VIBRATION CHARACTERISTICS OF A PIEZOELECTRIC
DISC COUPLED WITH AN ELASTIC BODY
Chunguang Piao and Jin Oh Kim
Department of Mechanical Engineering, Soongsil University, Dongjak-gu, Seoul 156-743, Korea
e-mail: jokim@ssu.ac.kr
The vibration characteristics of a piezoelectric disc coupled with an elastic body were
investigated in this paper. It was assumed that the vibration modes were coupled in the axial
and radial directions. Theoretically the differential equations of piezoelectric motions were
derived in terms of the axial and radial displacements and the electric potential. The
differential equations of elastic motion were expressed in terms of the axial and radial
displacements. Solving the governing equations and boundary conditions of the coupled
structure produced the characteristic equations providing the natural frequencies. The results
of the theoretical analysis were compared with those of the finite-element analysis and
experiments, and the theoretical analysis was verified.
1. Introduction
A piezoelectric disc is one of the main elements of ultrasonic sensors and actuators which adopt
electromechanical transduction mechanisms [1]. The transducers using piezoelectricity [2] are
widely used, especially in ultrasonic sensors [3] which convert mechanical vibrations into electric
signals, and actuators [4] which convert electric signals into mechanical vibrations. Many ultrasonic
devices, such as ultrasonic cleaners [5], utilize resonances to use the power of mechanical waves.
Piezoelectric transducers for these ultrasonic devices are designed on the basis of the vibration
characteristics of resonance [6].
It is well known that the natural frequencies of a piezoelectric disc depend on geometric shapes
and dimensions as well as material properties and boundary conditions [7]. In some cases the
fundamental frequency of a piezoelectric disc is too high to be used for the devices and it is to be
reduced severely. The natural frequency of a thickness mode would be reduced by increasing the
thickness of a disc. This is, however, not a desirable method because the larger distance between the
electrodes on the parallel flat faces would result in the smaller electric field and thus the smaller
piezoelectric effect also. Another and promising method is to add elastic blocks on each flat face of
the disk [8].
The purpose of this paper is to investigate the effect of an elastic body laminated on a
piezoelectric disc polarized in the thickness direction. Theoretically the differential equations of
piezoelectric motions are derived in terms of the radial and axial displacements and the electric
potential. The differential equations of elastic motion are expressed in terms of the radial and axial
displacements. The governing equations satisfying boundary conditions are solved to produce the
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characteristic equations providing the natural frequencies. Experimentally the natural frequencies
are measured by using an impedance analyzer. The results of the theoretical analysis are compared
with those of the finite-element analysis and experiments, and the theoretical analysis is verified.
2. Theoretical analysis
A piezoelectric disc coupled with an elastic body is schematically shown in Fig. 1, where a is
the radius and l 1 and l 2 are the thicknesses of the piezoelectric and elastic disks, respectively. The
piezoelectric disc has uniform electrodes on the top and bottom surfaces ( z = 0, l 1) .

piezoelectric disc
piezoelectric
elastic body dis




elastic body
(a) coupled disc
(b) coordinates
Figure 1. Schematic diagram of a piezoelectric disc coupled with an elastic body.
2.1 Constitutive equations
The electromechanical relations were well formulated, and the constitutive equations of the
piezoelectric material are expressed as follows [1,2]:
T
(1)
{T} = éëc E ùû {S} - [e] {E}
= [e ]{S} + éëε S ùû {E}
where {T} , {S} , {D} , and {E} are the matrices of stresses, strains, electric displacements, and
(2)
{D}
electric fields, respectively. In addition, éëc E ùû , [e ] , and éëε S ùû are the coefficient matrices of the
stiffness with constant electric field, the piezoelectric stress constant, and the permittivity with
constant strain, respectively.
The axisymmetric vibration in a disc can be formulated in general in terms of radial
displacement u ( r , z , t ) and axial displacement w ( r , z , t ) with cylindrical coordinates ( r , q , z ) and
time ( t ) . In the piezoelectric disc, the electric field ( E z ) is related with the electric potential
f ( z, t ) . Eqs. (1) and (2) can be rewritten as follows [7]:
(3a)
(3b)
æ¶ ö
æ ö
æ¶ ö
æ ¶f ö
s r1 = c11E ç u1 ÷ + c12E ç u1 ÷ + c13E ç w1 ÷ + e31 ç ÷
è ¶r ø
èrø
è ¶z ø
è ¶z ø
æ¶ ö
æ ö
æ¶ ö
æ ¶f ö
s q 1 = c12E ç u1 ÷ + c11E ç u1 ÷ + c13E ç w1 ÷ + e31 ç ÷
è ¶r ø
èrø
è ¶z ø
è ¶z ø
æ¶ ö
æ ö
æ¶ ö
æ ¶f ö
s z1 = c13E ç u1 ÷ + c13E ç u1 ÷ + c33E ç w1 ÷ + e33 ç ÷
(3c)
è ¶r ø
èrø
è ¶z ø
è ¶z ø
æ ¶u ö
æu ö
æ ¶w ö
æ ¶f ö
Dz = e31 ç 1 ÷ + e31 ç 1 ÷ + e33 ç 1 ÷ - e 33S ç ÷
(3d)
è ¶r ø
èrø
è ¶z ø
è ¶z ø
Similarly, the axisymmetric vibration in the isotropic elastic disk can be formulated without electric
potential f ( z , t ) and electric displacement Dz as follows [9]:
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(4a)
(4b)
(4c)
(4d)
¶ ö
¶
æ¶
s r 2 = l ç u 2 + u 2 + w2 ÷ + 2G u 2
r
¶z ø
¶r
è ¶r
¶ ö
æ¶
s 2 = l ç u 2 + u 2 + w2 ÷ + 2G u 2
r
¶z ø
r
è ¶r
¶ ö
¶
æ¶
s z 2 = l ç u 2 + u 2 + w2 ÷ + 2G w2
r
¶z ø
¶z
è ¶r
¶ w2 ö
æ ¶u 2
+
t zr 2 = G ç
÷
¶r ø
è ¶z
where l and G are Lame constants and especially G is shear modulus.
The axisymmetric motions in the stacked structure of a piezoelectric disc and an elastic body
are coupled with radial and axial modes. For simplicity two kinds of modes are separately
considered in the theoretical analysis and the results are compared with three-dimensional finiteelement analysis and experiments.
2.2 Radial in-plane modes
As described in detail in Reference [7], the governing equation for the radial motion of the
piezoelectric disc is
¶ 2 u1 1 ¶u1 u1
1 ¶ 2 u1
+
=
(5a)
2
2
2
¶r 2
r ¶r
C R1 ¶t
r
1/2
where C R1 = éëc11p / r 1ùû . Similarly, the governing equations for the elastic body is
¶ 2 u 2 1 ¶u 2 u 2
1 ¶2 u 2
+
=
(5b)
2
2
2
¶r 2
r ¶r
C R 2 ¶t
r
1/2
where C R 2 = éël 1p / r 2 ùû . C R1 and C R 2 in the wave Eq. (15) are the propagation speeds of the
wave.
The radial displacement at the center is zero and the circumferential faces are free of traction,
and
(6a)
at r = 0
(i = 1, 2)
u i ( 0, z , t ) = 0
(6b)
at r = a
(i = 1, 2)
s ri ( a, z , t ) = 0
The interface has continuity conditions as
(6c)
at z = 0
u r1 ( r , 0, t ) = u r 2 ( r , 0, t ) and t zr1 ( r , 0, t ) = t zr 2 ( r , 0, t ) = 0
When the voltage applied to the electrodes is a harmonic function of time t with frequency w ,
the displacement u and the electric potential f are regarded as harmonic functions of time with the
same frequency. Satisfying the boundary conditions (6a-c) yields the solutions of Eqs. (5a) and (5b)
as follows:
æ r ö jwt
(i = 1, 2)
u i ( r , z, t ) = A J 1 ç q n ÷ e
è aø
Here the variable q n ( n =1,2,3,…) is determined as 2.07, 5.40, 8.58, … from the characteristics
(7)
equation.
2.3 Axial out-of-plane modes
The governing equations in the thickness direction for the piezoelectric disc and the elastic body
are as follows:
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¶ 2 w1
1 ¶ 2 w1
=
2
2
¶z 2
C z1 ¶t
2
¶ 2f
e 33 ¶ w1
=
S
¶z 2
¶z 2
e 33
(8a)
(8b)
¶ 2 w2
1 ¶ 2 w2
=
2
2
¶z 2
C z 2 ¶t
1/2
1/2
2
æ
ö
E
S
where C z1 ç = éê c 33
+ e 33 / e 33
/ r 1ùú ÷ and C z 2 = éë ( l + 2G ) / r 2 ùû in the wave Eqs. (8a,c) are
û ø
è ë
the propagation speeds of the wave.
The boundary conditions are
(9a)
s z1 ( r , l 1, t ) = 0 , f ( r , l 1, t ) = V 0 e jwt at z = l 1
(8c)
(
)
(9b)
at z = - l 2
s z 2 ( r , -l 2, t ) = 0
and the interface conditions are
(9c)
w1 ( r , 0, t ) = w2 ( r , 0, t ) , s z1 ( r , 0, t ) = s z 2 ( r , 0, t ) , f ( r , 0, t ) = 0 at z = 0
When the voltage applied to the electrodes is a harmonic function of time t with frequency w ,
the displacement w and the electric potential f are regarded as harmonic functions of time with the
same frequency. The solutions of Eqs. (8a-c) have the following forms:
jwt
(10a)
w1(r , z , t ) = ( A1 cos k 1z + B1 sin k 1z ) e
( A2 cosk 2z + B 2 sin k 2z )
jwt
(10b)
w2(r , z , t ) =
(10c)
é
ù
f ( z, t ) = ê e33S ( A1 cos k 1z + B1 sin k 1z ) + a z + b ú e jwt
ë e 33
û
e
By satisfying the boundary and interface conditions (9a-c), the unknown constants are
determined and the characteristics equation is obtained.
3. Finite-element analysis
The theoretical analysis described in the previous section has advantages in the expressions to
calculate the natural frequencies and mode shapes. However, it has disadvantages in that the
analysis is based on some assumptions to simplify the real physical phenomenon. Before comparing
the theoretically-calculated results with experimental ones, the theoretical analysis is enhanced by a
finite-element analysis.
The natural frequencies and mode shapes are obtained by using the commercial software
ANSYS. Modal analysis and harmonic analysis were carried out with the boundary conditions that
all outer surfaces are mechanically free and the electric field is established in the thickness direction
by the electrodes on the top and bottom surfaces of the piezoelectric disc.
Harmonic analysis result can be displayed in the form of an impedance curve. In the impedance
magnitude curve, the minimum points represent the resonances of the transducer. The natural
frequencies obtained by the finite-element analysis are compared with the theoretically-calculated
values.
4. Experiments
The results of the theoretical analysis and finite-element analysis are verified by experiments. By
using an impedance gain/phase analyzer (Agilent Technology 4192A), impedance curves were
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measured for the specimen. The measurements were repeated with three pieces of the specimen. In
the impedance magnitude curve, the minimum points represent the resonances of the transducer.
The fundamental frequencies are compared with the other two results.
5. Conclusion
The paper investigated the vibration characteristics of piezoelectric hollow-disc transducers. It
was assumed that the vibration modes were coupled in the axial and radial directions. Theoretically
the differential equations of piezoelectric motions were derived in terms of the axial and radial
displacements and the electric potential. The boundary-value problem was solved to yield the
natural frequencies and mode shapes of the transducers. The theoretical results were verified by
comparing them with the results of finite-element analyses and experiments. It appeared that the
first mode of piezoelectric hollow-disc transducers was the radial mode and the fundamental natural
frequency was determined by the diameter and the radial boundary conditions of the circumferential
surface.
ACKNOWLEDGEMENTS
This research was supported by the MSIP (Ministry of Science, ICT & Future Planning), Korea,
under the Convergence-ITRC (Convergence Information Technology Research Center) support
program (NIPA-2014-H0401-14-1005) supervised by the NIPA (National IT Industry Promotion
Agency).
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7 Kim, D. J., Oh, S. H., Kim, J. O., Measurements of Radial In-plane Vibration Characteristics
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(Some Tables and Figures are intentionally not included in this paper.)
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