Proceedings of Meetings on Acoustics

Transcription

1 Proceedings of Meetings on Acoustics Volume 19, ICA 213 Montreal Montreal, Canada 2-7 June 213 Engineering Acoustics Session 3aEA: Computational Methods in Transducer Design, Modeling, Simulation, and Optimization II 3aEA3. Investigation of vibrations of piezoelectric spherical shells with axisymmetric holes David Brown* and Colton T. Brown *Corresponding author's address: Advanced Technology and Manufacturing Center, BTech Acoustics LLC, University of Massachusetts Dartmouth, Fall River, Massachusetts 2723, An experimental investigation of the vibration of radially polarized thin-walled piezoelectric ceramic spherical shells with axisymmetric holes is presented. Piezoelectric spherical electroacoustic transducers having holes at their poles to permit passage of wires, cables, or structural members is of interest in underwater acoustics. The coupled vibrations follow three resonance branches corresponding to () azimuthal extensional modes, (1) bending flexural modes, and (2) meridional extensional modes. The resonances and effective electromechanical coupling coefficient for each mode as a function of the hole-to-sphere diameter ratio have been determined from admittance measurements. In the limit that the hole-to-sphere diameter ratio approaches zero, the () mode is dominant corresponding to the spherically symmetric breathing mode with coupling coefficient of.546, which is consistent with the planar coupling coefficient for PZT-4 (Type I) material. In the limit that ratio approaches unity, the element in nearly cylindrical and the lowest () mode corresponds to the breathing mode of a ring having a coupling coefficient of about.33, which is consistent with the transverse (31) coupling coefficient for the material. The focus is on determining the resonances and electromechanical coupling in the intermediate region and obtaining corresponding vibration mode shapes with a non-contact optical fiber displacement sensor. Published by the Acoustical Society of America through the American Institute of Physics 213 Acoustical Society of America [DOI: / ] Received 22 Jan 213; published 2 Jun 213 Proceedings of Meetings on Acoustics, Vol. 19, 365 (213) Page 1

2 INTRODUCTION Piezoelectric spherical transducers are common as underwater sound projectors and receivers and their operation has been covered in detail in the literature [1,2]. Generally spherical transducers have desirable omnidirectional radiation characteristics over a broad frequency range. Thus the resonance frequency and effective electromechanical coupling factor are important characteristics in determining their frequency response In some implementations it becomes necessary to drill or grind a hole (or holes) in the polar cap(s) of the piezoelectric shell. This may be needed to pass wires, cables, structural members or to reduce the frequency of the element. It then important to know how the resonance frequency and effective electromechanical coupling factor change as a function of the hole diameter in relation to diameter of the piezoelectric sphere. This paper summarizes these experimental data. It is also desirable to obtain the vibration mode shapes, which makes it is possible to calculate the properties of the piezoelectric element using the energy method and synthesis results in the form of an equivalent electrical circuit for predicting transducer and subsystem performance as outlined in Ref. 3. FIGURE 1. Illustration of a spherical shell with holes in the polar caps. In [Ref. 4.] APPROACH Electrical impedance data was obtained on piezoelectric spherical shells with holes of increasing size by grinding successively larger holes in the shell and obtaining resonance and antiresonance frequency data using an HP4192 impedance analyzer. Vibration mode shapes were measured point-by-point along a meridian using a non-contact optical displacement sensor. SUMMARY OF TEST DATA ON SPHERICAL SHELLS WITH POLAR HOLES A summary of the test data follows. Fig. 2 presents the resonance frequency data for a sphere with a mean diameter of 35mm and thickness of 1.9mm. There are three resonance branches that trend. The branch is the extensional breathing mode. The branch 2 belongs to the meridional extensional vibrations when the shell has been opened. Branch 1 are the resonance frequencies associated with the lowest order flexural vibrations in the shell wall. The resonances are coupled and in general are analogous to the coupled vibrations seen in piezoelectric hollow cylinders [5,6]. Also note that the strength of the resonance data is shown by the size of the data marker used and are proportional to the effective electromechanical coupling coefficient, which is shown separately for each branch in Fig. 3. The solid trend line in Fig. 2 corresponding to the breathing mode and has a coupling coefficient of.546 in the limit when the hole size vanishes; this agrees well with the expected planar coupling coefficient for a piezoelectric shell made of PZT-4 Navy Type I. Mode shape data will be presented at the meeting. The coupling coefficient is calculated using the resonance-antiresonance method according to the formula, k eff = [1-(f r /f a ) 2 ]. Proceedings of Meetings on Acoustics, Vol. 19, 365 (213) Page 2

3 Resonance Frequency (khz) Hole Diameter (mm) FIGURE. 2. Resonance frequencies of a piezoelectric open-sphere as a function of hole diameter as obtained by admittance data. The spherical shell is comprised of Navy Type I (PZT-4); mean diameter 2a=35mm; thickness 1.9mm. Holes of equal diameter are located each pole. Three branches are shown as best-fit trend-lines to the corresponding experimental data. The data for Branch are indicated by open circles whose diameters are in proportion to the strength of the resonance or equivalently the corresponding effective coupling coefficient where data are shown separately in Fig. 2. Branch corresponds to the gravest extensional (or radial breathing) mode of the spherical shell and has resonance frequency of 48.5 khz for the complete sphere. The resonance asymptotically approaches the frequency for a short ring of 28. khz which occurs as the hole diameter approaches 35mm corresponding to thin equatorially belt. The data for Branch 2 are indicated by diamonds whose size is also in proportion to the corresponding effective coupling coefficient. Branch 2 corresponds to extensional vibrations along the meridian. The data for Branch 1 are indicated by triangles and correspond to a bending mode. There are additional weak resonances (effective coupling coefficients below about.1) not shown but observable in the admittance data and that correspond to higher order bending and extensional modes. Proceedings of Meetings on Acoustics, Vol. 19, 365 (213) Page 3

4 2 1 FIGURE. 3. Effective coupling coefficient for resonance branches of a piezoelectric open-sphere as a function of hole diameter obtained using resonance-antiresonance data method. The spherical shell is comprised of Navy Type I (PZT-4); mean diameter 2a=35mm; thickness 1.9mm. The effective coupling coefficient of three branches are shown as best-fit trend-lines to the corresponding experimental data. The effective coupling coefficient data for Branch are indicated by open circles and is bounded by the planar k eff value of.55 for a sphere and the 31-mode k eff value of.33 for a short ring; for Branch 2 by diamonds ; for Branch 1 by triangles. 4 2 Ym&Yp_mode_19.5mm_35-45kHz.dat 1 Modulus of Y [ms] Phase of Y [deg] Frequency [khz] FIGURE 4. Admittance Data (Magnitude and Phase) plot for the spherical piezoelectric shell of mean diameter 2a=35mm, thickness 1.9mm, having holes at each pole of 19.5mm diameter. At this hole-to-sphere diameter (19.5/35), the resonance of the branch and 1 branch are close and strongly coupled. Note two strong resonances, at a frequency of 38.1kHz (with anti-resonance of 38.9kHz) for the branch and the resonance with frequency of about 4.kHz (with anti-resonance of 41.8kHz) for the 1 branch (flexural). Proceedings of Meetings on Acoustics, Vol. 19, 365 (213) Page 4

5 Ym&Yp_mode_21.3mm_35-45kHz.dat 1 5 Modulus of Y [ms] Phase of Y [deg] Frequency [khz] FIGURE 5. Admittance Data (Magnitude and Phase) plot for the spherical piezoelectric shell having holes at each pole of 21.3 mm diameter. At this hole-to-sphere diameter (21.3/35), the resonance of the branch and 1branch have further separated and are less coupled. The resonance for the branch is at about 37.2 khz (with anti-resonance of 39.4 khz) and resonance of the 1 branch (flexural) is at about 4.15kHz (with anti-resonance of 42.5 khz). 25 Radial Displacment (arb. units) Angle (deg) 3mm (9 ) 8.3mm (25 ) 11mm (34 ) 15.3mm (47 ) FIGURE 6. Mode shapes data as function of hole opening angle. Radial displacement [normal to the shell surface] vs meridianol angle for opening diameters of 3, 8.3, 11, and 15.3mm corresponding to angles. 9.; 15.1; 25.2; 33.6; Proceedings of Meetings on Acoustics, Vol. 19, 365 (213) Page 5

6 SUMMARY REMARKS The resonance frequencies, electromechanical coupling coefficients, and representative modes shapes were taken on a piezoelectric spherical shell with axisymmetric holes located at each pole. The data is very useful for the design of transducers employing such elements and can be generalized for hole diameter to spherical shell diameter. The results for the branch and 2 branch are expected to be relatively independent on thickness, at least for thin shells, while the same can not be said of the resonance of the 1 branch, which are flexural in nature. As can be expected, the experimental results on spherical shells is similar to recent results obtained by the author (DAB) and collaborators on piezoceramic cylinders (or tubes ) as explained in Ref. 5 and 6. A complimentary theoretical analysis of the vibration mode shapes, resonance frequencies and coupling coefficients on open spherical shells with axisymmetric holes on the poles has recently been undertaken but is beyond the scope of this proceedings paper, which precedes the conference by about 5 months. These results may be compared with experimental results at the meeting. ACKNOWLEDGMENTS This work was funded by BTech Acoustics LLC. The author (DAB) is also with the Advanced Technology and Manufacturing Center (ATMC) and the Electrical and Computer Engineering (ECE) Dept. at the University of Massachusetts Dartmouth. Inquires to dbacoustics@cox.net. The author (CTB) is an undergraduate Physics and Computer Science student from Duke University working at the time as a summer intern at BTech Acoustics and the ATMC, University of Massachusetts Dartmouth. REFERENCES 1. Boris Aronov, David A. Brown, Corey Bachand, Xiang Yan, Analysis of unidirectional broadband piezoelectric spherical shell transducers for underwater acoustics, J. Acoust. Soc. Am. 131(3), (212). 2. Boris Aronov, David A. Brown, Xiang Yan, and Corey L. Bachand Modal analysis of the electromechanical conversion in piezoelectric ceramic spherical shells,, J. Acoust. Soc. Am. 13(2), (211). 3. Boris Aronov, B. S. Aronov, The energy method for analyzing the piezoelectric electroacoustic transducers, J. Acoust. Soc. Am., 117 (1), 21-22, (25). 4. Photo taken from the analysis of heat conduction in spherical shell with holes in chemical-engineering/transport-phenomena/heat-conduction-in-spherical-shell-fig.-11b.4-spherical (213). 5. Boris Aronov, David A. Brown, and Sundar Regmi, Experimental investigation of coupled vibrations in piezoelectric cylindrical shells,, J. Acoust. Soc. Am. 12(3), (26). 6. Boris Aronov, Coupled vibration analysis of the thin-walled cylindrical piezoelectric ceramic transducers, J. Acoust. Soc. Am. 125(2), (29). Proceedings of Meetings on Acoustics, Vol. 19, 365 (213) Page 6