resonant frequency, only random distributed speckles are displayed, and no fringe patterns
will be shown. However, if the frequency of vibration is in the neighborhood of the resonant frequency, stationary distinct fringe patterns will be observed. Then, as the func-tion generator is carefully and slowly adjusted, the number of fringes will increase and the fringe pattern will become more clear as the resonant frequency is approached. From the aforementioned experimental procedure, the resonant frequencies and the correspondent mode shapes can be de-termined at the same time.
From the AF-ESPI optical system and the experimen-tal procedure mentioned above, the resonant frequencies and corresponding mode shapes for the transverse and ex-tensional vibrations can be determined. In addition to the theoretical analysis and experimental measurement, finite element calculation is also performed using the commer-cially available software ABAQUS (Hibbit, Karlsson, and Sorensen, Inc., Pawtucket, RI) finite element package [19] in which 20-node, three-dimensional, solid piezoelectric el-ements (C3D20E) are selected to analyze the problem. Fig. 4 shows the experimental and numerical results for the first eight-mode shapes of transverse vibrations. Fig. 5 shows the first four-mode shapes of extensional vibrations. In Figs. 4 and 5 we indicate the phase of displacement in finite element results as solid or dashed lines, with the solid lines 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 is the brightest fringe in the experimental results, represents the nodal lines of the vibrating piezo-ceramic disk at resonance. The mode shapes obtained by experimental results can be checked by the nodal lines and fringe patterns with the numerical calculations. Excellent agreements of the experimental measurement and numer-ical calculation are found for both the transverse and ex-tensional vibration modes.
A point-wise optical technique, LDV also was used to validate the resonant frequencies of the transverse vibra-tion modes, and the gain spectrum of which is shown in Fig. 6. For the LDV system, a built-in dynamic signal ana-lyzer (DSA) composed of dynamic signal anaana-lyzer software and a plug-in waveform generator board can provide the piezoceramic disk with the swept-sine excitation signal, whose gain at corresponding frequencies is analyzed by the DSA software. The peaks appearing in the frequency response curve are resonant frequencies for transverse vi-brations. Table II shows the first eight transverse resonant frequencies of the piezoceramic disk obtained by using the AF-ESPI, LDV, theoretical prediction, and the FEM cal-culation. It is noteworthy that the theoretical predictions [i.e., from (17) and (18)] agree very well with the finite element results, with the discrepancies within 3.5%. In ad-dition, the experimental measurements by AF-ESPI and LDV are in excellent agreement, with differences generally below 1%. The comparison of the resonant frequencies ob-tained from the analytical method and experimental mea-surements are also in good agreement.
Because the electrical impedance of piezoceramic mate-rials drops to a local minimum when it vibrates at a
reso-nant frequency, the resoreso-nant frequency also can be deter-mined by an impedance analysis. Experimental impedance measurements of the piezoceramic disk are obtained by us-ing an HP4194A impedance/gain-phase analyzer (Hewlett Packard). The impedance curve for the piezoceramic disk measured from HP4194A is shown in Fig. 7 (top) and the simulated impedance curve calculated by (35) is shown in Fig. 7 (bottom). We can see that both results are well correlated. The local minima and maxima appearing in the impedance curve correspond to resonance and antires-onance, respectively. As we expected, only the resonant frequencies of the radial extensional modes are indicated in Fig. 7 (top), i.e., those of the transverse and tangen-tial modes cannot be obtained by the impedance analy-sis. This phenomenon can be explained qualitatively by means of the characteristics of piezoelectricity. When the piezoceramic plate vibrates at a resonant frequency, the charge will be strongly induced on the electrode surfaces due to the vibration deformation, called the direct piezo-electric effect, and the impedance will drop to a local minimum value. This is the reason that the resonant fre-quencies of piezoceramic plates can be determined by us-ing the impedance analyzer. However, if the summation of the induced charge distributed over the electrode sur-faces is zero, we are not able to find the large variation of impedance at the resonant frequency. Table III shows the first four extensional resonant frequencies of the piezo-ceramic disk obtained by using the AF-ESPI, impedance analysis, theoretical prediction, and FEM calculation. The discrepancy of resonant frequencies between AF-ESPI and impedance analysis is smaller than 0.3%. The theoretical predictions [i.e., from (36a) and (37)] are in excellent agree-ment with the finite eleagree-ment results and the discrepancies are within 1%, which is better than that of transverse vi-bration modes. The difference between the experimental measurements and analytical results may result from the determination of the material properties or the defects of the piezoceramic disk.
In view of the experimental techniques mentioned previ-ously, we find that the tangential vibration modes cannot be obtained experimentally. Hence, only the theoretical predictions and finite element calculations are performed. Table IV shows the comparison between the theoretical predictions [i.e., from (26) and (27)] and FEM calcula-tions for the first three tangential resonant frequencies of the piezoceramic disk. The discrepancies of both results are within 0.01%.
The electromechanical coupling coefficient is an impor-tant characteristic of piezoceramic elements used to mea-sure the energy conversion efficiency. The dynamic elec-tromechanical coupling coefficient proposed by Mason [20] can be computed by the expression:
kd= # f2 a− fr2 fa , (40)
where fr is the resonant frequency and fa is the
antires-onant frequency. This coefficient is of special interest in designing piezoelectric elements. Eq. (40) usually is used
Fig. 4. Mode shapes of the transverse vibration obtained by AF-ESPI and FEM for the piezoceramic disk.
TABLE II
Results of Resonant Frequencies Obtained by Theory, FEM, LDV, and AF-ESPI for the Transverse Vibration.
(a) (b) (c) (d)
Transverse Theory FEM LDV AF-ESPI Error Difference mode (Hz) (Hz) (Hz) (Hz) (a)(b) (c)(d) 1 3224 3193 3210 3180 0.97% 0.94% 2 7040 6990 6610 6580 0.72% 0.46% 3 7638 7499 7610 7495 1.85% 1.53% 4 13602 13238 13310 13190 2.75% 0.91% 5 15358 15095 14410 14380 1.74% 0.21% 6 21094 20371 20410 20230 3.55% 0.89% 7 25924 25165 24310 24160 3.02% 0.62% 8 28786 27925 26610 26600 3.08% 0.04% TABLE III
Results of Resonant Frequencies Obtained by Theory, FEM, Impedance Analysis, and AF-ESPI for the Extensional Vibration.
(a) (b) (c) (d)
Extensional Theory FEM Impedance AF-ESPI Error Difference
mode (Hz) (Hz) (Hz) (Hz) (a)(b) (c)(d)
1 64397 64368 69860 69745 0.05% 0.16%
2 167789 167404 179130 179250 0.23% −0.07% 3 266661 265147 283335 283740 0.57% −0.14% 4 364878 360988 385558 386735 1.08% −0.30%
Fig. 5. Mode shapes of the extensional vibration obtained by AF-ESPI and FEM for the piezoceramic disk.
to determine the electromechanical coupling coefficient by measuring the resonant and antiresonant frequencies. From Figs. 7(a) and (b), we can evaluate the value of kd
individually for each mode, and the results are shown in Table V for the extensional mode.
The theoretical derivation in this study is based on the assumption that the piezoceramic plat is thin, hence the theoretical predictions of resonant frequencies are accu-rate only if the thickness-to-diameter ratio (h/D) is small. To understand the validity of the theoretical results on the practical applications, the discrepancy between the theoretical analysis and finite element results of the res-onant frequencies for transverse and extensional vibra-tions are shown in Figs. 8 and 9, respectively. The cal-culations are performed for values of h/D ranging from 0.01 to 0.1. In general, the theoretical predictions of res-onant frequencies for the extensional vibration are better
Fig. 6. LDV output gain spectrum of transverse vibrations for the piezoceramic disk.
TABLE IV
Results of Resonant Frequencies Obtained by Theory and FEM for the Tangential Vibration.
Tangential Theory FEM
Mode (Hz) (Hz) Error
1 92002 92007 −0.005% 2 150791 150801 −0.007% 3 208163 208187 −0.01%
than those for the transverse vibration. The discrepancies of the first five modes for transverse vibration are within 5% for h/D < 0.05 and within 2% (h/D < 0.06) for the first three modes for the extensional vibration. The theoretical prediction for the first mode of extensional vibration is ex-cellent, with discrepancy for h/D = 0.1 of less than 0.5%. The displacement distribution of the mode shape based on the theoretical analysis also is evaluated to compare with that of the FEM. For simplicity, only the first two axisym-metric modes of transverse and extensional vibrations are presented in Fig. 10. It is shown that two results are highly coincident for the normalized vibration amplitude in the middle plane.
V. Conclusions
Most of the previous studies for vibration analysis of piezoelectric disks are analytical and numerical results for radial extensional modes. There are very few experimen-tal results available for the full-field configuration of mode shapes for vibrating piezoelectric disks. In this study, a
Fig. 7. Impedance variation curve of the piezoceramic disk by (top) impedance analyzer and (bottom) theoretical analysis.
TABLE V
Results of the Dynamic Electromechanical Coupling CoefficientkdObtained by Theory and FEM for the
Extensional Vibration.
Extensional kd kd Difference
vibration (Theory) (Impedance) (%) Mode 1 0.5832 0.4507 −22.71 Mode 2 0.2597 0.2121 −18.33 Mode 3 0.1665 0.1573 −5.56 Mode 4 0.1224 0.1479 20.86
Fig. 8. Discrepancies of resonant frequency predicted by theory and numerical calculation for the transverse vibration.
Fig. 9. Discrepancies of resonant frequency predicted by theory and numerical calculation for the extensional vibration.
Fig. 10. Normalized displacements calculated by theory and FEM for the (top) transverse and (bottom) extensional vibrations.
self-arranged, AF-ESPI optical setup with good visibility and noise reduction has been established to simultaneously obtain both the resonant frequencies and the correspond-ing mode shapes of transverse and extensional vibrations. This study investigates the vibration of piezoceramic disks with free-boundary conditions for transverse, tan-gential, and radial extensional modes by theoretical analy-sis, numerical simulation, and experimental measurement. Due to the symmetry of the material constants for piezo-ceramic materials, the nonaxisymmetric modes are ana-lyzed for the transverse vibration, and tangential and ex-tensional vibrations are restricted to axisymmetric modes.
The theoretical predictions for resonant frequencies and mode shapes are presented in detail. Two optical tech-niques (AF-ESPI and LDV) and electrical impedance mea-surement are used to validate the theoretical results. Only the resonant frequencies of radial extensional vibration can be measured by the impedance analysis, and the res-onant frequencies of transverse vibration are determined by the LDV system. However, both resonant frequencies and mode shapes for transverse and radial extensional vibrations are simultaneously measured by the AF-ESPI method. Good agreement of both the resonant frequencies and mode shapes are obtained for experimental measure-ments and theoretical predictions. The accuracy of the res-onant frequencies predicted by the theoretical analysis is verified by the FEM method with the h/D ratios ranging from 0.01 to 0.1.
References
[1] D. A. Berlincourt, D. R. Curran, and H. Jaffe, “Piezoelectric and piezomagnetic materials and their function as transduc-ers,” Phys. Acoust., vol. 1(A), pp. 169–270, 1964.
[2] J. Zelenka, Piezoelectric Resonators and Their Application. Prague, Czech Republic: Academia/Prague, 1986.
[3] H. F. Tiersten, Linear Piezoelectric Plate Vibrations. New York: Plenum, 1969.
[4] E. P. Eer Nisse, “Variational method for electroelastic vibration analysis,” IEEE Trans. Sonics Ultrason., vol. SU-14, no. 4, pp. 153–160, 1967.
[5] E. A. G. Shaw, “On the resonant vibrations of thick barium titanate disks,” J. Acoust. Soc. Amer., vol. 28, no. 1, pp. 38–50, 1956.
[6] R. Holland, “Contour extensional resonant properties of rectan-gular piezoelectric plates,” IEEE Trans. Sonics Ultrason., vol. SU-15, no. 2, pp. 97–105, 1968.
[7] H. A. Kunkel, S. Locke, and B. Pikeroen, “Finite-element anal-ysis of vibrational modes in piezoelectric ceramics disks,” IEEE
Trans. Ultrason., Ferroelect., Freq. Contr., vol. 37, no. 4, pp.
316–328, 1990.
[8] N. F. Ivina, “Numerical analysis of the normal modes of circular piezoelectric plates of finite dimensions,” Sov. Phys. Acoust., vol. 35, no. 4, pp. 385–388, 1990.
[9] N. Guo, P. Cawley, and D. Hitchings, “The finite element anal-ysis of the vibration characteristics of piezoelectric discs,” J.
Sound Vib., vol. 159, no. 1, pp. 115–138, 1992.
[10] U. Minoni and F. Docchio, “An optical self-calibrating technique for the dynamic characterization of PZT’s,” IEEE Trans.
In-strum. Meas., vol. 40, no. 5, pp. 851–854, 1991.
[11] M. Ohki, N. Shima, and T. Shiosaki, “Optical measurement of piezoelectric vibration in circular rod and disk ceramics,” Jpn.
J. Appl. Phys., vol. 31, no. 9B, pp. 3272–3275, 1992.
[12] M. Chang, “In-plane vibration displacement measurement using fiber-optical speckle interferometry,” Preci. Eng., vol. 16, no. 1, pp. 36–41, 1994.
[13] B. Koyuncu, “The investigation of high frequency vibration modes of PZT-4 transducers using ESPI techniques with ref-erence beam modulation,” Opt. Lasers Eng., vol. 1, pp. 37–49, 1980.
[14] J. R. Oswin, P. L. Salter, F. M. Santoyo, and J. R. Tyrer, “Elec-tronic speckle pattern interferometric measurement of flexten-sional transducer vibration patterns: In air and water,” J. Sound
Vib., vol. 172, no. 4, pp. 433–448, 1994.
[15] C. C. Ma and C. H. Huang, “The investigation of three-dimensional vibration for piezoelectric rectangular paral-lelepipeds by using the AF-ESPI method,” IEEE Trans.
Ul-trason., Ferroelect., Freq. Contr., vol. 48, no. 1, pp. 142–153,
2001.
[16] C. H. Huang and C. C. Ma, “Vibration characteristics for piezo-electric cylinders using amplitude-fluctuation electronic speckle
pattern interferometry,” AIAA J., vol. 36, no. 12, pp. 2262–2268, 1998.
[17] N. N. Rogacheva, The Theory of Piezoelectric Shells and Plates. Boca Raton, FL: CRC Press, 1994.
[18] W. C. Wang, C. H. Hwang, and S. Y. Lin, “Vibration measure-ment by the time-averaged electronic speckle pattern interfer-ometry methods,” Appl. Opt., vol. 35, no. 22, pp. 4502–4509, 1996.
[19] ABAQUS User’s Manual, ver. 5.5.. Pawtucket, RI: Hibbit, Karlsson, and Sorensen, Inc., 1995.
[20] W. P. Mason, Piezoelectric Crystals and Their Application to
Ultrasonics. New York: Van Nostrand, 1950.
Chi-Hung Huang received his B.S., M.S., and Ph.D. degrees in mechanical engineering from the National Taiwan University, Taipei, Taiwan, Republic of China, in 1987, 1992, and 1998, respectively. Since 1998, he has worked at the Mechanical Engineering Department, Ching Yun University, Chung-Li, Taiwan, Re-public of China, as an associate professor.
His current research interests include ex-perimental optical measurement and dynamic behavior of piezoelectric materials.
Yu-Chih Lin received the B.S. degree in me-chanical engineering from the National Tai-wan University, TaiTai-wan, Republic of China, and the M.S. degree in biomedical engineering from National Cheng Kung University, Tai-wan, Republic of China, in 1994 and 1996, respectively. She is currently a candidate for the Ph.D. degree in the Department of Me-chanical Engineering, National Taiwan Uni-versity.
She currently is engaged in research on res-onant characteristics of piezoelectric disks un-der water. Her research interests are in the fields of vibration analysis of piezoelectric materials and photomechanics.
Chien-Ching Ma received the B.S. degree in agriculture engineering from the National Tai-wan University, TaiTai-wan, Republic of China, in 1978, and the M.S. and Ph.D. degrees in mechanical engineering from Brown Univer-sity, Providence, RI, in 1982 and 1984, re-spectively. From 1984 to 1985, he worked as a postdoc in the Engineering Division, Brown University. In 1985, he joined the faculty of the Department of Mechanical Engineering, National Taiwan University, Taiwan, Repub-lic of China, as an associate professor. He was promoted to full professor in 1989.
His research interests are in the fields of wave propagation in solids, fracture mechanics, solid mechanics, piezoelectric material, and vibration analysis.
