Accepted Manuscript Title: Theoretical analysis and experimental measurement of flexural vibration and dynamic characteristics for piezoelectric rectangular plate Authors: Yi-Chuang Wu, Yu-Hsi Huang, Chien-Ching Ma PII: DOI: Reference:
S0924-4247(17)30832-4 http://dx.doi.org/doi:10.1016/j.sna.2017.07.034 SNA 10229
To appear in:
Sensors and Actuators A
Received date: Revised date: Accepted date:
8-5-2017 16-7-2017 18-7-2017
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Theoretical analysis and experimental measurement of flexural vibration and dynamic characteristics for piezoelectric rectangular plate
Yi-Chuang Wu a, Yu-Hsi Huang b , and Chien-Ching Ma a, *
* Corresponding author (E-mail:
[email protected]). aDepartment
of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic
of China. b
Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan 106, Republic of China.
1
Highlights
The new contributions of the paper to the field are: 1. Vibration characteristics of a piezoelectric plate is investigated from theoretical analysis, numerical calculation and experimental measurement 2. The analytical solutions based on the superposition method are compared with two different techniques to confirm the validity 3. Two experimental methods are used to validate the theoretical result 4. Special configuration of electrodes for piezoelectric plates in experimental measurement is designed to improve the electromechanical coupling efficiency of transverse vibration 5. A full-field and non-contact experimental technique called AF-ESPI is developed to obtained resonant frequencies and mode shapes
ABSTRACT In this study, we investigate the out-of-plane vibration characteristics of a piezoelectric rectangular plate from theoretical analysis, numerical calculation and experimental measurement. The resonant frequencies and mode shapes of the piezoelectric rectangular plate with free boundary condition are analyzed in detail. The analytical solutions based on the superposition method are compared with two different techniques to confirm the validity. One is the numerical computation base on the finite element method (FEM), the other is experimental measurement. For experimental technique, two methods are used to validate the theoretical result, one is amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI) technique, and the other is laser Doppler vibrometry (LDV). This study uses a special configuration of electrodes for piezoelectric plates in experimental measurement of AF-ESPI to improve the electromechanical coupling efficiency of transverse vibration. It can be seen that specific vibration modes are easily excited by manipulating the configuration of electrodes. The superposition method is used to solve the out-of-plane displacement field and resonant frequencies for the piezoelectric plate. According to the explicit out-of-plane displacement field solution, the analytical results of mechanical and electric fields are also presented in explicit forms. It is shown that the comparison between theoretical analysis, numerical computations and experimental 2
measurement in terms of the resonant frequencies and corresponding mode shapes are with good agreement.
3
1. Introduction Piezoelectric materials have been widely used in many engineering applications, such as ultrasonic transducers, actuators, sensors, control systems, and non-destructive testing devices because of the dynamic characteristics and the excellent interaction between electrical and mechanical fields. Especially for the piezoelectric ceramics made of lead zirconate titanate (PZT) that contain both great pyroelectric effect and piezoelectric effect, have been applied to mechanical devices, medical equipment, and civil structures. In order to increase the efficiency and application of these piezoelectric structures in various engineering fields, the investigation of free vibration for rectangular piezoelectric plates is necessary. Methodologically, the study of piezoelectric rectangular plate from earlier works was based on the analysis for the dynamic characteristic of elastic rectangular plate. The analysis of free vibration problem for rectangular plate has been done by numerous researchers in the literatures. Family of vibration problems for rectangular plate with at least one pair of opposite edges simply supported have been solved [1]. The closed-form solution of these vibration problems can be obtained by the Lévy solution directly and easily. Beam functions have been utilized to present reasonably accurate results for free vibration problem of rectangular plate with 21 combinations of classical boundary conditions (i.e., clamped, simply-supported, or free) [2]. This study also provides both effects of changing the boundary conditions upon the frequencies and upon their accuracies, and the effects of changing Poisson’s ratio upon the resonant frequencies. Although some vibration problems have the exact solutions, the Lévy type can only satisfy the
4
problem that has one pair of opposite simply supported boundary conditions. The remaining rectangular plate problems are much more difficult to solve, because the Lévy type solution cannot be used directly. It is impossible to choose characteristic functions of one variable which not only satisfy the boundary conditions at two opposite edges completely but permit separation of variables as submitting the solution into the governing differential equation. Even though a series of products of beam eigenfunctions have been used to solve the mixed boundary value problems, the free-free beam eigenfunctions, unfortunately do not fully satisfy the free boundary conditions of the rectangular plate. The plate governing equation cannot be fulfilled with these same product functions individually, so that the eigenvalues of these beam functions for free edge conditions are considered to be approximate. In order to deal with the vibration problem of rectangular plate with free edges, the superposition method was applied to analyze the dynamic problems for cantilever plate and completely free rectangular plate [3,4]. Above-mentioned studies have laid a favorable foundation for the analysis of dynamic characteristic of rectangular plate, but the problem becomes more complicated as the piezoelectric effect is involved and must be taken into consideration in the mathematical derivation. Theoretical analysis and experimental measurements of piezoelectric materials have been investigated by several researchers for a long period of time. The finite element method was used to assess the flexural-extensional coupling effect of metal-ceramic transducers, and predict the performance of the transducers coupled acoustically to resonator systems [5]. To measure the
5
piezoelectric constants, the fiber-optic technique was utilized to evaluate the vibration amplitude of piezoelectric circular rod and disk locally, and used the measurement result to determine the associated piezoelectric coefficients [6]. The linear theory of piezoelectric plate and shell vibrations was derived [7]. In this representative work, the static and dynamic analyses on distributed sensing and control system for piezoelectric continua were provided, and the governing differential equations of different piezoelectric structures were presented. Recently various numerical procedures and approximate methods have been developed because of the high-speed computers. A finite-element model was developed base on the classical laminated plate theory for analyzing active vibration control of a piezoelectric laminated composite plate [8]. Finite-element analysis was used to predict the vibration characteristic of metal-piezoceramic composite thin plate with fully simply-supported boundary conditions [9]. Approximate solution was provided to the periodic equations of motion for layered thin circular and thick discs composed entirely or in part by piezoelectric materials under simply-supported or clamped types of boundary conditions [10]. There are some literatures provided theoretical analyses and experimental measurement of the three-dimensional vibration characteristic of piezoelectric devices. For three-dimensional vibration measurement, the coupled out-of-plane (transverse) and in-plane (tangential and extensional) vibration characteristic of piezoelectric rectangular parallelepipeds was investigated by using the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) method, the comparison between experimental result and finite-element calculations was in good agreement [11]. The
6
analytic solutions of three-dimensional vibration problems were applied to describe the dynamic characteristic of the functionally graded piezoelectric rectangular plate [12,13]. The finite-element method can predict the resonant frequencies and mode shape of piezoelectric rectangular plate in great accuracy. Nevertheless, the analytical method was restricted for the plate with fully simply-supported at edges. Recently, lots of research provided not only theoretical analysis but also experimental result to give the whole information of vibration characteristic for piezoelectric plate. The thin piezoceramic circular plates [14], annular disks [15], and composite rectangular plate [16] were investigated by theoretical analysis, numerical simulation, and experimental measurement, the vibration characteristics were discussed in detail in terms of natural frequencies and mode shapes. The closed-form solution for bimorph cantilever was derived and utilized several experimental measurements to confirm the validity of the solution [17]. The superposition method was used to evaluate the dynamic electromechanical coupling factor (DEMCF), and provided the way to maximize the excitation efficiency of in-plane mode in a rectangular piezoelectric resonator [18]. Vibration characteristics of piezoelectric bimorphs with different electrical connections was analyzed, and compared the measurement result with numerical and theoretical analysis [19]. Many recent studies ([18], [20]) have addressed the design of partial electrodes for piezoelectric materials using theoretical analysis, finite element method, and experimental measurements.
In previous studies,
electrode design was used to enhance or reduce the specific out-of-plane vibration mode of piezoelectric quartz plate [20] and in-plane vibration mode of piezoceramic plate [18].
7
The
application of the piezoelectric actuator generates out-of-plane actuation in MEMS were presented in the literature [21, 22]. Even through a large amount of literature have analyzed the out-of-plane vibration problem of piezoelectric rectangular plate, few of them provided appropriate theoretical solution to deal with the mixed boundary value problem. Previous analysis methods using beam mode shapes as the fundamental function overlook the two-dimensional coupling properties of thin plates, which leads to an overestimation of resonant frequencies, greater rigidity in the solid structure, and an increase in errors with regard to the prediction of frequencies. The theoretical approach presented in this study considers the two-dimensional properties of actual thin plates; therefore, the results are more accurate and more consistent with those obtained using the finite element analysis and experimental measurements. Furthermore, the superposition method used in this paper is easy to extend to analyze the more difficult Mindlin plate problem. In this study, a simple and straightforward analytic solution for the out-of-plane vibration problem of completely free piezoelectric rectangular plate is developed based on Kirchhoff–Love plate theory. First of all, the procedure to translate a piezoelectric material into an equivalent isotropic material must be developed. During the translation process, the relation between electrical and mechanical field can be obtained easily and directly. Next, the superposition method is applied to solve the vibration problem of rectangular plate, the analytical solutions of resonant frequencies and associated electrical and mechanical fields can be obtained without difficulty. The experimental
8
techniques of AF-ESPI and laser Doppler vibrometry (LDV), and the numerical computations based on the finite-element method, are used to ensure the accuracy of the analytic solution. The out-of-plane resonant frequencies and mode shapes can be measured by using AF-ESPI, which is a full-field and non-contact experimental technique, and the results can be obtained directly and immediately in the measurement. The LDV technique is based on the detection of Doppler shifts. As the resonance phenomenon occurs, the peak value in the sweep-frequency response curve is a local maximum point, so the resonant frequency of the out-of-plane vibration can be determined by the LDV system. Excellent agreements of resonant frequencies and mode shapes are obtained for the theoretical analysis, experimental measurement, and numerical calculations.
This study uses
special designs of electrode for various configurations to enhance the excited efficiency of the thin piezoelectric plate.
2. Formulations of the thin piezoelectric plate Figure 1 shows the dimension of the piezoelectric thin rectangular plate, the thickness of the electrode is assumed to be extremely small and is not considered in the analysis. The polarized direction is along x3 axis and x1 - x2 plane is the middle plane of the thin plate. The constituent equation of piezoelectric material is written as follows: E Tij cijkl Skl ekij E k ,
D i eikl Skl ikS E k ,
(1)
where Tij , Skl , E k and D i represent the components of strain, stress, electric field, and electric S E displacement, respectively. The coefficients of cijkl , ekij and ik are the elastic stiffness,
9
piezoelectric, and dielectric coefficients, respectively. Because the polarized piezoceramic material with a C6mm crystal symmetry class, the constituent equation could be rewritten in matrix form as: E c11 T 11 E T c12 22 E T33 c13 0 T23 T13 0 T12 0 D 1 0 D2 0 D 3 e31
E c12
E c13
0
0
0
0
0
E c11
E c13
0
0
0
0
0
E E c13 c33
0
0
0
0
0
E c44
0
0
0
e15
0
E c44
0
e15
0
0
0
0
0
0
0
0
0
0
0
E c66
0
0
0
e15
0
11S
0
0
0
e15
0
0
0
11S
e31 e33
0
0
0
0
0
e31 S11 e31 S 22 e33 S 33 0 2 S23 E E c11 c12 E 2 S c . (2) 0 13 , and 66 2 0 2 S12 0 E1 E 0 2 E 3 S 33
The strain-displacement relation is
Sij
1 Ui, j U j,i , 2
(3)
where U1 , U2 , and U 3 are the displacement field in the x1 , x2 , and x3 directions. The relation between electric field and electrostatic potential v x1 , x2 , x3 is
E1 v , E 2 v , E 3 v . x1 x2 x3
(4)
To obtain the equations of motion and the boundary conditions, Hamilton’s principle is used. The electric enthalpy H Sij , E j for the piezoelectric continuum is written as
1 1 H Sij , E j cijkl Sij Skl eijk E i S jk ij E i E j , 2 2
(5)
and Hamilton’s principle states for a piezoelectric continuum could be expressed as
dt U jU j H Sij , E j dV dt tjkUk Q jv d j 0 . t V 2 t t1
1
t1
0
0
(6)
j
In the result of Hamilton’s principle indicated in (6), is the mass density; tjk is the surface traction; Q j is the surface charge per unit area; V is the volume of piezoelectric materical 10
considered; j is the surface over the volume in the x j direction. Submitting (2), (3), (4) and (5) into (6) and collecting the coefficients of U1 , U2 , U3 and v , we could obtain the equations of motion and the charge equation of electrostatics as: T11 T12 T13 2U 21 , x1 x2 x3 t T21 T22 T23 2U 2 , x1 x2 x3 t 2
(7)
T31 T32 T33 2U 23 , x1 x2 x3 t D 1 D 2 D 3 0. x1 x2 x3
Because the rectangular plate is thin and the deformation is very small, some hypotheses to simplify the theoretical analysis are indicated as below:
The rectilinear element normal to the mid-surface remain normal after deformation and the thickness of the plate does not change during the deformation, i.e., S13 S23 0
The thin plate is traction-free on the top and the bottom surface, so the normal stress T33 can be neglected relative to the other stress components. Thus we set T33 0 in the whole rectangular plate.
The lateral surface is free of electrodes, the electrical displacement component D 1 and D 2 , i.e., D D 0 . can be neglected relative to D 3 1 2
Electrical potential varies along x3 direction is a quadratic function in term of variables x3 , i.e., v v x3v x32v ; where v , v and v 0
1
2
0
1
parameters. 11
2
are unknown electrical potential
According to the first hypothesis, the displacement fields for the thin plate theory (or Kirchhoff–Love plate theory) can be expressed as:
U1 x1 , x2 , x3 , t u1 x1 , x2 , t x3 U 2 x1 , x2 , x3 , t u2 x1 , x2 , t x3 U 3 x1 , x2 , x3 , t u3 x1 , x2 , t .
u3 x1 , x2 , t , x1
u3 x1 , x2 , t , x2
(8)
The components of u1 , u2 and u3 are the displacement field of middle surface in the x1 , x2 , and x3 directions, respectively. From second and third hypothesis and displacement assumption (8), the stress and electrical displacement components can be expressed as follows: E 2 E 2 c c cE 13 13 T11 c11E E u1,1 x3u3,11 c12E E u2,2 x3u3,22 e31 13 e E , E 33 3 c33 c33 c33 2 2 c13E c13E cE E E T22 c12 E u1,1 x3u3,11 c11 E u2,2 x3u3,22 e31 13 e E , E 33 3 c33 c33 c33 cE cE T12 11 12 u1,2 u2,1 2 x3u3,12 , 2 e332 cE S D 3 e31 13 e 33 u u x u u 2,2 3 3,11 3,22 E3 . E 33 1,1 E c33 c33
(9)
We suppose that voltage of magnitude Vo is supplied to the electrode pair. The boundary condition of the electrical potential on the electrode-covered surfaces is
v
x3
h 2
Vo .
(10)
On the basis of fourth hypothesis and (4), (7), (9), and (10), the electric potential is
v
e e c c u 2 e c 31
2 33
E 33 13
E 33
E 33
S 33
2 h 2 2Vo u x3 , 3,11 3,22 x3 4 h
and the electrical field along x3 direction can be expressed as 12
(11)
E 3
e e e c
E 33 13
31
2 33
c
E 33
c33E S 33
u
3,11
u3,22 x3
2Vo . h
(12)
Submitting (8), (9), (12) into (7) and integrating the result over the plate thickness, we could obtain the equations of equilibrium and transverse shear forces Q13 and Q23 as follows:
N11 N12 2u h 21 , x1 x2 t
(13)
N12 N 22 2u h 22 , x1 x2 t
(14)
Q13 Q23 2u h 23 , x1 x2 t M11 M12 M12 M 22 Q13 , Q23 , x1 x2 x1 x2
(15) (16)
where N 11 , N12 , N 22 are the in-plane forces; M 11 , M12 , M 22 are the bending moments. The definition of in-plane force and bending moment components are shown below: h
N11 , N12 , N 22 x1 , x2 2h T11 , T12 , T22 x1 , x2 , x3 dx3 ,
(17)
2
h
M 11 , M 12 , M 22 x1 , x2 2h T11 , T12 , T22 x1 , x2 , x3 x3dx3 .
(18)
2
There are two different dynamic characteristics for plate vibration problem, one is the out-of-plane (transverse or flexural) vibration, and the other is the in-plane (tangential and extensional) vibration. The dominant displacement field of out-of-plane vibration is u3 , and u1 , u2 are associated to the in-plane vibration. Equations of equilibrium (13) and (14) are related to the in-plane vibration, the out-of-plane vibration electromechanical equation is given by (15), apparently. From the equations of equilibrium we could also perceive that the out-of-plane vibration and in-plane vibration are assumed to be decoupled as the plate is thin. To obtain the governing equation of out-of-plane vibration, the displacement field u3 may be expressed as the produce of two functions; 13
one involving only the space coordinates x1 and x 2 , the other involving the variable time. The result can be written as
u3 x1 , x2 , t w x1 , x2 eit ,
(19)
where is the angular frequency. From (15), (16), and (19) the governing equation can be expressed as: 4
4w 4w 4w 2 w0, x14 x12x22 x24 a where 4
(20)
a 4 h 2 . The equivalent plate flexural rigidity D1 is defined by: D1 D1
h3 E E 2 c c c33E k p2 , 11 13 12
(21)
with
k
2 p
e e c c e c E 33 13
31
2 33
E 33
E 2 33
S 33
.
(22)
Finally, the coupled mechanical and electrical properties of flexural vibration problem have been instituted clearly and the mathematical solution of out-of-plane vibration problem will be constructed by using the superposition method in the next section.
3. Analytical solutions of flexural vibration To obtain the resonant frequencies and associated mode shapes, the superposition method is used to solve the boundary-value problem of flexural vibration. Before dealing with the vibration problem for the piezoelectric plate, we should express the governing equation and boundary
14
conditions in dimensionless form. The dimensionless displacement field W , space variables , , and plate aspect ratio are defined as follows:
W
w x x a , = 1 , = 2 , = . a a b b
(23)
The dependent and independent variables of governing equation (20) may be multiplied and divided by the quantities a and b, the dimensionless governing differential equation is 4 4W 4W 2 4 W 2 4 4W 0 . 4 2 2 4
(24)
For the completely free rectangular plate vibration problem, the dimensionless boundary conditions are expressed as follows: 2W 2W M 11a 2 12 0, D1 0 2 0
V13a 2 D1
3W * 3W 3 12 0 , (25) 2 0
0
2W 2W M 11a 2 12 0, D1 1 2 1
V13a 2 D1
3W * 3W 3 12 0 , (26) 2 1 1
2W M 22b2 2W 2 1 2 0, 2 0 D1a 0
3W V23b3 3W 3 1* 2 0 ,(27) 2 0 D1a 0
2W 2W M 22b 2 2 1 2 0, D1a 1 2 1
3W 3W V23b3 3 1* 2 0 ,(28) D1a 1 2 1
where M11 , M22 are the bending moments involving only the space variables, V13 , V23 are the vertical edge reactions involving only the space variables, 1
E c12E c13E 2 c33 k p2 E k p2 c11E c13E 2 c33
is the equivalent
* Poisson ratio for the piezoelectric plate, and 1 2 1 .
To solve the boundary-value problem, the solution must satisfy not only the governing equation but all boundary conditions presented in (25)-(28); nevertheless, it is impossible to obtain the 15
closed-form solution of this mathematical problem. The superposition method provides an analytical solution to deal with the boundary-value problem, the basic concept of the method is shown in Fig. 2. Consider four appropriate plate forced vibration problems, whose solutions can be easily obtained and superimposed together. The constants appearing in their own boundary condition formulation can be adjusted, so that the combination of each solution provides boundary conditions that satisfy the requirements specified in the original problem, these solutions of the forced vibration problems are known as the building block. For the first building block as shown in Fig. 2, the problem will be solved in dimensionless form, thus the solution is referred to as W1 , . For first building block, there is a pair of small circles along the edge 0 , 1 , and 0 , respectively; the boundary condition is referred to as slip-shear boundary condition, the vertical edge reaction and slope taken normal to the edge is everywhere zero along the edge. At the edge 1 is free of vertical edge reaction, but is driven by an enforced edge rotation em distributed along the edge, therefore the boundary conditions of
W1 can be written as: W1 W1
3W * 3W1 0, 0 , 31 12 2 0
(29)
3W1 1* 3W1 , 0 3 2 2 0 , 1 1
(30)
0
3 3W W1 * 2 W1 0, 0 , 31 1 2 0 0
16
(31)
3 3W W1 * 2 W1 0, em , 31 1 2 1 1
(32)
The solution of first building block is taken in the Lévy-type solution, that is
W1 ( , )
Y
m
m 0,1..
( ) cos m ,
(33)
that each trigonometric term in (33) satisfies exactly the slip-shear boundary conditions at the edge
0 , and 1 . The solution is expressed in cosine series, the enforced edge rotation em can be also rewritten in the trigonometric form:
em
m 0,1..
Em cos m .
(34)
Submitting (34) into the governing differential equation and solve the ordinary forth order homogenous differential equation, then determine the constants relation by using boundary conditions (29)-(32), the solution of first building block is
W1 ( , )
k
m 0,1..
Em 11m cosh m 13m cos m cos m (35)
E
m k 1
m
22 m
cosh m 23m cosh m cos m ,
where 1 2 2 2 2 2 ( m ) , ( m ) ; 2 2 m (m ) , m 1 ( m ) 2 2 2 , 2 ( m ) 2 ,
1 2
(36)
For 2 (m )2 , the solution of first building block always involving the trigonometric term, whereas as 2 (m )2 , the solution involving two hyperbolic terms. For the second building block, the associated Lévy solution can be easily obtained from developed for the first building block, the trigonometric functions and the forced edge rotation of 17
second building block can be express as:
W2 , W2
Y ( ) cos n ,
n 0,1..
n 0,1..
1
n
(37)
E n cos n .
(38)
Following steps analogous to those used in analysis the first building block, we can extract
W2 , form that developed for W1 , easily, the solution of second building block can be shown as below:
W2 ( , )
r
E
n 0,1..
n
11n
E
n r 1
n
22 n
cosh n 13 n cos n cos n (39)
cosh n 23 n cosh n cos n ,
where 1 1 2 2 2 2 2 2 2 ( ) n 1 , ( n ) ; 1 n (n )2 2 2 2 , n 1 1 ( n ) 2 2 2 2 , 2 2 ( n ) 2 .
(40)
In (39), the case before the rth term indicates the situation 2 2 (n )2 , whereas the case after the rth term represents the situation 2 2 (n )2 . Similarly, the vertical edge reaction for W3 and W4 all equal zero, the enforced edge rotations at 0 and 0 are W3 Fm cos m , 0 m 0,1..
W4
0
n 0,1..
Fn cos n .
(41)
(42)
Utilizing the same way, the solution of the third and fourth building blocks can be derived, so that
18
linear superposition can be applied to all four building blocks of the solutions: W ,
k
m 0,1..
m k 1
Em 22 m cosh m 23m cosh m cos m
r
En 11 n cosh n 13 n cos n cos n
n 0,1...
E n
n r 1
k
m 0,1...
m k 1
r
n 0,1...
Em 11m cosh m 13m cos m cos m
22 n
n cosh n cos n cosh n 23
Fm 11m cosh m 1 13m cos m 1 cos m
Fm 22 m cosh m 1 23m cosh m 1 cos m Fn 11 n cosh n 1 13 n cos n 1 cos n
F
n r 1
n
22 n
n cosh n 1 cos n , cosh n 1 23
(43)
where Em , En , Fm , and Fn are undetermined constants, the rest of coefficients expressed in (43) are explicitly presented in Appendix A. In order to construct the eigenvalue matrix and determine the resonant frequency, the out-of-plane deflections solution (43) must satisfy the following original boundary conditions: 2W , 1 2W , 2 0 2 2
at =0, 1 ;
(44)
2 2W , 2 W , 0 at =0, 1 . 1 2 2
(45)
Using K terms in each building block and then submitting (43) into (44) and (45), the orthogonality of Fourier series is utilized to generate a set of 4K homogeneous algebraic equations related to the constants Em , En , Fm , and Fn . The homogeneous algebraic equations can be
19
formulated in matrix forms as follows:
AE = 0 ,
(46)
where A is a 4 K 4 K eigenvalue matrix, and E is a 4 K 1 matrix composed of the undetermined constants Em , En , Fm , and Fn . Figure 3 is the schematic representation of the matrix A in which each row represents the coefficients in one of the algebraic equations relating the unknown Em , En , Fm , and Fn , the detail of components of the matrix A are presented in Appendix B. Increment the input frequency to establish the natural frequency which causes the determinant of matrix A to vanish, the determinant indicates that a nontrivial solution exists for the quantities Em , En , Fm , and Fn . As the coefficient matrix E is determined, the associated mode shape can also be obtained. After constructing the result of out-of-plane deflections, the electrical potential can be derived by (11). The surface normal stress and electrical field components can be obtained from (9) and (12) with v 0 for the short-circuited resonator, the forms of these components are presented as below:
hY 2 hY 2 3 1 , T11 1 W field , T22 1 W field ; E 3 R pW field 2a 2a where R p
h e31 c13e33 c33 2 2a 33 e
2 31
c33
i , and W field (i=1, 2, 3) are expressed as:
20
(47)
i
W field
Em 1i cosh m 2i cos m cos m i i m 0,1.. F cosh 1 cos 1 m m 2 m 1 k
i i Em 3 cosh m 4 cosh m cos m i i m k 1 F cosh 1 cosh 1 m m 4 m 3
En 1i cosh n 2i cos n cos n i i n 0,1.. F cosh 1 cos 1 2 n 1 n n r
(48)
En 3i cosh n 4i cosh n cos n. i i n r 1 F cosh 1 cosh 1 n n 4 n 3
On the basis of the principal stress components and displacement solution, the equivalent of shear stress components can also be obtained by submitting (18) and (43) into (16). The components of the equivalent shear stress can be presented as follows:
4 4 Em m 1 cosh m 2 cos m D1 k Q13 2 sin m a m 0,1.. F m 4 cosh 1 4 cos 1 2 m m 1 m
Em m 34 cosh m 44 cosh m sin m 4 4 m k 1 F m cosh 1 cosh 1 3 4 m m m
En 1 4 sinh n 24 sin n cos n 4 4 n 0,1.. F sinh 1 sin 1 2 n n n 1 r
4 4 En 3 sinh n 4 sinh n cos n ; 4 4 n r 1 F sinh 1 sinh 1 4 n n n 3
21
(49)
5 5 Em 1 sinh m 2 sin m D1a k Q23 3 cos m b m 0,1.. F 5 sinh 1 5 sin 1 2 m m m 1
Em 35 sinh m 45 sinh m cos m 5 5 m k 1 F sinh 1 sinh 1 4 m m m 3
En n 15 cosh n 25 cos n r sin n 5 n 0,1.. F n 1 25 cos n 1 cosh 1 n n
(50)
5 5 En n 3 cosh n 4 cosh n sin n . 5 5 n r 1 F n cosh 1 cosh 1 4 n n 3 n
The coefficients in (48), (49) and (50) are listed in Appendix C.
4. Experimental techniques In this study, we utilize two experimental techniques to measure the vibration characteristics of rectangular piezoelectric plate, i.e., AF-ESPI and LDV. A brief introduction of these two experimental techniques will be provided in this section.
4.1. Out-of-plane vibration measurement by AF-ESPI For the out-of-plane vibration measurement as shown in Fig. 4, the first image is recorded as a reference after the plate vibrates periodically. While the specimen vibrates at a specific frequency, the digital images of vibration are recorded by a charge-coupled device (CCD, Pulnix TM-7CN, Pulnix America, Inc., Sunnyvale, CA) and the monitor will display the speckle fringe pattern in real time. According to the time-averaging method, the light intensity of the reference image can be described by the distribution I ( x , y ) and is expressed as
22
I
2 I I I I A t 2 cos (1 cos ) cos O R O R R dt . 0 L 1
(51)
The coefficients in (51) are indicated as follows: is the CCD refresh time, I O is the object light intensity, I R is the reference light intensity, R is the phase difference between the object and reference light, L is the wavelength of the laser, A is the vibration amplitude, is the angular frequency, and is the angle between the object light and observation direction of specimen for the out-of-plane setup. It should be noted that R and A are functions of the position on the vibration surface. The time-averaged method demonstrates that the vibration measurement includes many periods of object motion during the camera frame period and assume the vibration period is much smaller than the CCD refresh time (i.e., 2 ). If we set
2 (1 cos )
L
, then (51) can be reduced
to a simplified form as I1 I O I R 2 I O I R (cos R ) J 0 ( A) ,
(52)
where J 0 is a zero-order Bessel function of the first kind. The existence of dc term ( IO I R ) will cause the poor fringe pattern resolution. Hence, the subtraction is applied to eliminate the dc term by capturing a second vibrating image. The light intensity of the second image can be represented as
I2
I 1
0
O
I R 2 I O I R cos R ( A A) cos t dt .
(53)
Utilizing the Taylor series expansion and neglecting the higher order terms, (53) can be rewritten as follows:
23
1 I 2 IO I R 2 I O I R 1 2 ( A)2 J 0 ( A) . 4
(54)
The difference in light intensity between images I1 and I 2 can be rewritten as:
I I 2 I1
IO I R (cos R ) 2 ( A)2 J 0 ( A) . 2
(55)
The result of (55) indicates that the fringes patterns for the out-of-plane vibrations obtained by AF-ESPI are dominated by a zero-order Bessel function J 0 . For A 0 (the nodal line), the zero-order Bessel function has the maximum value that is related to the brightest fringe observed in the AF-ESPI measurement. The related amplitude, Ai , for the ith dark fringe in the AF-ESPI experimental results can be quantitatively determined from the roots of J 0 (A) 0 . The light source in the self-arranged optical setup is a He-Ne laser with a wavelength of L 632.8 nm. The vibration magnitudes of mode shapes obtained by the AF-ESPI method are on the sub-micrometer order.
4.2.
LDV system The LDV system (Model name: AVID, Ahead Optoelectronics Inc., Chung-Ho, Taipei, Taiwan)
provides a point-wise displacement measurement, which is perpendicular to the surface, with ultra-high resolution (on the order of nanometers) and has an extremely wide signal bandwidth (up to 20 MHz). The focus of the lens is 90 mm, and the size of the laser spot is 52 μm. The optical system is based on the principle of a Michelson interferometer and the Doppler effect. To avoid radio frequency electromagnetic interference caused by the high frequency signal driving the acoustic optical modulator, circular polarization interference is used. For the LDV system, a built-in dynamic
24
signal analyzer (DSA) composed of dynamic signal analyzer software and a plug-in waveform generator board can provide the specimen with the swept-sine excitation signal. In the analysis software, the swept-sine excitation signal is taken as input and the response measured by LDV is converted into the voltage signal and is taken as output. After the fast Fourier transform (FFT) processing of the input and output with the DSA software, the ratio of output/input (gain) is obtained. The frequency response curve can be obtained by the result chart, and the peaks appearing in the frequency response curve are resonant frequencies for out-of-plane vibration. Although the LDV system can precisely measure the wide-range vibrational displacements over several seconds at very high frequency, the measurement is time consuming for obtaining full-field vibrational displacements. Furthermore, the optical setup can only measure displacements perpendicular to the surface.
5. Experimental, numerical and theoretical results The vibration characteristics of resonant frequencies and mode shapes for piezoelectric rectangular plate are obtained by theoretical analysis and experimental measurements. The stress fields and electrical field are presented by analytical solutions and FEM calculation. In this study, a rectangular thickness-polarized piezoelectric plate with length a 50 mm , width b 30 mm , and thickness h 1 mm is selected for the investigation.
This piezoelectric plate is made of PZT
ceramics (APC-855, Physic Instrumente, Lederhose, Germany). Material properties for the piezoceramic are indicated in Table I, and the polarization is along the x3 -direction. The specimen
25
is completely coated by infinitesimally thin silver electrodes on the two major faces, and is excited by the application of an AC voltage across electrodes on the two surfaces. The schematic of optical AF-ESPI setup for measuring the out-of-plane resonant frequencies and corresponding vibration mode shapes of piezoelectric plate is shown in Fig. 4. The experimental measurement procedure of the AF-ESPI technique is performed as follows. For the out-of-plane measurement, the laser beam is divided into two parts by a beamsplitter, one is the object beam and the other is the reference beam. The object beam travels to the specimen, and then reflects to the CCD camera. The reference beam goes directly to the CCD camera via a mirror and a reference plate. Once the plate is excited into vibration, the interferogram recorded by the CCD camera is stored in an image buffer as a reference image. Then the next frame is grabbed and subtracted by the image processing system. If the input frequency is far from the resonant frequency, only randomly distributed speckles are displayed and no fringe patterns can be shown. However, if the input frequency is in the neighborhood of a resonant frequency, distinct stationary fringe patterns are observed on the monitor. At this time, adjust the function generator carefully and gradually, the number of fringes will increase and the fringe patterns will become clearest as the resonant frequency is approached. From the aforementioned experimental procedure, the resonant frequencies and the correspondent mode shapes can be determined at the same time. A point-wise optical technique, LDV is used to validate the resonant frequencies of the out-of-plane vibration modes, and the basic schematic of LDV setup is shown in Fig. 5. To avoid
26
selecting a point on the nodal line, two testing points on the surface of the piezoelectric rectangular plate are utilized to measure the vibration displacements. Figure 6 shows the response of vibration displacement with fully free boundary conditions measured by LDV for the frequency less than 8k Hz. The peaks appearing in the frequency response curve are resonant frequencies for the out-of-plane vibrations. 2 In theoretical analysis, resonant frequencies are determined from the roots i in (46), which is
reduced to a finite linear system with m 20 and n 20 . The system presented in (46) is a linear system involves 80 equations and 80 variables, therefore, can be handled easily. The associated mode shapes can be obtained by calculating the eigenvalues of matrix A , and finding the one, which is closest to zero. After obtaining the eigenvalue, the corresponding eigenvector is the nontrivial solution of the linear homogeneous system. With all the coefficients be established, the mode shape associated with the resonant frequency is determined. In addition to the theoretical analysis and experimental measurement, finite element calculation is also performed using the commercially available software ABAQUS in which 20-node, three-dimensional, solid piezoelectric element (C3D20E) are selected to analyze the problem. The element type of C3D20RE is a 20-node quadratic brick, reduced integration, three-dimensional continuum stress/displacement element. In this study, 1500 elements with 10903 nodes are utilized to calculate the vibration characteristics of the piezoelectric rectangular plate with completely free boundary conditions. From the aforementioned experimental measurement and numerical calculations, the first nine
27
vibrational modes of piezoelectric rectangular plate with fully free boundary conditions are shown in Fig. 7. For the FEM result, the dotted lines and the solid lines indicate the displacement in opposite moving directions; that is if the solid lines denote the displacements moving upward, then the dotted lines indicate the displacements moving downward. The transition region between dotted lines and solid lines, which is indicated as the bold-solid line in the mode shapes, correspond to a zero-displacement line or a nodal line. For the mode shape patterns presented by the theoretical analysis, contours of equal displacements are displayed by solid lines; the warm color and cold color regions represent the anti-phased (convex and concave) displacements, the transition region between these two regions is indicated as the nodal line. From the AF-ESPI experimental measurement results, the brightest region of the mode shapes are the nodal lines, and other dark fringes are contours of vibration displacements. The voltage supplied for excitation of a mode shape is also indicated in Fig. 7. Usually, it is difficult to excite the higher mode and more voltages should be applied on the piezoceramic specimen.
Furthermore, the maximum magnitude of the displacement should be
within 1 m in order to display good quality of the ESPI images. Base on the aforementioned reason, the specific voltages is chosen for applying on the PZT for different mode.
Initially, the
two major surfaces of the plate were completely covered with thin silver electrodes. To investigate the influence of partial electrodes, electrode configurations are designed in this study. The electrodes are slit by cutting off the conduction of adjacent regions. The voltage applied on the adjacent regions is out of phase. These regions are on the surfaces parallel to the centerlines of the rectangular plates.
28
For convenience, three electrodes configurations are named “full electrode”, “right-left electrode”, and “up-down electrode”.
Modes 1, 2, 3, 5, 7 and 9 are particularly suitable for exciting vibration
modes using the full electrode. Modes 4 and 6 are particularly suitable for exciting vibration modes using the right-left electrode. However mode 8 is particularly suitable for exciting vibration mode using the up-down electrode. Comparison of theoretical analysis, numerical calculations and experimental measurements for mode shapes can be directly performed by matching the distribution of nodal lines and contours of equal displacements. The experimental measurements, finite element calculations and theoretical analyses are in excellent agreement with the result of the mode shapes. Table II presents the first nine resonant frequencies obtained from FEM, theoretical analysis, AF-ESPI and LDV. The difference between FEM and theoretical analysis result is smaller than 1.96%. The discrepancy of resonant frequencies between FEM and AF-ESPI is less than 3.70%, and for the FEM and LDV result the discrepancy is within 4.64%. Because of the nodal line pass through the measuring point, the first, fourth, sixth and seventh modes are harder to obtain by LDV system at the measured point 1 than point 2 as indicated in Fig. 6. From the result of AF-ESPI mode shapes and corresponding voltage supplied for excitation as shown in Fig. 7, the eighth mode needs more voltage to obtain the clearer fringes. This phenomenon explains the eighth mode is also difficult to measure by LDV system at both point 1 and point 2. As the validity of the theoretical result for mode shape is confirmed, we utilize the out-of-plane displacement solution to derive the electrical field and stress fields. Figs. 8-10
29
present the first six electrical and mechanical fields obtained by finite element method and theoretical analysis. The numerical calculations and theoretical analysis results are also in excellent agreement to each other. From Fig. 7 and Fig. 8, we could easily find the distribution of the electrical field E 3 is similar to the out-of-plane displacement W . This means that the larger deflection occurs in the out-of-plane displacement field implies the larger electrical field E 3 in the piezoelectric plate. Although the result of normal stress fields (i.e., Fig. 9) obtained from FEM and theory is identical to each other, the shear stress fields (i.e., Fig. 10) calculated by FEM are not convergent well near the plate edges. Fig.9 and Fig. 10 show the smooth contour lines in the theoretical result for stress analysis near the plate edges, and the comparison result also provide a strong validity for the application of analytical method for the stress distribution on the surface of piezoelectric plate.
6. Conclusions Even through a large amount of literature have investigated the out-of-plane vibration problem of piezoelectric rectangular plate by finite element and experiments, few of them provided the complete theoretical solution for this problem. In this study, a simple and straightforward analytic solution for the out-of-plane vibration problem of completely free piezoelectric rectangular plate is developed. The proposed superposition method is applied to determine resonant frequencies and corresponding vibration mode shapes.
Furthermore, the analytical solutions for the associated
30
electrical and mechanical fields are presented without difficulty. The analytical solutions based on the superposition method are compared with two different techniques to confirm the validity. The experimental techniques of AF-ESPI and laser Doppler vibrometry (LDV), and the numerical computations based on the finite-element method, are used to ensure the accuracy of the analytic solution. Excellent agreements of resonant frequencies and mode shapes are obtained for the theoretical analysis, experimental measurement, and numerical calculations. This study also uses a special configuration of electrodes for piezoelectric plates in experimental measurement of AF-ESPI to improve the electromechanical coupling efficiency of transverse vibration. It can be seen that specific vibration modes are easily excited by manipulating the configuration of electrodes to enhance the excited efficiency of the thin piezoelectric plate.
Acknowledgements The authors gratefully acknowledge the financial support for this research by the Ministry of Science and Technology (Republic of China) grant MOST103-2221-E-002-031-MY3.
31
Appendix A. (Constants in the displacement solution) The constants in the out-of-plane solution W in (43) are listed as below: 2 2 ZZ 1 m m2 1* 2 m , ZZ 2 m m2 1* 2 m ,
(56)
2 2 ZZ 1 n n2 1* n 2 , ZZ 2 n n2 1* n 2 ,
(57)
1 , m ZZ 1 ZZ 2 m sinh m
(58)
ZZ 1 , ZZ 2 m ZZ 1 ZZ 2 m sin m
(59)
1 , m ZZ 1 ZZ 2 m sinh m
(60)
ZZ 1 , ZZ 2 m ZZ 1 ZZ 2 m sinh m
(61)
11m 11m 13m 13m
22 m 22 m 23m 23m
11 n 11 n
1
n ZZ 1 ZZ 2 n sinh n
ZZ1
13 n 13 n
,
(62) ,
ZZ 2 n ZZ 1 ZZ 2 n sin n 1 n 22 n 22 , n ZZ 1 ZZ 2 n sinh n ZZ1 n 23 n 23 . ZZ 2 n ZZ 1 ZZ 2 n sinh n
32
(63) (64) (65)
Appendix B. (Constants in eigenvalue matrix) The detail of eigenvalue matrix A in (46) is shown as below:
A 1,1 A 1, 2 A 1,3 A 1, 4 A 2,1 A 2, 2 A 2,3 A 2, 4 A A 3,1 A 3, 2 A 3,3 A 3, 4 A 4,1 A 4, 2 A 4,3 A 4, 4
(66)
The components in (66) are expressed as below: A 1,1 2 ( m ) 2 m 2 cosh m +13m m 2 2 ( m ) 2 cos m , as 2 m 2 ; (67) 11m 2 2 2 2 2 2 2 22 m ( m ) m cosh m 23m ( m ) m cosh m , as 2 m ,
2 2 cos m cos n 1 1 P1 m sinh m P2 m sin m , as 2 m ; n A 2,1 2 cos m cos n P 1 m sinh m P 1 m sinh m , as 2 m 2 , 3 4 n
(68)
2 m 2 2 2 m 2 2 , as 2 m 2 ; m m m m 11 13 A 3,1 2 2 2 22 m 2 m m2 23m 2 m m2 , as 2 m ,
(69)
2 2 cos n 1 1 P1 m sinh m P2 m sin m , as 2 m ; n A 4,1 n 1 2 cos 2 1 P3 m sinh m P4 m sinh m , as 2 m , n 2 cos m cos n 1 1 Q1 n sinh n Q2 n sin n , as 2 2 (n )2 ; m A 1, 2 2 cos m cos n Q 1 n sinh n Q 1 n sinh n , as 2 2 (n )2 , 3 4 m
(70)
(71)
A 2, 2 ( n )2 2 2 cosh 2 ( n )2 2 cos , as 2 2 ( n ) 2 ; (72) n n n 13n n 11n ( n )2 2 n2 cosh n 23 n ( n ) 2 2 n 2 cosh n , as 2 2 ( n )2 , 22 n
33
2 cos m 1 1 Q1 n sinh n Q2 n sin n , as 2 2 ( n )2 ; m A 3, 2 2 cos m Q 1 n sinh n Q 1 n sinh n , as 2 2 ( n )2 , 3 4 m
(73)
n 2 2 2 n 2 2 2 , as 2 2 ( n )2 ; n n 13n 11n A 4, 2 2 2 22 n n 2 n2 23 n n 2 n2 , as 2 2 ( n )2 ,
(74)
2 2 2 2 2 m 2 2 2 m 13m m m , as m ; 11m A 1,3 2 2 2 22 m 2 m m2 23m 2 m m2 , as 2 m ,
(75)
2 2 cos m 2 2 P1 m sinh m P2 m sin m , as 2 m ; n A 2,3 2 cos m P 2 m sinh m P 2 m sinh m , as 2 m 2 , 3 4 n
(76)
A 3, 3 2 2 2 2 m 2 2 cosh 2 as 2 m ; (77) 13m m m cos m , m m 11m 22 m 2 m 2 m2 cosh m 23m 2 m 2 m2 cosh m , as 2 m 2 ,
2 2 2 2 2 P1 m sinh m +P2 m sin m , as m ; n A 4,3 2 P 2 m sinh m P 2 m sinh m , as 2 m 2 , 3 4 n
(78)
2 cos n 2 2 Q1 n sinh n Q2 n sin n , as 2 2 ( n )2 ; m A 1, 4 2 cos n Q 2 n sinh n Q 2 n sinh n , as 2 2 ( n )2 , 3 4 m
(79)
n 2 2 2 n 2 2 2 , as 2 2 ( n ) 2 ; n n 13n 11n A 2, 4 2 2 22 n n 2 n2 23 n n 2 n2 , as 2 2 ( n )2 ,
(80)
2 2 2 2 2 2 Q1 n sinh n Q2 n sin n , as ( n ) ; m A 3, 4 2 Q 2 n sinh n Q 2 n sinh n , as 2 2 ( n )2 , 4 m 3
34
(81)
A 4, 4 n 2 2 2 cosh n 2 2 2 cos , as 2 2 ( n ) 2 ; 13n n n n n 11n 2 2 22 n n 2 n2 cosh n 23 n n 2 n2 cosh n , as 2 2 (n )2 ,
(82)
where the coefficients in (67)-(82) are: 2 if m 0 2 if n 0 , n ; m 1 if m 1 1 if n 1
(83)
11m ( m ) 2 m 2 2 13m ( m ) 2 m 2 2 1 1 , P2 ; P1 2 2 m 2 n m 2 n
(84)
1
P3
2 2 2 22 m ( m ) 2 m 2 2 1 23m ( m ) m ; , P 4 2 2 m 2 n m 2 n
Q1 1
1
Q3
11 n ( n ) 2 2 n2 n2 m
2
22 n ( n ) 2 2 n2 n2 m
2
, Q21
1
, Q4
(85)
13 n ( n ) 2 2 r2
;
(86)
23 n ( n ) 2 2 r2
.
(87)
n 2 m
2
n 2 m
2
11m m m2 2 13m m m2 2 2 2 ; P1 , P2 2 2 2 2 m n m n
(88)
22 m m m2 2 23m m m2 2 2 2 ; P3 , P4 2 2 2 2 m n m n
(89)
2
2
2
2
Q1
2
2 2 13 n n 2 n2 11 n n 2 n2 2 ; , Q2 2 2 2 2 n m n m
(90)
n n 2 n2 n n 2 n2 23 22 2 . , Q4 2 2 2 2 n m n m
(91)
2
2
Q3
2
35
Appendix C. The constants in (48)-(50) are presented as follows:
11 11m 2 m 2 m 2 , 1 1 11m
21 13m 2 m 2 m 2 1 2 13m
(92)
31 22 m 2 m 2 m 2 , 1 3 22 m
41 23m 2 m 2 m 2 1 4 23m
(93)
11 11 n 2 2 21 13 n 2 2 2 n n 2 n n , 1 1 2 13n 1 11n
(94)
31 22 n 2 2 41 23 2 2 n n , 1 n 2 n2 n 1 4 23n 3 22 n
(95)
1 2 11m 2 2 2 m 1 m , 1 11m
22 13m 2 2 2 m 1 m 2 13m
(96)
32 22 m 42 23m 2 2 m 2 1 m 2 (97) m 1 m , 2 2 4 23m 3 22 m 1 2 11 n 22 13 n 2 2 1 n 2 n2 1 n n , 2 2 2 13 n 1 11n
(98)
32 22 n 2 2 2 1 n n , 3 22 n
42 23 n 2 2 2 1 n n 4 23n
(99)
1 3 11m 1 2 m 2 m2 , 3 1 11m
23 13m 1 2 m 2 m2 3 2 13m
(100)
33 22 m 43 23m 2 2 2 1 2 m 2 m2 1 m m , 3 3 4 23m 3 22 m
(101)
1 3 11 n n 2 1 2 n2 , 3 1 11n
36
23 13 n n 2 1 2 n2 3 2 13n
(102)
3 3 22 n n 2 1 2 n2 , 3 3 22 n
43 23 n n 2 1 2 n2 3 4 23n
(103)
24 13m 2 2 4 m m 2 13m
(104)
34 22 m 44 23m 2 2 m 2 m 2 m m , 4 4 4 23m 3 22 m
(105)
1 4 11 n 24 13 n 2 3 n n 2 n3 n n n , 4 4 2 13 n 1 11n
(106)
3 4 22 n 2 3 4 n n n , 3 22 n
(107)
1 4 11m 2 2 4 m m , 1 11m
44 23 n 2 3 4 n n n 4 23n
15 11m 25 13m 2 2 3 m 2 m 2 m3 m m m , 5 5 1 11m 2 13m
(108)
35 22 m 45 23m 2 2 3 m 2 m 2 m3 m m m , 5 5 4 23m 3 22 m
(109)
15 11 n 2 2 n n 2 , 5 1 11n
25 13 n 2 2 n n 2 5 2 13n
(110)
35 22 n 2 2 n n 2 , 5 n 3 22
45 23 n 2 2 n n 2 5 n 4 23
(111)
37
References [1]
S.P. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells, 2nd ed. New York, NY,
[2] [3]
USA: McGraw-Hill, 1959. A.W. Leissa, The free vibration of rectangular plates, J. Sound Vibrat. 31, (1973) 257-293. D.J. Gorman, Free vibration analysis of cantilever plates by the method of superposition, J. Sound Vibrat. 49 (1976) 453-467.
[4] [5] [6] [7] [8] [9]
D.J. Gorman, Free vibration analysis of the completely free rectangular plate by the method of superposition, J. Sound Vibrat. 57, (1978) 437-447. W. Denkmann, R. Nickell, D. Stickler, Analysis of structural-acoustic interactions in metal-ceramic transducers, IEEE Trans. Audio Electroacoust. 21 (1973) 317-324. M. Ohki, N. Shima, T. Shiosaki, Optical measurement of piezoelectric vibration in circular rod and disk ceramics, Jpn. J. Appl. Phys. 31(1992) 3272-3275. H.S. Tzou, Piezoelectric Shells: distributed sensing and control of continua, Norwell, MA: Kluwer Academic Publishers, 1993. K.Y. Lam, X.Q. Peng, G R. Liu, J.N. Reddy, A finite-element model for piezoelectric composite laminates, Smart Mater. Struct. 6 (1997) 583-591. S.Y. He, W.S. Chen, Z.L. Chen, A uniformizing method for the free vibration analysis of metal–piezoceramic composite thin plates, J. Sound Vibrat. 217 (1998) 261-281.
[10] P.R. Heyliger, G. Ramirez, Free vibration of laminated circular piezoelectric plates and discs, J. Sound Vibrat. 229 (2000) 935-956. [11] C.C. Ma, C.H. Huang, The investigation of three-dimensional vibration for piezoelectric rectangular parallelepipeds using the AF-ESPI method, IEEE Trans.Ultrason., Ferroelectr.,Freq. Control. 48 (2001) 142-153. [12] W.Q. Chen, H.J. Ding, On free vibration of a functionally graded piezoelectric rectangular plate, Acta Mech. 153 (2002) 207-216. [13] Z. Zhong, E.T. Shang, Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate, Int. J. Solids Struct. 40 (2003) 5335-5352. [14] C.H. Huang, Y.C. Lin, C. C. Ma, Theoretical analysis and experimental measurement for resonant vibration of piezoceramic circular plates, IEEE Trans.Ultrason., Ferroelectr., Freq. Control. 51 (2004) 12-24. [15] C.H. Huang, C.C. Ma, Y.C. Lin, Theoretical, numerical, and experimental investigation on resonant vibrations of piezoceramic annular disks, IEEE Trans.Ultrason., Ferroelectr.,Freq. Control. 52 (2005) 1204-1216. [16] C.C. Ma, H.Y. Lin, Y.C. Lin, Y.H. Huang, Experimental and numerical investigations on resonant characteristics of a single-layer piezoceramic plate and a cross-ply piezolaminated composite plate, J. Acoust. Soc. Amer. 119 (2006) 1476-1486. [17] A. Erturk, D.J. Inman, An experimentally validated bimorph cantilever model for piezoelectric 38
energy harvesting from base excitations, Smart Mater. Struct. 18 (2009) 025009. [18] A. Krushynska, V. Meleshko, C.C. Ma, Y.H. Huang, Mode excitation efficiency for contour vibrations of piezoelectric resonators, IEEE Trans.Ultrason., Ferroelectr.,Freq. Control. 58 (2011) 2222-2238. [19] Y.H. Huang, C.C. Ma, C.K. Chao, High-frequency resonant characteristics of triple-layered piezoceramic bimorphs determined using experimental measurements and theoretical analysis, IEEE Trans.Ultrason., Ferroelectr.,Freq. Control. 59 (2012) 1219-1232. [20] Y.H. Huang, C.C. Ma, Forced vibration analysis of piezoelectric quartz plates in resonance, Sens Actuators A 149 (2009) 320-330. [21] X. Xie, Y. Zaitsev, L.F. Velásquez-García, S. Teller, C. Livermore, Scalable, MEMS-enabled, vibrational tactile actuators for high resolution tactile displays, J. Micromech. Microeng. 24 (2014) 125014. [22] X. Xie, C. Livermore, Passively self-aligned assembly of compact barrel hinges for high-performance, out-of-plane MEMS actuators, Proc. MEMS 2017, Las Vegas (2017) 813-816.
39
Biographies Yi-Chuang Wu received the B.S. degree and the M.S. degree both from department of mechanical engineering, National Taiwan University, Taipei, Taiwan, Republic of China, in 2007 and 2009. He is currently the PhD candidate of mechanical engineering department, National Taiwan University. His research interests are in the field of piezoelectric material and vibration analysis for thick plates.
Yu-Hsi Huang received the B.S., M.S., and Ph.D. degrees in mechanical engineering from the National Taiwan University, Taipei, Taiwan, Republic of China, in 2001, 2003 and 2009, respectively. He has received the First Prize Award of Students Paper Competition from the Society of Theoretical and Applied Mechanics of Taiwan in 2003. During 2009’s winter to fall he is a visiting scholar at the Department of Mechanical Engineering of the University of Michigan. In 2011, he joined the faculty of National Taiwan University of Science and Technology, where he is currently an associate professor at Mechanical Engineering Department. Professor Huang is engaged in investigations on the dynamic characteristics of piezoelectric materials and applications. His research interests are in the field of piezoelectric sensors and actuators, vibration analysis, and signal analysis.
Chien-Ching Ma received the B.S. degree in agriculture engineering from the National Taiwan University, Taiwan, in 1978, and the M.S. and Ph.D. degrees in mechanical engineering from Brown University, Providence, RI, in 1982 and1984, respectively. 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, as an associate professor. He was promoted to full professor in 1989. Professor Ma has received the Distinguished Research Award from the National Science Council (NSC) of Taiwan three times. He has been elected as a fellow of the American Society of Mechanical Engineers. He is currently a distinguished professor of the Department of Mechanical Engineering, National Taiwan University. His research interests are in the fields of wave propagation in solids, fracture mechanics, solid mechanics, piezoelectric material, and vibration analysis.
40
41
Table 1. Material constants of piezoelectric APC-855. Material properties
Values
Density ( kg m3 )
7400
Elastic constant ( N m2 )
c11E
12.18 1010
c12E
8.02 1010
c13E
8.53 1010
c33E
11.64 1010
c44E
2.30 1010
Electrical Relative dielectric constant
Vacuum permittivity ( F m )
11S 0
2720
33S 0
2600
0
8.854 10 12
e31
10.48
e33
16.60
e15
14.26
Piezoelectric piezoelectric strain constant ( N Vm )
42
Table 2. The resonant frequencies obtained from finite element method (FEM), theoretical analysis, AF-ESPI and LDV for the piezoelectric rectangular plate. Resonant frequency (Hz) Mode no.
FEM
1
1154.8
2 3 4 5 6 7 8 9
1195.7 2662.1 3262.1 3602.8 4473.0 4905.9 6019.4 7115.3
Theory
AF-ESPI
LDV-(P1)
LDV-(P2)
(error)
(error)
(error)
(error)
1168.4
1153
1150
1150
(1.18 %)
(-0.16 %)
(-0.42 %)
(-0.42 %)
1201.6
1215
1210
1210
(0.49 %)
(1.61 %)
(1.20 %)
(1.20 %)
2702.1
2701
2680
2680
(1.50 %)
(1.46 %)
(0.67 %)
(0.67 %)
3293.2
3241
3310
3300
(0.95 %)
(-0.65 %)
(1.47 %)
(1.16 %)
3641.4
3736
3770
3770
(1.07 %)
(3.70 %)
(4.64 %)
(4.64 %)
4541.8
4572
--
4670
(1.54 %)
(2.21 %)
--
(4.40 %)
4993.9
4949
4970
4970
(1.79 %)
(0.88 %)
(1.31 %)
(1.31 %)
6137.6
6028
6090
--
(1.96 %)
(0.14 %)
(1.17 %)
--
7234.6
7359
7390
7440
(1.68 %)
(3.43 %)
(3.86 %)
(4.56 %)
43
x3
x2
x1
VAC
h
b a
Fig. 1. Geometric dimensions and coordinate system of the piezoelectric thin plate
W
W1
W2
W3
W4
Fig. 2. Schematic representation of building blocks employed in the analysis of free vibration for completely free rectangular plate
44
1 0 0 0 0
Em 2 0 0 0 0
En
Fm
3 1 2 3 1 2 0 0 0 0 0 0
0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0
0 0
Fn
3 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Fig. 3. Schematic representation of eigenvalue matrix A for completely free plate using three-term expansions
45
Fig. 4. Optical setup of the AF-ESPI technique for the out-of-plane measurement.
46
Displacement (nm)
Fig. 5. Schematic of laser Doppler vibrometry (LDV) system.
Fig. 6. Frequency response of out-of-plane vibration displacement obtained by laser Doppler vibrometry (LDV) for the piezoelectric rectangular plate within 8 kHz.
47
Mode
1
Error (%)
1154.8
Theory
1168.4
1.18
ESPI
1153
-0.16
76 V
FEM
1195.7
Theory
1201.6
0.49
ESPI
1215
1.61
Voltage
3
Resonant Frequency(Hz)
FEM
Voltage
2
Mode Shape W
80 V
FEM
2662.1
Theory
2702.1
1.50
ESPI
2701
1.46
Voltage
100 V
48
Mode
4
Error (%)
3262.1
Theory
3293.2
0.95
ESPI
3241
-0.65
250 V
FEM
3602.8
Theory
3641.4
1.07
ESPI
3736
3.70
Voltage
6
Resonant Frequency(Hz)
FEM
Voltage
5
Mode Shape W
175 V
FEM
4473
Theory
4541.8
1.54
ESPI
4572
2.21
Voltage
250 V
49
Mode
7
Error (%)
4905.9
Theory
4993.9
1.79
ESPI
4949
0.88
200 V
FEM
6019.4
Theory
6137.6
1.96
ESPI
6028
0.14
Voltage
9
Resonant Frequency(Hz)
FEM
Voltage
8
Mode Shape W
300 V
FEM
7115.3
Theory
7234.6
1.68
ESPI
7359
3.43
Voltage
225 V
Fig. 7. First nine resonant frequencies and associated mode shapes of the piezoelectric rectangular plate obtained from finite element method, theoretical analysis and AF-ESPI.
50
Mode no.
Electrical Field E 3 Theory
FEM
1168.4
1154.8
1201.6
1195.7
2702.1
2662.1
3293.2
3262.1
3641.4
3602.8
4541.8
4473
Mode 1
Shape Frequency
2
Mode Shape Frequency
3
Mode Shape Frequency
4
Mode Shape Frequency Mode
5
Shape Frequency
6
Mode Shape Frequency
Fig. 8. Comparison of first to sixth electrical field contours for piezoelectric rectangular plate from theoretical analysis and finite element method.
51
Normal Stress Fields
Mode Theory
no.
FEM
T11
T11
T22
T22
1
Frequency
1168.4
1154.8
T11
T11
T22
T22
2
Frequency
1201.6
1195.7
T11
T11
T22
T22
3
Frequency
2702.1
2662.1
52
Mode no.
Normal Stress Fields Theory
FEM
T11
T11
T22
T22
4
Frequency
3293.2
3262.1
T11
T11
T22
T22
5
Frequency
3641.4
3602.8
T11
T11
T22
T22
6
Frequency
4541.8
4473
Fig. 9. Comparison of first to sixth normal stress contours for piezoelectric rectangular plate from theoretical analysis and finite element method.
53
Shear Stress Fields
Mode no.
Theory
FEM
Q13
T13
Q23
T23
1
Frequency
1168.4
1154.8
Q13
T13
Q23
T23
2
Frequency
1201.6
1195.7
Q13
T13
Q23
T23
3
Frequency
2702.1
2662.1
54
Mode no.
Shear Stress Fields Theory
FEM
Q13
T13
Q23
T23
4
Frequency
3293.2
3262.1
Q13
T13
Q23
T23
5
Frequency
3641.4
3602.8
Q13
T13
Q23
T23
6
Frequency
4541.8
4473
Fig. 10. Comparison of first to sixth shear stress contours for piezoelectric rectangular plate from theoretical analysis and finite element method.
55