An indirect method to measure the variation of elastic constant c33 of piezoelectric ceramics shunted to circuit under thickness mode

Sensors and Actuators A 218 (2014) 105–115

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

An indirect method to measure the variation of elastic constant c33 of piezoelectric ceramics shunted to circuit under thickness mode Yang Sun a , Zhaohui Li a,∗ , Qihu Li b a b

Department of Electronics, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, China Institute of Advanced Technology, Peking University, No.5 Yiheyuan Road, Haidian District, Beijing 100871, China

a r t i c l e

i n f o

Article history: Received 9 May 2014 Received in revised form 25 July 2014 Accepted 25 July 2014 Available online 9 August 2014 Keywords: Measurement method Shunt damping Piezoelectric Elastic constant

a b s t r a c t Academic interest in the use of piezoelectric shunt damping technology in the field of vibration and noise control has been on the rise in recent years. The experimental observation of the variation of elastic constants of piezoelectric materials shunted to different circuits is a fundamental problem of this technology. The existing methods to measure the elastic constants of piezoelectric ceramics are either unsuited or complicated for such a purpose. In this paper, a modified resonator measurement method is proposed to indirectly measure the variation of elastic constant c33 of piezoelectric ceramics shunted to circuit under thickness mode. The main idea is transforming the change of c33 into the change of the electrical resonant frequency of a single-oscillator model which is easier to be observed. Mason equivalent circuit and ANSYS finite element simulation are employed for the analysis of the admittance and resonant frequency of the proposed system. Based on the proposed method, the variation of the elastic constant c33 has been investigated when the piezoelectric ceramic wafer (PZT-5H) is shunted to a resistor, a capacitor, and a resistor connected in series with an inductor respectively. The variations of the resonant frequency measured in the experiment are highly consistent with those of c33 . Besides, the measurement of variation of mechanical loss factor with a shunt resistor is also discussed in this paper. © 2014 Elsevier B.V. All rights reserved.

1. Introduction As early as 1979, Forward [1] first proposed piezoelectric transducer shunted to electronic circuit for vibration control in optical system. The piezoelectric shunt damping technology has gained fruitful achievements in the field of vibration control [2–13] and noise elimination [14–17] in recent years. In the vibration control applications, a piezoelectric transducer shunted to the passive electrical circuits is attached to a host structure, which can be a beam [2,3,7], a vibrator [4,6,9] or a plate [5,11,17]. The mechanical energy produced by the structural vibration is converted into electrical energy through the piezoelectric effect. Then, a portion of the electrical energy is converted into heat and dissipated through the shunt circuits. The design and the implementation of the efficient shunt circuits are two of main focuses associated with the piezoelectric shunt damping technology. [18] Much early attention has been paid to the investigation of the basic shunt circuits including a resistor [2–5], a

∗ Corresponding author. Tel.: +86 13301052091. E-mail addresses: [email protected] (Y. Sun), [email protected] (Z. Li), [email protected] (Q. Li). http://dx.doi.org/10.1016/j.sna.2014.07.021 0924-4247/© 2014 Elsevier B.V. All rights reserved.

capacitor [6], a series R–L [2,8,10] and a parallel R–L [7–9], each of which can suppress the vibration in only one mode. Subsequently, shunt circuits aiming at multiple mode vibration control have been proposed and improved, including Hollkamp’s circuit [19], Wu’s blocking circuit [20], Moheimani’s current-flowing circuit [21] and negative capacitor circuit [22,23]. Another switching shunt circuit [17,24–27] belonging to the semi-passive shunt approach makes it possible to adjust the passive damping property of the vibration control system. In order to design the shunt impedance more systematically, a design method of the shunt circuit parameters based on minimizing the H2 norm of the damped system [28] is presented. And the synthetic impedance consisting of a voltage-controlled current source [28] makes the implement of the shunt network easier and more flexible. The utilization of the piezoelectric shunt damping technology is mainly based on the changes of two variables of the vibration control system—damping factor and resonant frequency [2–12]. The former reflects the damping effect whereas the latter decides the frequency range of the shunt damping system. It is observed that resonant frequency of the system with shunt resistance is slightly lower than that of the open-circuit system, but higher than that of the short-circuit system. [2,4,5,11,12] The change of resonant frequency is substantially caused by the change of elastic constants of

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piezoelectric materials. Therefore the experimental observation of the variation of elastic constants of piezoelectric materials shunted to different circuits is of significant research value. Several classical methods used to measure the elastic constants of piezoelectric ceramics have long been applied in practice. [29] The static or quasistatic methods [30] directly measuring strain of the material under an exertion of stress are the early and basic techniques. But they are seldom applied in the measurement of elastic constants in recent years since it is difficult to control the electrical boundary conditions. The pulse-echo methods [31–33] are based on the fact that the plane-wave velocities are related to the fundamental material constants of piezoelectric materials and can be determined from pulse transit time measurements. However, inaccuracies may occur under various cases especially for small-size samples such as the phase shift in coupling the transducer to the sample, the distortion of the pulses during reflection, the detection errors of wave front of the pulses and so on. The resonator measurement method [34] which directly measures the impedance or admittance of the piezoelectric material is based on the electrical properties of a piezoelectric vibrator which are dependent on the elastic, piezoelectric, and dielectric constants. The material constants can be derived from the measured resonance frequency, antiresonance frequency and the motional capacitance. However, directly measuring the impedance or admittance of a piezoelectric ceramic wafer shunted to different circuits cannot be applied to obtain the variation of c33 of the piezoelectric materials shunted to different circuits. Because the shunt circuits and the piezoelectric ceramics are in the same circuit loop, the change of c33 due to the shunt circuit cannot be separated from the electrical effects on the resonant frequency or other electrical variables of the equivalent circuit that can be measured. In the work of Law et al. [4], a mechanical mass-spring-dashpot model is established to measure the variations of the resonant frequency and damping ratio as the shunt resistance changes. The resonant frequency and damping ratio were determined from the frequency response function (FRF) which is obtained by measurement of the force and the acceleration of the model under the excitation of an external vibration exciter. To measure the FRF, a series of indispensable instruments are employed. Hence the method is complicated and costly. Moreover, uncertainty may also be introduced by the variables measured by those instruments. In this paper, an indirect method, namely a modified resonator measurement method to measure the variation of c33 of piezoelectric ceramics shunted to a circuit is proposed. The measurement system is a single-oscillator model consisting of two identical piezoelectric ceramic wafers and a mass block. The mass block is introduced to make the piezoelectric ceramics in the quasi-static state. The shunt circuit is connected to one ceramic wafer while the admittance measurement is conducted on the other ceramic wafer. The variation of c33 is turned into the variation of the system resonant frequency that can be easily measured. Compared with the measurement of the mechanical FRF, the measurement of the electrical admittance in this paper does not need any complex apparatus but an impedance analyzer, which achieves good performance on simplifying the experiment, reducing the introduction of measurement uncertainty and lowering the cost. The variation of the elastic constant c33 has been investigated based on Mason equivalent circuit theory in which the piezoelectric ceramic wafer is under different electrical boundary conditions. The finite element analysis is also carried out using ANSYS and the simulation numerical results show good agreement with the theoretical analysis. The experiment is conducted to measure the variations of the resonant frequency, which show good coincidence with the variations of c33 . Besides, the measurement of variation of mechanical loss factor is also discussed in this paper.

For the thickness mode is one of the essential modes for the covering layer structure that needs the reduction of structural vibrations, and c33 is the elastic constant representing the thickness mode, this paper limits discussion to the elastic constant c33 under thickness mode. However, the proposed method can be extended to investigate the variations of other elastic constants of piezoelectric ceramics shunted to circuit under other modes. Three basic kinds of shunt circuits, namely resistive, inductive, and capacitive shunt circuits, are chosen to be studied in this paper. Likewise, the method can be extended to study other shunt circuit types. In addition, this measurement method can also be applied to measure the elastic constant variation of the piezoelectric composites with high electromechanical coupling factor such as 1-3 [35] and 1-3-2 composites [36–38], which are promising materials with attractive performance for transducers applied in medical ultrasonic and underwater acoustics. 2. Model description An indirect method is proposed for the purpose of avoiding the direct measurement of the admittance of the piezoelectric wafer itself shunted to circuit. The proposed single-oscillator system consists of two identical piezoelectric ceramic wafers a, b and a co-axial cylindrical mass block which are tightly bonded together. The underside of a is fixed, while the upper surface of the mass block is free, as shown in Fig. 1(b). If b is under different electrical boundary conditions such as short-circuit, shunted to impedance or open-circuit, the elastic constant c33 of b will change as well as the resonant frequency of the single-oscillator system. By observing the changes of the resonant frequency, the variation of c33 can be indirectly obtained. The piezoelectric ceramic material PZT-5H (Lead zirconate titanate) is chosen as an example in the research work. Apart from material, proper thickness and radius of the ceramic wafer are selected. A mass block is introduced to make the resonant frequency of the system as low as several tens kHz, which is far lower than the resonant frequency of the ceramic wafer itself to provide a quasi-static state [4]. For the purpose of approaching the condition of an ideal single-oscillator system, tungsten alloy is chosen as the material of the mass block because of its high density and hardness. The principle of the proposed method can be explained by a single-oscillator system. The ideal single-oscillator system similar to the above proposed system is shown in Fig. 1(a). It consists of a mass block Mm and two identical springs connected in series which are equivalent to the two identical piezoelectric ceramic wafers a, b. The bottom spring is fixed and its stiffness k0 is a constant, just like that ceramic wafer a is in open-circuit state and its elastic constant . The stiffness c33 is a constant—the open-circuit elastic constant c D 33 of the upper spring k is a variable, just as the elastic constant c33 of the ceramic wafer b is a variable, since b is under different electrical boundary conditions. For this ideal single-oscillator system, since the two springs are connected in series, the stiffness of the system is: k¯ =

kk0 k + k0

(1)

Therefore, the resonant frequency is:



1 fsi = 2

k¯ Mmi

(2)

Here, ‘i’ represents the ideal single-oscillator system. Then, the stiffness of the upper spring k is obtained as, k=

Mmi (2fsi )

2

1 − (Mmi /k0 )(2fsi )

2

(3)

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Fig. 1. Single-oscillator system model: (a) ideal single-oscillator system; (b) schematic drawing of real single-oscillator system; (c) finite element model (clamped in radial direction); (d) finite element model (free in radial direction); (e) symmetrical experimental sample.

If the stiffness k and k0 are high enough and the resonant frequency is comparatively low, Eq. (3) can be further simplified to be: k ≈ Mmi (2fsi )

2

(4)

Hence, the following relationship can be derived. 2fsi k = k fsi

(5)

It is concluded that k/k is proportional to fsi /fsi . Therefore, the variation of the stiffness k can be obtained according to the variation of the resonant frequency fsi of the ideal single-oscillator system. 3. Theoretical equivalent circuit analysis 3.1. The relationship between c33 and resonant frequency The Mason equivalent circuits of the ceramic wafer and the mass block stretching in thickness mode are shown in Fig. 2(a) and (b) respectively. In Fig. 2(a), F1 and F2 are the pressures applied on the two end faces of the ceramic wafer. V is the voltage emerging between the two electrodes of the wafer because of piezoelectric effect. U1 and U2 are the particle velocities on the two end faces. In Fig. 2(b), F1 and F 2 are the pressures applied on the two end faces of the mass block. U1 and U2 are the particle velocities on the two end faces. By paralleling the equivalent circuits of the ceramic wafers a, b and the mass block, the electro-mechanical equivalent circuit of the whole model is obtained and depicted in Fig. 2(c). The underside of wafer a is fixed, which means its velocity U1 equals to 0 and the branch circuit is open-circuited. The upper surface of the mass block is free, which means its pressure F 2 equals to 0 and the branch

circuit is short-circuited. The ceramic wafer a is open-circuited. The ceramic wafer b is shunted to an impedance Zx , where Zx = 0 and Zx = ∞ indicate that it is short-circuited and open-circuited, respectively. For convenience of the theoretical analysis, in this part, the ceramic wafer b is assumed in open-circuit state, i.e. Zx = ∞, while its elastic constant c33 is assumed to be a variable in order to derive the relationship between c33 and the resonant frequency. open open open open open = jp cp Sp tan(kp tp /2), Z2 = (p cp In Fig. 2, Z1 open Sp /j sin(kp tp ), Z1 = jp cp Sp tan(kp tp /2), Z2 = (p cp Sp /jsin(kp tp ), Z1 = jm cm Sm tan(km tm /2) and Z2 = (m cm Sm /j sin(km tm ) are the mechanical impedances, where subscript ‘p’ and ‘m’ represent the piezoelectric ceramics and the mass block respectively. Accordingly, p , m are the densities, tp , tm are the thicknesses and Sp , Sm are the areas of the ceramic wafers and the mass block respectively. C0 = (εS33 Sp /tp ) is the inherent capacitance and n = (e33 Sp /tp ) denotes the electromechanical conversion factor of the piezoelectric ceramics. εS33 is the dielectric constant and e33 is the piezoelectric stress constant. Besides, kp = (ω/cp ), km = (ω/cm ) are the wave numbers.  cm is the longitudinal  wave velocity of the open D / and c = mass block. cp = c˜33 c˜33 /p are the longitup p D dinal wave velocities of ceramic wafer a and b respectively. c˜33 (open-circuit elastic constant) and c˜33 are the elastic constants of ceramic wafer a and b. The elastic constant is actually a complex value that consists of the real part and the imagine part. The ratio of the imagine part to the real part represents the mechanical loss factor whose reciprocal is the mechanical quality factor Qm . They are expressed as follows: 

D D D c˜33 = c33 + jc33

(6)

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Fig. 2. Electro-mechanical equivalent circuits of (a) the ceramic wafer; (b) the mass block; (c) the whole model. 

D c33 D c33

=

1

(7)

D Qm

 c˜33 = c33 + jc33  c33

c33

=

(8)

1 Qm

(9)

It is assumed that Sp = Sm = S. The electrical admittance of the system calculated from the two electrodes of wafer a can be obtained according to the equivalent circuit and simplified based on the low frequency approximation conditions kp tp  1 and km tm  1, namely: Ya =

1−

 Aω2 (Bc

jωC0 n2

 

2 2 2 33 ω −(c33 +c 33 )S) +(c D D−ω2 C) 2 33 (Bω2 −c33 S) +c  2 S 2 33

+j

c  ABω4  33 +c D D 2 33 (ω2 B−c33 S) +c  2 S 2 33



(10)

A = C0 S(Mm +(Mp /2)), B = tp (Mm +(Mp /2)), C = C0 Mp , Where, D = (SC0 /tp ) are constants. Mp and Mm are the masses of the ceramic wafer and the mass block respectively. The equivalent circuit of thickness mode only takes into account the piezoelectric ceramic wafer and the mass block with radial dimensions much larger than their thicknesses or they are radially clamped. The radiuses of the ceramic wafers and the mass block are set as 14 mm. The thicknesses of the ceramic wafers and mass block are chosen to be 2 mm and 10 mm respectively. The parameters of the piezoelectric ceramics (PZT-5H) and the mass block are listed in Table 1. Let the admittance Ya = Ga + jBa , therefore Ga is the conductance and Ba is the susceptance. For c33 with certain value, according to Eq. (10), the conductance of the system calculated from the two electrodes of wafer a can be calculated. For example, when ceramic wafer b is open-circuited, namely c 33 = c D , 33 the conductance is calculated and shown in Fig. 3. The peak of the conductance is Gamax = 17.36 ms, the resonant frequency of the system is fs = 59.96 kHz. The two frequencies f1 = 59.65 kHz and f2 = 60.26 kHz are corresponding to the condition that the √ conductance equals to Ga max / 2. The quality factor of the conductance is defined as Qe = fs /(f2 − f1 ) = (fs /f) and is calculated to be 98.94. Then, the electrical loss factor which is defined as 1/Qe = (f2 − f1 )/fs = (f/fs ) can be obtained to be 0.01. With the change of c33 , the relationship between c33 /c33 D and fs /fs D (fs D is the open-circuit resonant frequency) is calculated

Fig. 3. Electrical conductance of the system calculated from the two electrodes of wafer a (ceramic wafer b is open-circuited, c33 = c33 D ).

according to Eq. (10) and is shown in Fig. 4. It can be seen that c33 /c33 D nearly varies linearly with fs /fs D in a certain range, which is nearly the same as the relationship between k/k and fsi /fsi shown in Eq. (5) for the ideal single-oscillator system. Therefore, the variation of c33 of wafer b can be indirectly captured according to the variation of the resonant frequency of the system.  , the relationship between the mechanical With the change of c33 loss factor 1/Qm and the electrical loss factor of the system 1/Qe can also be obtained according to Eq. (10) and is shown in Fig. 5. It can be seen that 1/Qm varies linearly with the change of 1/Qe in a certain range. So the mechanical loss factor of wafer b 1/Qm can be indirectly derived based on the variation of the electrical loss factor of the system 1/Qe . 3.2. The electrical conductance in different shunt circuit cases In this part, the ceramic wafer b is shunted to a circuit, where Zx is arbitrary. The equivalent circuit of the model is also shown in Fig. 2. The ceramic wafer a is in open-circuit state. Under this

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Table 1 Parameters of piezoelectric ceramics and mass block [39]. The piezoelectric ceramics (PZT-5H) Density (kg/m3 )

Elastic constant(1010 N/m2 )

p

c11 E

c12 E

c13 E

c33 E

7500

12.6 c11 D 13.0

7.95 c12 D 8.28

8.41 c13 D 7.22

11.7 c33 D 15.7

Mechanical quality factor

Relative dielectric constant

Piezoelectric stress constant (C/m2 )

c44 E

ε11 S /ε0

ε33 S /ε0

e31

e33

e15

Qm D

2.3 c44 D 4.22

1700

1470

−6.5

23.3

17.0

65

The mass block (Tungsten Alloy) Density m (kg/m3 )

Young modulus (N/m2 )

Poisson’s ratio

18,700

35.4

0.35

from the two electrodes of wafer a can be obtained according to the equivalent circuit as Eq. (11). Ya =



1

2

1



open )// 1

(Z  //Z  +Z  +Z

1 n2 (Zx //

1 − 1 )+Z open jωC0 jωC0 2

n2

Fig. 4. The relationship between c33 /c33 D and fs /fs D .



open open +Z 1 2

+2Z

−

1 jωC0

(11) 1 // jωC

0

Where ‘’ denotes parallel. The real part of the electrical admittance is the electrical conductance. The obtained electrical conductance of the system calculated from the two electrodes of wafer a when wafer b is shunted to a resistor, a capacitor, and a resistor (100 ) connected in series with an inductor are shown in Fig. 6(a1),(b1),(c1) respectively. Under each condition, several conductance curves corresponding to different shunt resistances, capacitances and inductances are chosen as examples and depicted in the figures. It can be seen in Fig. 6(a1) that from short circuit to open circuit, with the increase of the shunt resistance, the resonant frequency gradually rises accordingly and the width and height of the resonant peak (represented by the electrical loss factor 1/Qe ) also undergo certain changes. In Fig. 6(b1), from open circuit to short circuit, the resonant frequency gradually declines with the growth of the shunt capacitance. In Fig. 6(c1), from the condition of Lx = 0 to open circuit, with the increase of the shunt inductance, the resonant frequency first decreases, then jumps to the peak value and continues to decrease. From the calculated electrical conductance, the variations of the resonant frequency and electrical loss factor 1/Qe with the change of shunt impedances can be solved. Therefore, the variations of c33 and 1/Qm with the change of shunt impedances can be obtained. 4. Finite element analysis

Fig. 5. The relationship between the mechanical loss factor of wafer b 1/Qm and the electrical loss factor of the system 1/Qe .

condition, the change of the shunt impedance is set as the factor that causes the change of the resonant frequency of the system. The elastic constant c33 of both wafer a and b is the open-circuit elasD . The electrical admittance of the system calculated tic constant c˜33

For verification of the theoretical analytical results, the finite element simulation software ANSYS is also adopted to solve the electrical admittance. A 3-D finite element model is established given that the shunt circuit element should be connected to a volume block which is consistent with the actual experimental condition. According to the rotational symmetry of the model, an arbitrarily chose fraction of the whole cylindrical model is enough to fulfill the goals, which can also considerably improve the calculating speed. In this paper a 1/12-model is established in ANSYS as shown in Fig. 1(c). The piezoelectric ceramics is polarized in z-axis direction and the whole model vibrates in z-axis direction under thickness mode. The rotation axis is fixed in the radial direction. The side faces are fixed in the rotation direction. The outer surfaces are fixed in radial direction. The bottom of wafer a is fixed in z-axis direction. The circuit element CIRCU94, connected to the piezoelectric ceramic wafer b, is employed to build the shunt resistor, capacitor and

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Fig. 6. Electrical conductance of the system calculated from the two electrodes of wafer a when wafer b is shunted:(a1) a resistor (equivalent circuit analysis); (a2) a resistor (finite element analysis); (b1) a capacitor (equivalent circuit analysis); (b2) a capacitor (finite element analysis); (c1) a resistor (100 ) and an inductor in series (equivalent circuit analysis); (c2) a resistor (100 ) and an inductor in series (finite element analysis).

inductor elements. As this is a 1/12-model, the shunt resistance and inductance must be twelve times of those connected to the full model. While the shunt capacitance must be one-twelfth of that connected to the full model. Two electrodes are respectively

defined on the two surfaces of the piezoelectric ceramic wafer a. Voltage 1 V is loaded on one node of the upper surface and voltage 0 V is loaded on one node of the underside. After taking harmonic response analysis, the node charge value Q on the upper surface

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of the piezoelectric ceramic wafer a is picked up. The electrical admittance of 1/12-model is derived as follows: Ya(1/12) = −j2f

Q V

(12)

where V(V = 1 V) is the excitation voltage and f is the frequency. The admittance of the whole model should be twelve times of Ya(1/12) . The obtained electrical conductance of the system calculated from the two electrodes of wafer a when wafer b is shunted to a resistor, a capacitor, and a resistor (100 ) connected in series with an inductor are shown in Fig. 6(a2),(b2),(c2) respectively. By taking use of the same calculating process of the theoretical research, the variation of the elastic constant c33 can also be obtained with finite element method. When the piezoelectric ceramic wafer b is shunted to a resistor, a capacitor, and a resistor connected in series with an inductor respectively, the simulation results are compared with the theoretical analytical ones in Fig. 7, which indicates that they are consistent with each other. From Fig. 7, it indicates that for piezoelectric ceramics shunted to a resistor or a capacitor, the elastic constant c33 varies between E (117 GPa) and the open-circuit the short-circuit elastic constant c33 D (157 GPa). With the rise of the shunt resistance, elastic constant c33 E , the value c starting from c33 33 first climbs rapidly and then keeps a D . On the contrary, with the rise of slow rising trend until it reaches c33 D , first goes down sharply the shunt capacitance, c33 , starting from c33 E . While for and then declines gradually until the value reaches c33 the resistor (100 ) and inductor in series shunt circuit condition, E to c D . With the rise of the c33 varies beyond the range from c33 33 E ,c shunt inductance, almost starting from c33 33 first declines slowly, D . then makes a soar to the peak value before drops slowly to c33 The consistent variations of the effective stiffness can be found in references [2,4,6] For the resistor case [2,4], with the increase of the resistance, the effective stiffness increases monotonously between the minimum and the maximum. While for the resistor and inductor in series case [2], with the increase of the inductance, the effective stiffness first decreases, then jumps to the peak value but decreases again. For the capacitor case [6], the tunable stiffness decreases steadily with the increase of the shunt capacitance. The pure resistance is the sole means to dissipate energy while both capacitive and inductive elements can only store but cannot dissipate energy. So it is worthy to investigate the damping behavior of the shunt resistance case. The electrical loss factor 1/Qe of the system for the shunt resistance case has been calculated by equivalent circuit method and finite element method respectively. The mechanical loss factor of wafer b 1/Qm can be indirectly derived based on the variation of the electrical loss factor of the system 1/Qe as shown in Fig. 8. The concerned general trends of 1/Qm are almost the same for the two analysis methods with only slightly difference between the two curves. It can be seen that when ceramic wafer b is shunted to an optimal resistance load (about 600 ), the mechanical loss factor of wafer b reaches to its maximum. Correspondingly, the effect of piezoelectric shunt damping is the best. 5. Experiment To validate the theoretical and finite element results, an experimental sample has been manufactured to measure the admittance. To realize the underside fixed boundary condition of the piezoelectric ceramic wafer a, wafer a should be affixed on a base made of material whose stiffness is much higher than that of piezoelectric ceramics. Since the stiffness of the piezoelectric ceramics is fairly high (c33 D = 157 GPa), there are no such materials that can meet the demand. To avoid the difficult problem of the fixed condition on the underside of wafer a, a symmetrical system model is adopted. By

Fig. 7. The variation of c33 when wafer b is shunted to:(a) a resistor;(b) a capacitor;(c) a resistor (100 ) and an inductor in series.

making two identical mass blocks and four piezoelectric ceramic wafers b, a, a and b stick together, a symmetrical system is implemented as shown in Fig. 1(e). The end surfaces of the two mass blocks are free. The dimensions of each component are listed in Table 2. The symmetrical model is equivalent to the single-oscillator model with the underside fixed boundary condition. The resonant frequencies of the two models are the same, while the maximum

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Compared with the finite element results, the variation trends of the resonant frequency measured in experiment are almost consistent with the variation trends of c33 when wafer b is shunted to certain type of shunt circuit. It verifies that the variation trend of c33 can be indirectly obtained by observing the variation of the resonant frequency. The electrical loss factor 1/Qe of the system for the shunt resistance case measured in the experiment with the finite element results of the mechanical loss factor of wafer b 1/Qm as shown in Fig. 11. It can be seen that when ceramic wafer b is shunted to an optimal resistance load (about 600 ), the electrical loss factor of the system and the mechanical loss factor of wafer b reach to their maximums. Correspondingly, the effect of piezoelectric shunt damping is optimal. 6. Comparison and discussion

Fig. 8. The variation of mechanical loss factor 1/Qm when wafer b is shunted to a resistor.

Table 2 Dimensions of experimental apparatus.

Piezoelectric ceramics Mass

To illustrate the significant advantages of the method proposed in this paper, the comparisons with the static method [30] and the mechanical impedance measurement method [4] are given and discussed below, respectively. 6.1. Comparison with the static method

Material

Radius (mm)

Thickness (mm)

PZT-5H Tungsten alloy

14.00 13.00

2.00 10.00

conductance of the symmetrical model is one half of that of the single-oscillator model since the volume of the symmetrical model is twice of that of the single-oscillator model. It is needed to connect two same shunt circuit elements to wafer b and b respectively. An impedance analyzer is employed to measure the electrical admittance of the system from the two electrodes of wafer a. Because the real piezoelectric ceramic wafer with a limited radius in the experiment cannot fully satisfy the strict radially fixed condition of the theoretical analysis, influenced by the finite radial dimension and the radially free boundary condition, the thickness resonant frequency will be different from that of the ideal case. Corresponding to the dimensions of the experimental sample, a 1/12-model is also established in ANSYS as shown in Fig. 1(d). The only difference of the boundary conditions from that of the ideal case is that the outer surfaces of the ceramic wafers and the mass block are free. For the shunt resistance, capacitance and inductance conditions, the electrical conductance calculated by finite element method and the electrical conductance measured in experiment are shown in Fig. 9. For each condition, several conductance curves corresponding to different shunt resistances, capacitances and inductances are chosen as examples. Compared with the conductance of the ideal case in Fig. 6, the resonant frequency of the model with radially free boundary condition is lowered by about 15 kHz, which is caused by the finite radial dimension and the radially free boundary condition of the model. It can be seen in Fig. 9 that for certain shunt circuit case, the resonant frequencies of the single-oscillator model in finite element analysis and the symmetrical model used in experiment are nearly the same. The maximum conductance of the symmetrical model is about one half of that of the single-oscillator model. The variations of c33 obtained by finite element analysis and the resonant frequency measured in experiment are compared in Fig. 10. c33 values calculated by finite element method are marked with triangle symbols and shown on the left y-axis. While the experimental results of the resonant frequencies are marked with olid square symbols and shown on the right y-axis.

In order to measure the elastic constant c33 by the static method [30], a force should be perpendicularly applied on one surface of a ceramic wafer and the other surface of the wafer is fixed in its thickness direction. Under certain electrical boundary conditions, the resulting displacement needs to be measured to obtain the strain component in thickness direction. The ratio between the applied force and the strain component in thickness direction is the elastic constant c33 . Eqs. (13) and (14) are the definitions of c33 under ‘constant electrical field’ condition and ‘constant electrical displacement’ condition respectively.



E c33 =

 D c33 =

∂T3 ∂S3 ∂T3 ∂S3



(13)



E

(14) D

where T3 and S3 are the stress and strain component in the thickness direction of the wafer respectively, ()E and ()D denote the ‘constant electrical field’ and ‘constant electrical displacement’ conditions respectively. Since the elastic constant of the piezoelectric ceramics is very high and the resulting strain caused by the applied force within the tolerance of the ceramics is very slight, the error will be introduced to the result of displacement by the measurement instrument. Furthermore, a fixed boundary condition on either surface of the wafer can hardly be satisfied. Therefore, the elastic constant c33 is hard to be obtained accurately since the error occurs in estimation of the strain component of the wafer. Another problem to measure the elastic constant c33 by the static method is the control of the electrical boundary condition. The elastic constant accuracy by this method essentially depends on how the ‘constant electrical field’ and ‘constant electrical displacement’ conditions are satisfied during the measurement. On the contrary, the aim of this paper is to measure the variation of the elastic constant c33 when the piezoelectric ceramic wafer is shunted to different circuits, namely under different electrical boundary conditions. Therefore, the static method is obviously not a proper choice for such a purpose. Being different from the static method, the proposed method in this paper is simpler and more accurate because it only needs to measure the electrical conductance which can be obtained accurately.

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Fig. 9. Electrical conductance of the system measured from the two electrodes of wafer a when wafer b is shunted to: (a1) a resistor (finite element analysis); (a2) a resistor (experiment); (b1) a capacitor (finite element analysis); (b2) a capacitor (experiment); (c1) a resistor (100 ) and an inductor in series (finite element analysis); (c2) a resistor (100 ) and an inductor in series (experiment).

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Fig. 11. The variation of mechanical loss factor of wafer b 1/Qm and electrical loss factor of the system 1/Qe when wafer b is shunted to a resistor.

of the ratio of the acceleration a to the force F, which reflects the mechanical impedance or the mechanical admittance of the system.



a

F

FRF =

(15)

In order to measure the FRF in Eq. (15), a series of indispensable instruments are employed. A signal generator, a power amplifier and a shaker are needed to excite the system into vibration. A force sensor is needed to measure the input force and an accelerometer is needed to measure the acceleration response. Besides, two charge amplifiers and a two-channel FFT analyzer are employed to analyze the output signals from the sensors. Thus, the measurement process is complicated and errors may be introduced by the measurement of various variables. While for method in this paper, both the resonant frequency and damping ratio are determined from the electrical conductance of the model. The electrical conductance is the only measured variable and an impedance analyzer is the only necessary instrument. Therefore, it is comparatively simple and low-cost. 7. Conclusions

Fig. 10. The variation of c33 and resonant frequency of PZT-5H shunted to:(a) a resistor;(b) a capacitor;(c) a resistor (100 ) and an inductor in series.

6.2. Comparison with the mechanical impedance measurement method In the work of Law etc. [4], a mechanical mass-spring-dashpot model consisting of two resistor-shunted piezo-material rings and two mass blocks is established to measure the variations of the resonant frequency and damping ratio as the shunt resistance changes. The system is excited into vibration and then both the input force and the acceleration response are measured to form the frequency response function (FRF), from which the resonant frequency and damping ratio are determined. The FRF is defined as the modulus

This paper has proposed a modified resonator measurement method to experimentally study the variation of elastic constant c33 of the piezoelectric ceramics shunted to circuits under thickness mode. Since c33 /c33 D nearly varies linearly with fs /fs D , it is only needed to measure the variation of the resonant frequency which are easier to be observed, then the variations of c33 is indirectly obtained. Similarly, the variation of mechanical loss factor is linearly with 1/Q of the resonant conductance, so it can be reflected through the Q-value of the resonant conductance. When the radial dimensions of the piezoelectric ceramic wafers and the mass block are much larger than their thicknesses or they are radially clamped, the single-oscillator system vibrates in pure thickness mode and can be analyzed based on Mason equivalent circuit theory. The variations of the elastic constant c33 have been indirectly derived according to the variation of the resonant frequency of the system when the piezoelectric ceramic wafer is shunted to a resistor, a capacitor, and a resistor connected in series with an inductor respectively. The finite element analysis is also used to validate the theoretical analytical results, with the FEM results showing good agreement with the theoretical results. A symmetrical model sample that is equivalent to the singleoscillator model has been conceived in the experiment. The variation of the resonant frequency has been measured by an impedance analyzer. Besides, another finite element analysis is

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conducted under radially free boundary condition to simulate the experimental situation. The experimental results have good coincidence with the finite element results, which confirms the validity of this indirect method to measure the variation of elastic constant c33 of piezoelectric ceramics shunted to circuit under thickness mode. Although this paper only discusses the elastic constant c33 under thickness mode with three basic shunt circuit types, and the mechanical loss factor with a shunt resistor, the proposed method can be extended to investigate the variations of elastic constants and the mechanical loss factor of piezoelectric composites, or elastic constants of piezoelectric materials shunted to different kinds of shunt circuits under other modes. References [1] R.L. Forward, Electronic damping of vibrations in optical structures, Appl. Opt. 18 (1979) 690–697. [2] N.W. 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Biographies

Yang Sun was born in Jilin Province, China, in 1987. She received the B.S. from College of Information Technical Science, Nankai University, Tianjin, China, in 2010. She is currently working on her Ph.D. in Department of Electronics, Peking University, Beijing, China. Her research areas are piezoelectric shunt damping and under water acoustic absorption materials.

Zhaohui Li was born in Henan Province, China, in 1966. He received the B.S. and M.S. degrees in acoustics and the Ph.D. degree in communication from the Department of Electronics, Peking University, China, in 1987, 1992 and 2005, respectively. He is currently pursuing his research work at Acoustics Laboratory, School of Electronics and Computer Science, Peking University, China. His research interests included ultrasonics, underwater acoustics, signal processing, and the theory of transduction and transducers.

Qihu Li was born in Wenzhou, Zhejiang Province, China, in 1939. He graduated from the Digital Mechanics Department of Beijing University. He was appointed to be the fourth director of Institute of Acoustics, Chinese Academy of Sciences (IACAS). He was elected member of the Academic Division of Science and Technology, the Chinese Academy of Sciences (academician) in 1997. Based on his analysis of the features of acoustic propagation in shallow sea of our country, Academician Li innovatively applied the traditional methods and theories to solve the new problems related to underwater acoustic signal processing. His successful researches have made great contributions to the modernization of naval sonar. He has been a part-time professor in Institute of Advanced Technology, Peking University from 2007.