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The resonance frequencies on mechanically pre-stressed ultrasonic piezotransducers
Ultrasonics 39 (2001) 1±5
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The resonance frequencies on mechanically pre-stressed ultrasonic piezotransducers F.J. Arnold a,*, S.S. M uhlen b b
a Superior Center of Technology Education, Campinas State University, P.O. Box 456, 13484-420 Limeira, SP, Brazil Faculty of Electric and Computation Engineering, Campinas State University, P.O. Box 6040, 13091-070 Campinas, SP, Brazil
Received 31 January 2000
Abstract The piezotransducers employed in high power ultrasound are composed of piezoelectric ceramics and metallic pieces. These transducers are mechanically pre-stressed in order to avoid the ceramic fractures when high voltage is applied under resonance. The resonance and anti-resonance frequencies are shifted depending on the level of applied mechanical pre-stressing. This paper discusses some causes of this shifting on a experimental study. The discussion takes into account the variations on characteristic parameters of the ceramics and the acoustic coupling between parts of the transducer. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 43.38.Ar Keywords: Transducers; Piezoelectrics; Pre-stressing; Ultrasound
1. Introduction The high power piezoelectric transducers used on medical and industrial applications are composed of piezoelectric ceramics and metallic pieces. These transducers are supplied with high voltage that causes large strains which can lead ceramics to fracture in the traction semi-cycle. When the transducer is mechanically pre-stressed, the ceramics are compressed and can develop strains of higher amplitudes [1]. However, the prestressing can yield variations on performance of the transducer, due to shiftings on the characteristic parameters of the ceramic and on the coupling between transducer pieces. The shifting on resonance and antiresonance frequencies is a detected phenomenon in such transducers, and it is studied in this paper. The scienti®c literature covering the mechanical prestressing shows experimental studies of the piezoelectric [2±9] and electrostrictive [7,10,11] ceramics submitted to high stress levels. These works show that mechanical stress can yield domains switching on grains of the ce-
*
Corresponding author. Tel.: +55-19-4407139; fax: +55-194513939. E-mail address: [email protected] (F.J. Arnold).
ramic [12] and, thus, the dipolar polarization is reoriented. The change of physical properties of the ceramics is a consequence of this mechanism and has strong dependence of the composition of the ceramics and the stress level applied. An interesting example of change of piezoelectric properties due to eects of an uniaxial stress [8] has shown that the piezoelectric coecients (d33 and d31 ) of PZT-5A remains constant up to 50 MPa, while ones of PZT-4 have a signi®cant increase in the same range [9]. The goal of this paper is to investigate the shifting on resonance and anti-resonance frequencies related to the mechanical pre-stressing up to 50 MPa. Two hypothesis are considered to explain these eects: the changes on the physical characteristic of the ceramics and the changes on the eective contact surface between transducer parts under mechanical pre-stressing. 2. Theoretical concepts and de®nitions Here, the characterization of the piezoelectric ceramics is limited to measurements and parameters related to the axial direction, and all coecients related to this are denoted by index 3. These parameters [13] contribute to describe the longitudinal vibration mode
0041-624X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 0 ) 0 0 0 4 7 - 0
2
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5
of the transducers used in the experiments. The characterization covers the following parameters: the elastic T S coecient cD 33 , the dielectric coecients e33 and e33 and the piezoelectric coecient d33 . Other vibration modes and their characteristic coecients are neglected. 2 The elastic stiness coecient cD 33 (N/m ) is the ratio between the stress and the strain for an electric displacement equal to zero. The dielectric coecients eT33 and eS33 (F/m) are the ratios between electric displacement and electric ®eld when the ceramic is free and clamped, respectively. The piezoelectric coecient d33 (C/N) is the ratio between the electric displacement and stress.
3. Experimental method The piezoelectric ceramics used in the experiments, produced by THORNTON-INPEC from Vinhedo, SP, Brazil, are annulars shaped, with 6.3 mm of thickness and 12.5 and 38.0 mm of internal and external diameter, respectively. They have silver-coated electrodes on plane surfaces. The ceramics used on experiments were characterized according to IEEE standard [14]. The experimental results are obtained with the construction of some composed transducers, shown in Fig. 1. These transducers are composed of two piezoelectric ceramics; brass electrodes with 0.3 mm of thickness; two pieces of commercial aluminum with the radial dimensions equal to ceramics and a steel bolt that pre-stresses the system when tightened. The dierence between transducers is the length of its aluminum pieces. The pre-stressing value is obtained through the calibration curve resulting from the experimental setup shown on Fig. 2. When the ceramics are compressed, they accumulate electric charges that can be converted into voltage with a storage capacitor, and measured with
Fig. 1. Schematic representation of the transducer.
Fig. 2. Experimental setup used to determine the calibration curve.
a high electronic voltmeter. So, it is possible to obtain a relation between applied force and voltage on capacitor [14], which can be plotted depicting the calibration curve. The piezoelectric coecient d33 is the ratio of the accumulated charges and applied force on ceramics, and can be obtained from slope of this curve. The transducers were assembled and the ceramics were connected to devices 3 and 4 in the setup shown on Fig. 2. When the bolt tightens the assembly, the voltmeter indicates the value that will be converted in to stress throughout calibration curve. After that, devices 3 and 4 were disconnected and the transducer was connected to a fasorial impedance meter (HP4192A) that supplies a 1.0 Vpp sinus excitation. The impedance meter also provides the resonance and anti-resonance frequencies of the transducer. In order to obtain the electric permittivity of the ceramics with the free and clamped conditions, the capacitances were measured with the impedance meter in two frequencies (800 Hz and 800 kHz). The choice of these frequencies is based on solutions of equations of forced oscillator systems [15]. When the transducer is supplied with frequencies smaller than the ®rst resonance (800 Hz), the electric excitation is approximately equal to assembly restoring force, thus, the transducer vibrates free following the external excitation signal. When the transducer is supplied with frequencies above the highest resonance and their principal harmonics (800 kHz), the displacements of the assembly becomes inversely proportional to its mass and the excitation frequency, thus, the resulting strain is approximately null. The values of capacitances in the above cited frequencies allow one to calculate the free and the clamped dielectric constant (eT33 and eS33 , respectively) through e Ct=A, where e is the electric permittivity (F/m), C is the capacitance (F), t is the thickness of the ceramic (m) and A is the transversal section of the ceramic.
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5
3
Fig. 3. Experimental setup for the measuring of propagation velocity as a function of mechanical pre-stressing in multilayer transducer using the pulse-echo method.
The measurement of the elastic coecient shifting was made by the pulse-echo method (Fig. 3). By considering that the aluminum Young modulus of the aluminum remains constant when the material is submitted to compression up to 50 MPa and by measuring the travel time of a pulse in the multi-layered transducer (transmitted pulse plus echo pulse), the wave velocity (v) in the transducer can be determined and the elastic coecient cD 33 derived. Four transducers called T1, T2, T3 and T4 have been assembled. They have aluminum pieces with 13, 26, 33 and 37 mm of thickness, respectively, on each end, with the same ceramics at the center. The resonance and antiresonance frequencies of each transducer were measured under dierent mechanical pre-stressing values. The frequencies were measured at the minimum (resonance) or maximum (anti-resonance) electric impedance modulus.
4. Results The measurements of characteristics of the non-prestressed ceramic obtained are eS33 11 nF/m, cD 33 13:9 1010 N/m2 , and v 4325 m/s. The relation between the applied force and the accumulated charges, when plotted, constitutes the calibration curve whose calculated angular coecient is d33 366 1012 C/N. This curve is used to determine the pre-stressing in the transducers by tightening the central bolt. The capacitance dielectric coecient presents a small increase proportional to applied mechanical pre-stressing, and can be neglected. The elastic and piezoelectric coecients did not present variations with the prestressing. Figs. 4±7 show the curves of resonance and anti-resonance frequencies as a function of mechanical prestressing obtained by proposed method. The upper and
Fig. 4. Experimental results of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T1 (aluminum piece thickness 13 mm).
Fig. 5. Experimental results of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T2 (aluminum piece thickness 26 mm).
lower curves represent the anti-resonance and the resonance frequency, respectively.
5. Discussion and conclusions The results obtained in the measurement show that mechanical pre-stressing (up to 50 MPa) does not yield remarkable changes on ceramic physical characteristics. The dielectric coecient showed a small increase related to pre-stressing. By replacing this small increase in a model of thickness mode of piezoelectric disc [14], the observed variation of resonance frequency is neglected. The pulse-echo method did not show changes in the
4
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5
Fig. 6. Experimental results of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T3 (aluminum piece thickness 33 mm).
ance media is similar to that obtained with shifting the cross-section along the propagation axis. In a multilayered medium is observed shifting on resonance frequency when the mechanical impedance of some layer is altered [16,17]. The experiments show that the characteristic physical of the transducer pieces are not changed when the mechanical pre-stressing is applied, thus, the mechanical impedance is not altered and the frequencies should not vary. The frequency variations observed shall be caused by dierence between eective cross-sections of the ceramic and the aluminum pieces. This means that each level of pre-stressing de®nes an eective matching surface and, thus, the behavior of the waves in the system can be considered similar to a wave propagation in a duct with changes in the cross-section along its way. When the mechanical pre-stressing is lower than 30 MPa, the contact acoustic between pieces is not perfect, i.e., the cross-sections of the ceramics and the aluminum parts on their interfaces are not equal. The tightness of the bolt yields higher compression in the central part of these pieces (near to hole) than on the periphery, thus, the wave transmission is concentrated in this region, because the minimum compression level to the eective matching is reached. When the pre-stressing increases, the eective matching region is expanded, so that when the pre-stressing is higher than 30 MPa, the whole interface is in eective matching. The experimental results allow one to conclude that the mechanical pre-stressing improves the acoustical contact between the transducer pieces on application range up to 50 MPa and does not alter the physical parameters of ceramics and metallic pieces. References
Fig. 7. Experimental reslts of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T4 (aluminum piece thickness 37 mm).
propagation velocity of the multilayered transducer related to the pre-stressing and, thus, the elastic coecient also did not change. The piezoelectric coecient did not shift either, the calibration curve shows that the voltage and the stress are linearly related. The experimental results show that the resonance and anti-resonance frequencies vary less on pre-stressing range between 30 and 50 MPa. The transmission coecient [15] between two or more media is related to energy ¯ux in the system. When the wavelength is greater than the diameter of the transducer pieces (this is the case of the transducer here), the expression of this coecient through dierent imped-
[1] S. van Randeraat, R. Setterington, Piezoelectric Ceramics, Philips application book, second edn., Mullard, London, 1974. [2] R. Nishi, Eects of one-dimensional pressure on the properties of several transducer ceramics, J. Acoust. Soc. Am. 40 (2) (1966) 486. [3] H. Krueger, Stress sensitivity of piezoelectric ceramics: part 1. Sensitivity to compressive stress parallel to the polar axis, J. Acoust. Soc. Am. 42 (3) (1967) 636. [4] H. Krueger, Stress sensitivity of piezoelectric ceramics: part 3. Sensitivity to compressive stress perpendicular to the polar axis, J. Acoust. Soc. Am. 43 (3) (1968) 583. [5] S. Meeks, R. Timme, Eects of one-dimensional stress on piezoelectric ceramics, J. Appl. Phys. 46 (10) (1975) 4334. [6] A. Tims, Eects of multidimensional stress on radially polarized piezoelectric ceramic tubes, J. Acoust. Soc. Am. 70 (1) (1981) 21. [7] H. Cao, A.G. Evans, Nonlinear deformation of ferroelectric ceramics, J. Am. Cer. Soc. 76 (4) (1993) 890. [8] Q.M. Zhang, J. Zhao, K. Uchino, J. Zheng, Change of the weak®eld properties of Pb(ZrTi)O3 piezoceramics with compressive uniaxial stresses and its links to the eect of dopants on the stability of the polarizations in the materials, J. Mat. Res. 12 (1) (1997) 226. [9] Q.M. Zhang, J. Zhao, Electromechanical properties of lead zirconate titanate piezoceramics underthe in¯uence of mechanical
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5 stresses, IEEE Trans. Ultrason. Ferroelec. Freq. Control. 46 (6) (1999) 1518. [10] Y. Nakajima, T. Hayashi, I. Haysai, K. Uchino, Eletrostrictive properties of a PMN stacked actuator, Jpn. J. Appl. Phys. 24 (2) (1985) 235. [11] C.L. Hom, S.M. Pilgrim, N. Shankar, K. Bridger, M. Massuda na, S.R. Winzer, Calculation of quasi-static electromechanical coupling coecients for electrostrictive ceramic materials, IEEE Trans. Ultrason. Ferroelec. Freq. Control. 41 (4) (1994) 542. [12] G. Arlt, Twinning in ferroelectric and ferroelastic ceramics: stress relief, J. Mat. Res. 25 (1990) 2655.
5
[13] D. Berlincourt, D. Curran, H. Jae, Piezoelectric and piezomagnetic materials and their function in transducers, Phys. Acoust. 1(A) (1964) 170. [14] ANSI/IEEE, Standard on Piezoelectricity, 1978, p. 176. [15] A. Kinsler, L. Frey, Fundamentals of Acoustics, Wiley, New York, 1962. [16] R. Hill, S. El Dardiry, A theory for optimization in the use of acoustic emission transducers, J. Acoust. Soc. Am. 67 (2) (1980) 673. [17] R. Hill, S. El-Dardiry, Variables in the use and design of acoustic emission transducers, Ultrasonics 19 (1981) 9.
www.elsevier.nl/locate/ultras
The resonance frequencies on mechanically pre-stressed ultrasonic piezotransducers F.J. Arnold a,*, S.S. M uhlen b b
a Superior Center of Technology Education, Campinas State University, P.O. Box 456, 13484-420 Limeira, SP, Brazil Faculty of Electric and Computation Engineering, Campinas State University, P.O. Box 6040, 13091-070 Campinas, SP, Brazil
Received 31 January 2000
Abstract The piezotransducers employed in high power ultrasound are composed of piezoelectric ceramics and metallic pieces. These transducers are mechanically pre-stressed in order to avoid the ceramic fractures when high voltage is applied under resonance. The resonance and anti-resonance frequencies are shifted depending on the level of applied mechanical pre-stressing. This paper discusses some causes of this shifting on a experimental study. The discussion takes into account the variations on characteristic parameters of the ceramics and the acoustic coupling between parts of the transducer. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 43.38.Ar Keywords: Transducers; Piezoelectrics; Pre-stressing; Ultrasound
1. Introduction The high power piezoelectric transducers used on medical and industrial applications are composed of piezoelectric ceramics and metallic pieces. These transducers are supplied with high voltage that causes large strains which can lead ceramics to fracture in the traction semi-cycle. When the transducer is mechanically pre-stressed, the ceramics are compressed and can develop strains of higher amplitudes [1]. However, the prestressing can yield variations on performance of the transducer, due to shiftings on the characteristic parameters of the ceramic and on the coupling between transducer pieces. The shifting on resonance and antiresonance frequencies is a detected phenomenon in such transducers, and it is studied in this paper. The scienti®c literature covering the mechanical prestressing shows experimental studies of the piezoelectric [2±9] and electrostrictive [7,10,11] ceramics submitted to high stress levels. These works show that mechanical stress can yield domains switching on grains of the ce-
*
Corresponding author. Tel.: +55-19-4407139; fax: +55-194513939. E-mail address: [email protected] (F.J. Arnold).
ramic [12] and, thus, the dipolar polarization is reoriented. The change of physical properties of the ceramics is a consequence of this mechanism and has strong dependence of the composition of the ceramics and the stress level applied. An interesting example of change of piezoelectric properties due to eects of an uniaxial stress [8] has shown that the piezoelectric coecients (d33 and d31 ) of PZT-5A remains constant up to 50 MPa, while ones of PZT-4 have a signi®cant increase in the same range [9]. The goal of this paper is to investigate the shifting on resonance and anti-resonance frequencies related to the mechanical pre-stressing up to 50 MPa. Two hypothesis are considered to explain these eects: the changes on the physical characteristic of the ceramics and the changes on the eective contact surface between transducer parts under mechanical pre-stressing. 2. Theoretical concepts and de®nitions Here, the characterization of the piezoelectric ceramics is limited to measurements and parameters related to the axial direction, and all coecients related to this are denoted by index 3. These parameters [13] contribute to describe the longitudinal vibration mode
0041-624X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 0 ) 0 0 0 4 7 - 0
2
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5
of the transducers used in the experiments. The characterization covers the following parameters: the elastic T S coecient cD 33 , the dielectric coecients e33 and e33 and the piezoelectric coecient d33 . Other vibration modes and their characteristic coecients are neglected. 2 The elastic stiness coecient cD 33 (N/m ) is the ratio between the stress and the strain for an electric displacement equal to zero. The dielectric coecients eT33 and eS33 (F/m) are the ratios between electric displacement and electric ®eld when the ceramic is free and clamped, respectively. The piezoelectric coecient d33 (C/N) is the ratio between the electric displacement and stress.
3. Experimental method The piezoelectric ceramics used in the experiments, produced by THORNTON-INPEC from Vinhedo, SP, Brazil, are annulars shaped, with 6.3 mm of thickness and 12.5 and 38.0 mm of internal and external diameter, respectively. They have silver-coated electrodes on plane surfaces. The ceramics used on experiments were characterized according to IEEE standard [14]. The experimental results are obtained with the construction of some composed transducers, shown in Fig. 1. These transducers are composed of two piezoelectric ceramics; brass electrodes with 0.3 mm of thickness; two pieces of commercial aluminum with the radial dimensions equal to ceramics and a steel bolt that pre-stresses the system when tightened. The dierence between transducers is the length of its aluminum pieces. The pre-stressing value is obtained through the calibration curve resulting from the experimental setup shown on Fig. 2. When the ceramics are compressed, they accumulate electric charges that can be converted into voltage with a storage capacitor, and measured with
Fig. 1. Schematic representation of the transducer.
Fig. 2. Experimental setup used to determine the calibration curve.
a high electronic voltmeter. So, it is possible to obtain a relation between applied force and voltage on capacitor [14], which can be plotted depicting the calibration curve. The piezoelectric coecient d33 is the ratio of the accumulated charges and applied force on ceramics, and can be obtained from slope of this curve. The transducers were assembled and the ceramics were connected to devices 3 and 4 in the setup shown on Fig. 2. When the bolt tightens the assembly, the voltmeter indicates the value that will be converted in to stress throughout calibration curve. After that, devices 3 and 4 were disconnected and the transducer was connected to a fasorial impedance meter (HP4192A) that supplies a 1.0 Vpp sinus excitation. The impedance meter also provides the resonance and anti-resonance frequencies of the transducer. In order to obtain the electric permittivity of the ceramics with the free and clamped conditions, the capacitances were measured with the impedance meter in two frequencies (800 Hz and 800 kHz). The choice of these frequencies is based on solutions of equations of forced oscillator systems [15]. When the transducer is supplied with frequencies smaller than the ®rst resonance (800 Hz), the electric excitation is approximately equal to assembly restoring force, thus, the transducer vibrates free following the external excitation signal. When the transducer is supplied with frequencies above the highest resonance and their principal harmonics (800 kHz), the displacements of the assembly becomes inversely proportional to its mass and the excitation frequency, thus, the resulting strain is approximately null. The values of capacitances in the above cited frequencies allow one to calculate the free and the clamped dielectric constant (eT33 and eS33 , respectively) through e Ct=A, where e is the electric permittivity (F/m), C is the capacitance (F), t is the thickness of the ceramic (m) and A is the transversal section of the ceramic.
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5
3
Fig. 3. Experimental setup for the measuring of propagation velocity as a function of mechanical pre-stressing in multilayer transducer using the pulse-echo method.
The measurement of the elastic coecient shifting was made by the pulse-echo method (Fig. 3). By considering that the aluminum Young modulus of the aluminum remains constant when the material is submitted to compression up to 50 MPa and by measuring the travel time of a pulse in the multi-layered transducer (transmitted pulse plus echo pulse), the wave velocity (v) in the transducer can be determined and the elastic coecient cD 33 derived. Four transducers called T1, T2, T3 and T4 have been assembled. They have aluminum pieces with 13, 26, 33 and 37 mm of thickness, respectively, on each end, with the same ceramics at the center. The resonance and antiresonance frequencies of each transducer were measured under dierent mechanical pre-stressing values. The frequencies were measured at the minimum (resonance) or maximum (anti-resonance) electric impedance modulus.
4. Results The measurements of characteristics of the non-prestressed ceramic obtained are eS33 11 nF/m, cD 33 13:9 1010 N/m2 , and v 4325 m/s. The relation between the applied force and the accumulated charges, when plotted, constitutes the calibration curve whose calculated angular coecient is d33 366 1012 C/N. This curve is used to determine the pre-stressing in the transducers by tightening the central bolt. The capacitance dielectric coecient presents a small increase proportional to applied mechanical pre-stressing, and can be neglected. The elastic and piezoelectric coecients did not present variations with the prestressing. Figs. 4±7 show the curves of resonance and anti-resonance frequencies as a function of mechanical prestressing obtained by proposed method. The upper and
Fig. 4. Experimental results of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T1 (aluminum piece thickness 13 mm).
Fig. 5. Experimental results of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T2 (aluminum piece thickness 26 mm).
lower curves represent the anti-resonance and the resonance frequency, respectively.
5. Discussion and conclusions The results obtained in the measurement show that mechanical pre-stressing (up to 50 MPa) does not yield remarkable changes on ceramic physical characteristics. The dielectric coecient showed a small increase related to pre-stressing. By replacing this small increase in a model of thickness mode of piezoelectric disc [14], the observed variation of resonance frequency is neglected. The pulse-echo method did not show changes in the
4
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5
Fig. 6. Experimental results of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T3 (aluminum piece thickness 33 mm).
ance media is similar to that obtained with shifting the cross-section along the propagation axis. In a multilayered medium is observed shifting on resonance frequency when the mechanical impedance of some layer is altered [16,17]. The experiments show that the characteristic physical of the transducer pieces are not changed when the mechanical pre-stressing is applied, thus, the mechanical impedance is not altered and the frequencies should not vary. The frequency variations observed shall be caused by dierence between eective cross-sections of the ceramic and the aluminum pieces. This means that each level of pre-stressing de®nes an eective matching surface and, thus, the behavior of the waves in the system can be considered similar to a wave propagation in a duct with changes in the cross-section along its way. When the mechanical pre-stressing is lower than 30 MPa, the contact acoustic between pieces is not perfect, i.e., the cross-sections of the ceramics and the aluminum parts on their interfaces are not equal. The tightness of the bolt yields higher compression in the central part of these pieces (near to hole) than on the periphery, thus, the wave transmission is concentrated in this region, because the minimum compression level to the eective matching is reached. When the pre-stressing increases, the eective matching region is expanded, so that when the pre-stressing is higher than 30 MPa, the whole interface is in eective matching. The experimental results allow one to conclude that the mechanical pre-stressing improves the acoustical contact between the transducer pieces on application range up to 50 MPa and does not alter the physical parameters of ceramics and metallic pieces. References
Fig. 7. Experimental reslts of resonance (lower curve) and anti-resonance (upper curve) frequencies versus mechanical pre-stressing of the transducer T4 (aluminum piece thickness 37 mm).
propagation velocity of the multilayered transducer related to the pre-stressing and, thus, the elastic coecient also did not change. The piezoelectric coecient did not shift either, the calibration curve shows that the voltage and the stress are linearly related. The experimental results show that the resonance and anti-resonance frequencies vary less on pre-stressing range between 30 and 50 MPa. The transmission coecient [15] between two or more media is related to energy ¯ux in the system. When the wavelength is greater than the diameter of the transducer pieces (this is the case of the transducer here), the expression of this coecient through dierent imped-
[1] S. van Randeraat, R. Setterington, Piezoelectric Ceramics, Philips application book, second edn., Mullard, London, 1974. [2] R. Nishi, Eects of one-dimensional pressure on the properties of several transducer ceramics, J. Acoust. Soc. Am. 40 (2) (1966) 486. [3] H. Krueger, Stress sensitivity of piezoelectric ceramics: part 1. Sensitivity to compressive stress parallel to the polar axis, J. Acoust. Soc. Am. 42 (3) (1967) 636. [4] H. Krueger, Stress sensitivity of piezoelectric ceramics: part 3. Sensitivity to compressive stress perpendicular to the polar axis, J. Acoust. Soc. Am. 43 (3) (1968) 583. [5] S. Meeks, R. Timme, Eects of one-dimensional stress on piezoelectric ceramics, J. Appl. Phys. 46 (10) (1975) 4334. [6] A. Tims, Eects of multidimensional stress on radially polarized piezoelectric ceramic tubes, J. Acoust. Soc. Am. 70 (1) (1981) 21. [7] H. Cao, A.G. Evans, Nonlinear deformation of ferroelectric ceramics, J. Am. Cer. Soc. 76 (4) (1993) 890. [8] Q.M. Zhang, J. Zhao, K. Uchino, J. Zheng, Change of the weak®eld properties of Pb(ZrTi)O3 piezoceramics with compressive uniaxial stresses and its links to the eect of dopants on the stability of the polarizations in the materials, J. Mat. Res. 12 (1) (1997) 226. [9] Q.M. Zhang, J. Zhao, Electromechanical properties of lead zirconate titanate piezoceramics underthe in¯uence of mechanical
F.J. Arnold, S.S. M uhlen / Ultrasonics 39 (2001) 1±5 stresses, IEEE Trans. Ultrason. Ferroelec. Freq. Control. 46 (6) (1999) 1518. [10] Y. Nakajima, T. Hayashi, I. Haysai, K. Uchino, Eletrostrictive properties of a PMN stacked actuator, Jpn. J. Appl. Phys. 24 (2) (1985) 235. [11] C.L. Hom, S.M. Pilgrim, N. Shankar, K. Bridger, M. Massuda na, S.R. Winzer, Calculation of quasi-static electromechanical coupling coecients for electrostrictive ceramic materials, IEEE Trans. Ultrason. Ferroelec. Freq. Control. 41 (4) (1994) 542. [12] G. Arlt, Twinning in ferroelectric and ferroelastic ceramics: stress relief, J. Mat. Res. 25 (1990) 2655.
5
[13] D. Berlincourt, D. Curran, H. Jae, Piezoelectric and piezomagnetic materials and their function in transducers, Phys. Acoust. 1(A) (1964) 170. [14] ANSI/IEEE, Standard on Piezoelectricity, 1978, p. 176. [15] A. Kinsler, L. Frey, Fundamentals of Acoustics, Wiley, New York, 1962. [16] R. Hill, S. El Dardiry, A theory for optimization in the use of acoustic emission transducers, J. Acoust. Soc. Am. 67 (2) (1980) 673. [17] R. Hill, S. El-Dardiry, Variables in the use and design of acoustic emission transducers, Ultrasonics 19 (1981) 9.