- Email: [email protected]
Analysis on radial vibration of a stack of piezoelectric shells
Available online at www.sciencedirect.com
CERAMICS INTERNATIONAL
Ceramics International 41 (2015) S856–S860 www.elsevier.com/locate/ceramint
Analysis on radial vibration of a stack of piezoelectric shells Likun Wanga,n, Lei Qina, Wenjie Lia, Bin Zhanga, Yu Lua, Guang Lib a
Beijing Key Laboratory for Sensors, Beijing Information Science & Technology University, No.35 North Fourth ring road, Chaoyang District, Beijing 100101, China b National Oceanic Technology Center, Tianjin 300111, China Received 26 October 2014; accepted 19 March 2015 Available online 30 March 2015
Abstract The vibration properties of layers of piezoelectric shells are investigated by theoretical analysis, finite element simulation, and experimental measurement. The equivalent admittance of a 2-layered shell is first analyzed based on the admittance equation of a single shell, and the resonant frequencies of shell stacks are then deduced by setting the equivalent admittance to infinity. The analytical expressions are calculated numerically, and the results are expressed in terms of resonant frequencies and bandwidths as a function of average radius. Comparisons with calculation, finite element analysis and measurement show general agreement, although the bandwidth figures show some variations. & 2015 Published by Elsevier Ltd and Techna Group S.r.l.
Keywords: Piezoelectric shell stack; Radial vibration; Mode analysis
1. Introduction Underwater transducers are devices which transform acoustic energy into electrical energy and vice versa, in this way receiving and launching acoustic signals. They have been widely applied to underwater communication, detection, navigation, and target location. Two common underwater acoustic transducers are the thin-wall spherical shell transducer [1] and the Tonpilz transducer [2,3]. Because the former generates an omnidirectional beam, and the latter, with its stack of piezoceramic discs, provides strong transmitting power, they have been widely applied to underwater acoustic detection. Recently, a novel piezoelectric shell stack transducer has been developed in our laboratory which combines the features of both [4]. It consists of two piezoelectric ceramic shells bonded to each other and connected in parallel electrically. The stack configuration allows the vibration of each shell to build on the other, enhancing transmitting power. As a result, the new n
Corresponding author. Tel.: +8613671055786. E-mail addresses: [email protected] (L. Wang), [email protected] (L. Qin), [email protected] (W. Li), [email protected] (B. Zhang), [email protected] (G. Li). http://dx.doi.org/10.1016/j.ceramint.2015.03.159 0272-8842/& 2015 Published by Elsevier Ltd and Techna Group S.r.l.
transducer is omnidirectional and also has high transmitting power, or equivalently, high sensitivity. In this paper the vibration frequency and resonant bandwidth of the shell stack are analyzed through theoretical calculations and verified by finite element simulation and experimental measurement. Because the working frequency of a transducer depends on its resonant frequencies, the most important aspect considered here is the resonance frequency of the shell stack, and analytical expressions for this are deduced.
2. Resonant frequency of a shell stack The structure of a shell stack piezoelectric vibrator is shown in Fig. 1, where the inner and outer walls of the shells are covered with silver and the shells are polarized in the radial direction. The thickness of each shell is t and the radius of the innermost and outermost shells are a and b respectively. Then the average radius of the shell stack is r ¼ (aþ b)/2. The z-axis (direction 3) is chosen in the radial direction, and the x- and yaxes are in the plane of the shells. Based on the constitutive equation for a piezoelectric and its kinematics, the equivalent
L. Wang et al. / Ceramics International 41 (2015) S856–S860
Fig. 1. Diagram of a shell stack piezoelectric vibrator.
admittance of a single shell can be shown to be [5] jω4πR2 εT33 k2P ω20 T 2 Y¼ þ ε33 ð1 k P Þ ; t ω20 ω2
ð1Þ
2
ω20 ¼
2
R
ρðsE11 þ sE12 Þ
; k2P ¼
;
sE11 and sE12 are the compliance constants under constant electric field, d31 is the piezoelectric constant, εT33 is the relative dielectric constant under constant stress, and R is the average radius of the shell. When the two shells are connected in parallel, the stack's equivalent admittance approaches infinity at the resonant frequency, so the stack's resonant frequency can be calculated by setting Y in Eq. (1) to infinity. Suppose the inner and outer admittance of shell stack are respectively Y1 and Y2, and thus I1 jω4πa2 εT33 k 2P ω20 T 2 Y1 ¼ ¼ þ ε ð1 k Þ 33 P V1 t ω20 ω2 and Y2 ¼
Because the two shells are connected in parallel, the total admittance of the shell stack is jω4π 2εT33 k2P 1 T 2 2 þ ε a ð1 k Þ Y ¼ Y1 þY2 ¼ 33 P t ρðsE11 þ sE12 Þ ω20 ω2 jω4π 2εT33 k2P 1 þ þ εT33 b2 ð1 k2P Þ E E 2 t ρðs11 þ s12 Þ ω1 ω2 jω4π 2εT33 k2P 1 1 ¼ þ t ρðsE11 þ sE12 Þ ω20 ω2 ω21 ω2 þ εT33 ð1 k 2P Þða2 þ b2 Þ jω4π 2εT33 k2P ω20 þ ω21 2ω2 ¼ t ρðsE11 þ sE12 Þ ðω20 ω2 Þðω21 ω2 Þ þ εT33 ð1 k 2P Þða2 þ b2 Þ Conversely, anti-resonance occurs when Y ¼ 0, so that ω20 þ ω21 2ω2a ¼ 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω20 þ ω21 a2 þ b2 ¼ ωa ¼ 2 2 a2 b ρðsE11 þ sE12 Þ
where 2d 231 ðsE11 þ sE12 ÞεT33
I2 jω4πb2 εT33 k 2P ω21 T 2 ¼ þ ε ð1 k Þ 33 P ; V2 t ω21 ω2
and the anti-resonance frequency, fa, can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 1 a2 þ b2 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2Þ fa ¼ ¼ 2π a2 b2 ρðsE11 þ sE12 Þ 2π ρðsE11 þ sE12 Þ ab For the piezoelectric ceramic PZT-4 sE11 ¼ 12.3 10–12 m /N, sE12 ¼ –4.05 10–12 m2/N, and ρ ¼ 7500 kg/m2. Substituting these parameters into Eq. (2) gives the relation between frequency and radius, and the results are shown in Fig. 2. 2
3. Finite element simulation of a shell stack Using a finite element analysis method (Ansys software), the vibration modes of the shell stack were simulated. PZT-4 was chosen as the shell material and its dimensions were the same as in the theoretical calculation. Because of the structure's symmetry, one-twelfth of a half-shell (one-twenty-fourth of a whole shell) was chosen as the modeling unit, as shown in Fig. 3.
where ω20 ¼
2
; a2 ρðsE11 þ sE12 Þ
a2 ω20 ¼ b2 ω21 ¼
ω21 ¼
2 b
2
ρðsE11 þ sE12 Þ
; and
2
: ρðsE11 þ sE12 Þ
Then jω4π 2εT33 k2P 1 T 2 2 Y1 ¼ þ ε33 a ð1 k P Þ t ρðsE11 þ sE12 Þ ω20 ω2 and Y2 ¼
S857
jω4π 2εT33 k2P 1 T 2 2 þ ε b ð1 k Þ 33 P : t ρðsE11 þ sE12 Þ ω21 ω2
Fig. 2. Anti-resonance frequency vs radius.
S858
L. Wang et al. / Ceramics International 41 (2015) S856–S860
Fig. 3. Finite element simulation of a double shell unit (1/24 of a whole shell).
Fig. 5. Resonant frequency of a single shell and of a stack against radius.
Fig. 4. Simulation plots of resonant frequency vs radius and thickness.
Fig. 6. Plots of resonant bandwidth against radius and stack number.
As the finite element, 3-D Coupled-Field Solid was used to mesh the unit and symmetric constraints were placed on each of the three cross-sections. The electrode voltage at the interface of the two shells was set to 1 V, while the electrodes on the inner and outer surfaces of the stack were set to 0 V. In the modal analysis, radial vibration was calculated according to the chosen structural parameters. Resonant frequencies were calculated as a function of average radius, which varied from 9 to 14 mm using a shell thickness of 1 mm; similarly, shell thickness was varied from 0.7 to 1.3 mm using a fixed average radius of 9 mm. The results are presented in Fig. 4. Fig. 4 shows that the resonant frequency varies markedly with radius but only little with thickness. The practical conclusion therefore is that the resonant frequency of the radial vibration mode of a spherical sensor depends in large measure on its average radius. To compare the vibration characteristics of single shell and a shell stack, the vibration of each was simulated using similar structure parameters, and the results are shown in Fig. 5. It can be seen that the resonant frequency of a single shell is higher
than that of a shell stack of the same average radius, a reasonable result because a shell stack is loaded more heavily than a single shell. A similar finite element simulation was done for multilayer shells. Harmonic analysis was used to compare the resonant bandwidth of a single shell with that of three-, four-, and fivefold shells, and the results are shown in Fig. 6. Here the resonant bandwidth of a single shell is compared with those of multi-layer stacks of the same average radius. The average radius of the shells was set to 9 mm and the thickness of each shell to 1 mm. Fig. 6 confirms that the bandwidths of a single shell, and of a stack, do not change appreciably with average radius, although that of the stack is somewhat higher than that of a single shell of the same average radius. The plot also shows that the bandwidth increases as the number of layers increases, a result with the practical implication that the bandwidth of a piezoelectric vibrator can be widened by controlling its stack structure.
L. Wang et al. / Ceramics International 41 (2015) S856–S860 Table 1 Structural parameters of five shell stack transducers.
S859
Table 2 Relative errors between calculated and measured values in Fig. 7.
Sample number
Inner radius of inner shell (mm)
Outer radius of outer shell (mm)
Average radius (mm)
Thickness of each shell (mm)
Sample number
Average Measured radius (mm) frequencies fs (kHz)
calculated frequencies fa (kHz)
Relative error (%)
1 2 3 4 5
8.7 9.8 10.7 11.6 13.3
11.0 11.9 12.6 13.8 15.4
10.0 10.9 11.7 12.7 14.3
1.1 1.0 0.9 1.1 1.0
1 2 3 4 5
10.0 10.9 11.7 12.7 14.3
90.87 83.00 77.50 71.50 63.00
14.7 0.95 4.8 3.97 5.9
Fig. 7. Plots of resonant frequencies against radius with measurments, calculations and simulations.
4. Experimental testing To confirm the validity of the theoretical calculations and the finite element simulations, five different shell stacks, with parameters shown in Table 1, were prepared and their performance tested. Two hemispherical shells were first glued together by epoxy resin to form a double-layer hemispherical stack, and then two identical hemispherical stacks were similarly bonded to form a sphere. Wire electrodes were welded on to the inner and outer surfaces of the stacks. An impedance analyzer was used to measure the samples' resonant frequency and bandwidth, and the results of frequencies are presented in Fig. 7 together with the corresponding theoretical calculation and simulations. It is clear from Fig. 7 that for all three curves the resonant frequency generally declines as average radius increases. However, the theoretical values and the simulation values are both slightly higher than the measured ones. The errors between them may be caused by the following reasons. First, the result of theoretical calculation by formula (2) is anti-resonance frequency because the analytic solution of resonance frequency is difficult to solved, but measured values is resonance frequency. The relationship between anti-resonance frequency fa and resonance frequency fs is fa/fs ¼ 1þ 1/2γ, γ¼ C0/C1, C0 is (static capacitance) smaller than C1 (dynamic capacity) a lot. Second, only 1/24 of a whole shell was chosen to simulate approximately whole as shown in Fig. 3 in finite element simulation, it can bring some errors. Moreover, it is possible that the PZT-4 parameters used in
79.22 82.22 73.95 68.77 59.49
Fig. 8. Resonant bandwidth of a single shell and of a stack.
both the theoretical calculations and the simulations may deviate from those of the actual shells [3]. The errors between calculated and measured values are presented in Table 2. It shows that the errors between calculated and measured values were less than 5.9% except for sample no. 1 (average radius of 10 mm), which reached 14.7%. This may be because in this case the shell radius is so small that the gap between the shells fails to totally fill with resin, which leads to both shells vibrating out of sync. Fig. 8 shows how the resonant bandwidth of both a single shell and a stack, derived from simulation and measurement, changes with average radius. It can be seen that the measured bandwidth of a stack is higher than that of a single shell, which agrees with the simulation results of Fig. 6; it also illustrates how stack structure can be used to expand bandwidth. However, the simulated bandwidth of the stack is much higher than indicated by measurement. Possible reasons for this are that the modeling unit (Fig. 3) ignore the epoxy resin layer, the PZT-4 stiffness coefficient used in the simulations may smaller than those of the actual shells, the smaller the stiffness coefficient of material the more bandwidth of it's resonance. 5. Conclusions In this paper, an equation for the resonance frequency of a piezoelectric shell stack has been theoretically deduced. The equation allows curves of how frequency and bandwidth change with stack configuration to be plotted and compared with finite element simulation and measurement. The
S860
L. Wang et al. / Ceramics International 41 (2015) S856–S860
comparison shows that, in terms of frequency, the three approaches accord with each other well and the relative errors between them are less than 5.9%. The frequency drops rapidly with an increase in average radius, so that frequency becomes approximately inversely proportional to radius. However, the frequency changes little with thickness, the curves follow similar trends but there are some deviations. The bandwidth of a multi-layered shell stack is higher than that of a single shell, although there are some unexplained variations as the number of layers increase. Conflict of interest We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 61302015, 61471047), and the
Foundation of Beijing College Innovation Capability Promotion Plan (NO.PXM2015_014224_000028).
References [1] J.F. Tressler, S. Alkoy, R.E. Newnham, Piezoelectric sensors and sensor materials, J. Electroceram. 2 (4) (1998) 257–272. [2] T. Inoue, T. Nada, T. Tsuchiya, T. Nakanishi, T. Miyama, M. Konno, Tonpilz piezoelectric transducers with acoustic matching plates for underwater color image transmission, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40 (1993) 121–130. [3] Q. Yao, L. Bjorno, Broadband tonpilz underwater acoustic transducers based on multimode optimization, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44 (5) (1997) 1060–1066. [4] L. Wang, L. Qin, L. Li, T. Xiao, Piezoelectric shell stack underwater acoustic transducer, CN patent 200810075184.2, 2008. [5] G. Luan, J. Zhang, Piezoelectric Transducers and Transducer Arrays, Beijing University Press, Beijing, 1990.
CERAMICS INTERNATIONAL
Ceramics International 41 (2015) S856–S860 www.elsevier.com/locate/ceramint
Analysis on radial vibration of a stack of piezoelectric shells Likun Wanga,n, Lei Qina, Wenjie Lia, Bin Zhanga, Yu Lua, Guang Lib a
Beijing Key Laboratory for Sensors, Beijing Information Science & Technology University, No.35 North Fourth ring road, Chaoyang District, Beijing 100101, China b National Oceanic Technology Center, Tianjin 300111, China Received 26 October 2014; accepted 19 March 2015 Available online 30 March 2015
Abstract The vibration properties of layers of piezoelectric shells are investigated by theoretical analysis, finite element simulation, and experimental measurement. The equivalent admittance of a 2-layered shell is first analyzed based on the admittance equation of a single shell, and the resonant frequencies of shell stacks are then deduced by setting the equivalent admittance to infinity. The analytical expressions are calculated numerically, and the results are expressed in terms of resonant frequencies and bandwidths as a function of average radius. Comparisons with calculation, finite element analysis and measurement show general agreement, although the bandwidth figures show some variations. & 2015 Published by Elsevier Ltd and Techna Group S.r.l.
Keywords: Piezoelectric shell stack; Radial vibration; Mode analysis
1. Introduction Underwater transducers are devices which transform acoustic energy into electrical energy and vice versa, in this way receiving and launching acoustic signals. They have been widely applied to underwater communication, detection, navigation, and target location. Two common underwater acoustic transducers are the thin-wall spherical shell transducer [1] and the Tonpilz transducer [2,3]. Because the former generates an omnidirectional beam, and the latter, with its stack of piezoceramic discs, provides strong transmitting power, they have been widely applied to underwater acoustic detection. Recently, a novel piezoelectric shell stack transducer has been developed in our laboratory which combines the features of both [4]. It consists of two piezoelectric ceramic shells bonded to each other and connected in parallel electrically. The stack configuration allows the vibration of each shell to build on the other, enhancing transmitting power. As a result, the new n
Corresponding author. Tel.: +8613671055786. E-mail addresses: [email protected] (L. Wang), [email protected] (L. Qin), [email protected] (W. Li), [email protected] (B. Zhang), [email protected] (G. Li). http://dx.doi.org/10.1016/j.ceramint.2015.03.159 0272-8842/& 2015 Published by Elsevier Ltd and Techna Group S.r.l.
transducer is omnidirectional and also has high transmitting power, or equivalently, high sensitivity. In this paper the vibration frequency and resonant bandwidth of the shell stack are analyzed through theoretical calculations and verified by finite element simulation and experimental measurement. Because the working frequency of a transducer depends on its resonant frequencies, the most important aspect considered here is the resonance frequency of the shell stack, and analytical expressions for this are deduced.
2. Resonant frequency of a shell stack The structure of a shell stack piezoelectric vibrator is shown in Fig. 1, where the inner and outer walls of the shells are covered with silver and the shells are polarized in the radial direction. The thickness of each shell is t and the radius of the innermost and outermost shells are a and b respectively. Then the average radius of the shell stack is r ¼ (aþ b)/2. The z-axis (direction 3) is chosen in the radial direction, and the x- and yaxes are in the plane of the shells. Based on the constitutive equation for a piezoelectric and its kinematics, the equivalent
L. Wang et al. / Ceramics International 41 (2015) S856–S860
Fig. 1. Diagram of a shell stack piezoelectric vibrator.
admittance of a single shell can be shown to be [5] jω4πR2 εT33 k2P ω20 T 2 Y¼ þ ε33 ð1 k P Þ ; t ω20 ω2
ð1Þ
2
ω20 ¼
2
R
ρðsE11 þ sE12 Þ
; k2P ¼
;
sE11 and sE12 are the compliance constants under constant electric field, d31 is the piezoelectric constant, εT33 is the relative dielectric constant under constant stress, and R is the average radius of the shell. When the two shells are connected in parallel, the stack's equivalent admittance approaches infinity at the resonant frequency, so the stack's resonant frequency can be calculated by setting Y in Eq. (1) to infinity. Suppose the inner and outer admittance of shell stack are respectively Y1 and Y2, and thus I1 jω4πa2 εT33 k 2P ω20 T 2 Y1 ¼ ¼ þ ε ð1 k Þ 33 P V1 t ω20 ω2 and Y2 ¼
Because the two shells are connected in parallel, the total admittance of the shell stack is jω4π 2εT33 k2P 1 T 2 2 þ ε a ð1 k Þ Y ¼ Y1 þY2 ¼ 33 P t ρðsE11 þ sE12 Þ ω20 ω2 jω4π 2εT33 k2P 1 þ þ εT33 b2 ð1 k2P Þ E E 2 t ρðs11 þ s12 Þ ω1 ω2 jω4π 2εT33 k2P 1 1 ¼ þ t ρðsE11 þ sE12 Þ ω20 ω2 ω21 ω2 þ εT33 ð1 k 2P Þða2 þ b2 Þ jω4π 2εT33 k2P ω20 þ ω21 2ω2 ¼ t ρðsE11 þ sE12 Þ ðω20 ω2 Þðω21 ω2 Þ þ εT33 ð1 k 2P Þða2 þ b2 Þ Conversely, anti-resonance occurs when Y ¼ 0, so that ω20 þ ω21 2ω2a ¼ 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω20 þ ω21 a2 þ b2 ¼ ωa ¼ 2 2 a2 b ρðsE11 þ sE12 Þ
where 2d 231 ðsE11 þ sE12 ÞεT33
I2 jω4πb2 εT33 k 2P ω21 T 2 ¼ þ ε ð1 k Þ 33 P ; V2 t ω21 ω2
and the anti-resonance frequency, fa, can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 1 a2 þ b2 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2Þ fa ¼ ¼ 2π a2 b2 ρðsE11 þ sE12 Þ 2π ρðsE11 þ sE12 Þ ab For the piezoelectric ceramic PZT-4 sE11 ¼ 12.3 10–12 m /N, sE12 ¼ –4.05 10–12 m2/N, and ρ ¼ 7500 kg/m2. Substituting these parameters into Eq. (2) gives the relation between frequency and radius, and the results are shown in Fig. 2. 2
3. Finite element simulation of a shell stack Using a finite element analysis method (Ansys software), the vibration modes of the shell stack were simulated. PZT-4 was chosen as the shell material and its dimensions were the same as in the theoretical calculation. Because of the structure's symmetry, one-twelfth of a half-shell (one-twenty-fourth of a whole shell) was chosen as the modeling unit, as shown in Fig. 3.
where ω20 ¼
2
; a2 ρðsE11 þ sE12 Þ
a2 ω20 ¼ b2 ω21 ¼
ω21 ¼
2 b
2
ρðsE11 þ sE12 Þ
; and
2
: ρðsE11 þ sE12 Þ
Then jω4π 2εT33 k2P 1 T 2 2 Y1 ¼ þ ε33 a ð1 k P Þ t ρðsE11 þ sE12 Þ ω20 ω2 and Y2 ¼
S857
jω4π 2εT33 k2P 1 T 2 2 þ ε b ð1 k Þ 33 P : t ρðsE11 þ sE12 Þ ω21 ω2
Fig. 2. Anti-resonance frequency vs radius.
S858
L. Wang et al. / Ceramics International 41 (2015) S856–S860
Fig. 3. Finite element simulation of a double shell unit (1/24 of a whole shell).
Fig. 5. Resonant frequency of a single shell and of a stack against radius.
Fig. 4. Simulation plots of resonant frequency vs radius and thickness.
Fig. 6. Plots of resonant bandwidth against radius and stack number.
As the finite element, 3-D Coupled-Field Solid was used to mesh the unit and symmetric constraints were placed on each of the three cross-sections. The electrode voltage at the interface of the two shells was set to 1 V, while the electrodes on the inner and outer surfaces of the stack were set to 0 V. In the modal analysis, radial vibration was calculated according to the chosen structural parameters. Resonant frequencies were calculated as a function of average radius, which varied from 9 to 14 mm using a shell thickness of 1 mm; similarly, shell thickness was varied from 0.7 to 1.3 mm using a fixed average radius of 9 mm. The results are presented in Fig. 4. Fig. 4 shows that the resonant frequency varies markedly with radius but only little with thickness. The practical conclusion therefore is that the resonant frequency of the radial vibration mode of a spherical sensor depends in large measure on its average radius. To compare the vibration characteristics of single shell and a shell stack, the vibration of each was simulated using similar structure parameters, and the results are shown in Fig. 5. It can be seen that the resonant frequency of a single shell is higher
than that of a shell stack of the same average radius, a reasonable result because a shell stack is loaded more heavily than a single shell. A similar finite element simulation was done for multilayer shells. Harmonic analysis was used to compare the resonant bandwidth of a single shell with that of three-, four-, and fivefold shells, and the results are shown in Fig. 6. Here the resonant bandwidth of a single shell is compared with those of multi-layer stacks of the same average radius. The average radius of the shells was set to 9 mm and the thickness of each shell to 1 mm. Fig. 6 confirms that the bandwidths of a single shell, and of a stack, do not change appreciably with average radius, although that of the stack is somewhat higher than that of a single shell of the same average radius. The plot also shows that the bandwidth increases as the number of layers increases, a result with the practical implication that the bandwidth of a piezoelectric vibrator can be widened by controlling its stack structure.
L. Wang et al. / Ceramics International 41 (2015) S856–S860 Table 1 Structural parameters of five shell stack transducers.
S859
Table 2 Relative errors between calculated and measured values in Fig. 7.
Sample number
Inner radius of inner shell (mm)
Outer radius of outer shell (mm)
Average radius (mm)
Thickness of each shell (mm)
Sample number
Average Measured radius (mm) frequencies fs (kHz)
calculated frequencies fa (kHz)
Relative error (%)
1 2 3 4 5
8.7 9.8 10.7 11.6 13.3
11.0 11.9 12.6 13.8 15.4
10.0 10.9 11.7 12.7 14.3
1.1 1.0 0.9 1.1 1.0
1 2 3 4 5
10.0 10.9 11.7 12.7 14.3
90.87 83.00 77.50 71.50 63.00
14.7 0.95 4.8 3.97 5.9
Fig. 7. Plots of resonant frequencies against radius with measurments, calculations and simulations.
4. Experimental testing To confirm the validity of the theoretical calculations and the finite element simulations, five different shell stacks, with parameters shown in Table 1, were prepared and their performance tested. Two hemispherical shells were first glued together by epoxy resin to form a double-layer hemispherical stack, and then two identical hemispherical stacks were similarly bonded to form a sphere. Wire electrodes were welded on to the inner and outer surfaces of the stacks. An impedance analyzer was used to measure the samples' resonant frequency and bandwidth, and the results of frequencies are presented in Fig. 7 together with the corresponding theoretical calculation and simulations. It is clear from Fig. 7 that for all three curves the resonant frequency generally declines as average radius increases. However, the theoretical values and the simulation values are both slightly higher than the measured ones. The errors between them may be caused by the following reasons. First, the result of theoretical calculation by formula (2) is anti-resonance frequency because the analytic solution of resonance frequency is difficult to solved, but measured values is resonance frequency. The relationship between anti-resonance frequency fa and resonance frequency fs is fa/fs ¼ 1þ 1/2γ, γ¼ C0/C1, C0 is (static capacitance) smaller than C1 (dynamic capacity) a lot. Second, only 1/24 of a whole shell was chosen to simulate approximately whole as shown in Fig. 3 in finite element simulation, it can bring some errors. Moreover, it is possible that the PZT-4 parameters used in
79.22 82.22 73.95 68.77 59.49
Fig. 8. Resonant bandwidth of a single shell and of a stack.
both the theoretical calculations and the simulations may deviate from those of the actual shells [3]. The errors between calculated and measured values are presented in Table 2. It shows that the errors between calculated and measured values were less than 5.9% except for sample no. 1 (average radius of 10 mm), which reached 14.7%. This may be because in this case the shell radius is so small that the gap between the shells fails to totally fill with resin, which leads to both shells vibrating out of sync. Fig. 8 shows how the resonant bandwidth of both a single shell and a stack, derived from simulation and measurement, changes with average radius. It can be seen that the measured bandwidth of a stack is higher than that of a single shell, which agrees with the simulation results of Fig. 6; it also illustrates how stack structure can be used to expand bandwidth. However, the simulated bandwidth of the stack is much higher than indicated by measurement. Possible reasons for this are that the modeling unit (Fig. 3) ignore the epoxy resin layer, the PZT-4 stiffness coefficient used in the simulations may smaller than those of the actual shells, the smaller the stiffness coefficient of material the more bandwidth of it's resonance. 5. Conclusions In this paper, an equation for the resonance frequency of a piezoelectric shell stack has been theoretically deduced. The equation allows curves of how frequency and bandwidth change with stack configuration to be plotted and compared with finite element simulation and measurement. The
S860
L. Wang et al. / Ceramics International 41 (2015) S856–S860
comparison shows that, in terms of frequency, the three approaches accord with each other well and the relative errors between them are less than 5.9%. The frequency drops rapidly with an increase in average radius, so that frequency becomes approximately inversely proportional to radius. However, the frequency changes little with thickness, the curves follow similar trends but there are some deviations. The bandwidth of a multi-layered shell stack is higher than that of a single shell, although there are some unexplained variations as the number of layers increase. Conflict of interest We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 61302015, 61471047), and the
Foundation of Beijing College Innovation Capability Promotion Plan (NO.PXM2015_014224_000028).
References [1] J.F. Tressler, S. Alkoy, R.E. Newnham, Piezoelectric sensors and sensor materials, J. Electroceram. 2 (4) (1998) 257–272. [2] T. Inoue, T. Nada, T. Tsuchiya, T. Nakanishi, T. Miyama, M. Konno, Tonpilz piezoelectric transducers with acoustic matching plates for underwater color image transmission, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40 (1993) 121–130. [3] Q. Yao, L. Bjorno, Broadband tonpilz underwater acoustic transducers based on multimode optimization, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44 (5) (1997) 1060–1066. [4] L. Wang, L. Qin, L. Li, T. Xiao, Piezoelectric shell stack underwater acoustic transducer, CN patent 200810075184.2, 2008. [5] G. Luan, J. Zhang, Piezoelectric Transducers and Transducer Arrays, Beijing University Press, Beijing, 1990.