A new design on broadband flextensional transducer

Applied Acoustics 72 (2011) 836–840

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A new design on broadband flextensional transducer Yaozong Pan ⇑, Xiping Mo, Yong Chai, Yongping Liu, Zheng Cui Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, PR China

a r t i c l e

i n f o

Article history: Received 26 February 2010 Received in revised form 27 January 2011 Accepted 10 May 2011

Keywords: Flextensional transducer Piezoelectric Free-flooded Cavity Transmitting voltage response

a b s t r a c t The new flextensional transducer presented is driven by a tube stacked by longitudinally polarized piezoelectric ceramic rings. The rings are compressed between two annulus steel end plates which are coupled by a dual convex aluminum shell with slotted gaps. The transducer is a free-flooded design with the interior of the tube open to the surrounding water. Three main vibrating modes including the cavity, the longitudinal and the radial can be utilized by appropriately coupling design to broaden the working bandwidth. A prototype is fabricated and measured. The results confirm the three vibrating modes mentioned above and the broad band of transmitting voltage response is gained successfully with difference less than 10 dB from 2200 Hz to 9000 Hz. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction

2. Design and analysis

Flextensional transducers are common underwater electroacoustic transducers, which convert small longitudinal displacements of a driver into relatively large flexural displacements of a shell [1–3]. Barrel-stave flextensional transducer is one popular type which is usually an air-backed design. But it can also be designed to be free-flooded with the interior of the driving stack being open to the surrounding medium mentioned by McMahon in 1990 [4]. Bayliss Clive developed another free-flooded design using a compliant material to partially fill the interior of the transducer in 1998 [5]. Flextensional transducer works mainly at frequency near the resonance of the shell. There is usually only one resonant mode of the transducer. This paper presents a broadband free-flooded transducer with three vibrating modes-the cavity, the longitudinal and the radial. The structure of the transducer is similar to that of the barrel-stave transducer except that the piezoelectric ceramic disks are replaced by piezoelectric ceramic rings. The cavity resonant frequency and equivalent electrical circuit are analyzed. The three modes are calculated with Finite Element Method. A prototype is developed and tested. The measured resonant frequencies of the modes show good agreement with that of the calculated. The transmitting voltage response (TVR) is broadband.

Broadband transducer has many advantages in signal processing and underwater application. Multi-mode coupling is one approach to broaden the bandwidth of transducers. Butler presented a broadband transducer with triple resonant design [6]. Jones developed a broadband barrel-stave flextensional transducer through coupling of the fundamental flexural and longitudinal vibration modes [7]. As mentioned above, barrel-stave flextensional transducer could be used as a free-flooded type when the driving element was replaced by piezoelectric ceramic rings. It is known that in open tubes there exist two resonances including cavity resonance and ring resonance [8]. When substituting the driving element for a tube stacked by piezoelectric ceramic rings and letting the tube open to the surrounding medium, barrel-stave flextensional transducer could introduce several vibration modes. The cylindrical water volume enclosed by the tube introduces a cavity resonance; the tube of piezoelectric ceramic rings can also excite a longitudinal vibration mode of the transducer when the end plates and the beams of the shell vibrate. Besides, the wall of the tube exhibits a radial vibration mode. A free-flooded flextensional transducer was then designed to broaden the working bandwidth with proper coupling of these three vibration modes. The structure of the designed flextensional transducer is shown in Fig. 1. It consists of two annulus steel end plates, a tube stacked by piezoelectric ceramic rings as the driving element compressed between the end plates and a dual slotted convex aluminum shell coupled one end plate to the other. The structure of the transducer is similar to that of the barrel-stave flextensional transducer except that the piezoelectric ceramic disks are replaced by piezoelectric

⇑ Corresponding author. Tel.: +86 10 82547682. E-mail address: [email protected] (Y. Pan). 0003-682X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2011.05.007

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Fig. 2. Behavior of the cavity resonant frequency as a function of h/a in water.

Fig. 1. The vertical section of the transducer.

ceramic rings. A sealing boot is covered on both inner and outer surface so that the water could freely flood through the ring stack. The piezoelectric ceramic ring stack is made up of a dozen axially polarized PZT-4 rings with outer diameters of 142 mm, inner diameters of 122 mm and thicknesses of 12 mm. They are bonded together by epoxy and connected electrically in parallel. The entire aluminum shell with one middle ring, two end rings and two convex parts is machined by a numerically controlled lathe. The two convex parts are equally divided into 12 pieces of beams along the circumferential direction with gap width of 4 mm. The curvature radius of the beams is 150 mm with an average wall thickness of 7 mm. The annulus steel end plates are fastened to the shell and the piezoelectric ceramic ring stack is compressed between the two end plates. The transducer has a full size (excluding the boot) of 206 mm in height, 122 mm in inner diameter and 202 mm in outer diameter.

Fig. 3. Sketch of piezoelectric ceramic ring.

2.1. Cavity resonant frequency The behavior of the cavity enclosed by cylindrical wall has been studied by McMahon [8]. The first cavity mode can be obtained by

Xðh=2a þ 0:633Þ  0:106X2 ¼ p=2

ð1Þ

where the dimensionless frequency parameter X equals xca/c0, xc is the angular frequency, c0 is the velocity of sound in the enclosed water column, h is the height of the cylinder, and a is the inner radius. Because of the finite stiffness of the stack wall, the velocity of sound c0 in the enclosed water column is less than the velocity of sound c in the open water, that is

c ¼ c0 ð1 þ 2Ba=Y 11 tÞ1=2

ð2Þ

where B is the bulk modulus of the water, Y11 is the transverse Young’s modulus of the piezoelectric ceramic at constant electric field and t is the wall thickness of the stack. Then the dimensionless frequency parameter for the cavity mode becomes Xc = xca/c. The behavior of the cavity resonant frequency is calculated as a function of h/a from Eqs. (1) and (2) which is shown in Fig. 2. Substituted the values of h and a, the cavity resonant frequency can be estimated at 2220 Hz. 2.2. Equivalent circuit of the transducer The 2D equivalent circuit of piezoelectric ceramic ring with coupled vibration has been studied by Lin [9,10] and Feng et al. [11]. A sketch of the piezoelectric ceramic ring polarized along z axis is shown in Fig. 3. The height of the ring is l, with inner and outer

radiuses a and b. n1, n2, n3, n4 represent the displacement amplitudes of the bottom and top surfaces and the inner and outer surfaces. F1, F2, F3, F4 represent the force amplitudes of the bottom and top surfaces and the inner and outer surfaces. The equivalent circuit of the piezoelectric ceramic ring with coupled vibration has been derived shown in Fig. 4a

!   ZT n2z ðU 1 þ U 2 Þ þ Z c U 03 þ U 04 F1 ¼  j sinðkz lÞ jxC 20   kz l U 1 þ nz V þ jZ T tg 2 !   ZT n2z F2 ¼  ðU 1 þ U 2 Þ þ Z c U 03 þ U 04 2 j sinðkz lÞ jxC 0   kz l U 2 þ nz V þ jZ T tg 2 F 03 ¼ ðZ a þ Z d ÞU 03 þ Z d U 04 þ Z c ðU 1 þ U 2 Þ þ nr V F 04 ¼ ðZ a þ Z d ÞU 04 þ Z d U 03 þ Z c ðU 1 þ U 2 Þ þ nr V   I ¼ jxC 0 V  nz ðU 1 þ U 2 Þ  nr U 03 þ U 04 The detailed description of the constants and variables in the equations above are defined as follows:

U 1 ¼ n_ 1 ;

U 2 ¼ n_ 2 ;

U 03 ¼ an_ 3 ;

U 04 ¼ bn_ 4 ;

F 03 ¼

F3 ; a

F 04 ¼

F4 b

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Y. Pan et al. / Applied Acoustics 72 (2011) 836–840

Fig. 4. 2D equivalent circuits of (a) piezoelectric ceramic ring, and (b) transducer.

pðb2  a2 ÞeS33

C0 ¼

l 2

;

nz ¼

Z T ¼ pðb  a Þqv z ; 2pcE13 Zc ¼ ; jx

2

pðb2  a2 Þe33

v

2 z

2cE11 l Zd ¼ ; jxabD

¼

l cD33

;

Table 1 Properties of the transducer materials.

nr ¼ 2pe31 ;

q

D ¼ J1 ðkr aÞY 1 ðkr bÞ  J1 ðkr bÞY 1 ðkr aÞ

Za ¼ j

 2plqv r 2p l  E ½Y 1 ðkr bÞJ 0 ðkr aÞ  J1 ðkr bÞY 0 ðkr aÞ  c  cE12  Z d jxa2 11 aD

Zb ¼ j

2plqv r ½Y 1 ðkr aÞJ 0 ðkr bÞ  J1 ðkr aÞY 0 ðkr bÞ  cE11  cE12  Z d 2 bD jxb 2pl 



In these equations, cEij is the elastic stiffness constants measured at constant electric field, eij is the piezoelectric stress constants, and eS33 is the dielectric constant measured at constant stress. The transducer driver consists of n piezoelectric ceramic rings connected electrically in parallel. Assuming the equivalent mechanical impedances of the shell is Zs and the equivalent radiation impedances of the inner and outer surfaces are Zr1 and Zr2. The 2D equivalent circuit of the transducer can be obtained which is shown in Fig. 4b, in which ke is the equivalent wave number. From the equivalent circuit of the transducer, the electrical input admittance is given as follows:

  nz nnr M þ N  nn þ Zc n I r z Y ¼ ¼ jxnC 0  2 V MN  Z c

ð3Þ

where

M¼

  ZT 1 nke l þ Z s þ Z r2 þ j Z T tg 2 2 j sinðnke lÞ

N ¼ nZ d þ nZ b ==ðnZ a þ Z r1 Þ

ð4Þ

PZT-4

Aluminum

Steel

Density (kg/m3) Young’s module (1010 N/m) Poison ratio Permittivity (109 F/m) Piezoelectric matrix (C/m2) Elasticity matrix (1010 N/m2)

7500

2790 6.85

7840 21.6

0.34

0.28

e11 = 6.46, e33 = 5.62 e13 = 5.2, e33 = 15.1, e52 = 12.7 c11 = 13.9, c12 = 7.78, c13 = 7.43, c33 = 11.5, c44 = 3.06, c66 = 2.56

branch, when Im(M) = 0 it is called longitudinal vibrating mode; and as to the radial vibration branch, it is called radial vibrating mode when Im(N) = 0. Zr1 is the complex impedance of cylindrical cavity enclosed by the inner surface of the tube, which consists of three components: resistance, acoustical mass reactance of the cavity and acoustical compliance reactance of the cavity [12,13]. At the frequency when the acoustical mass reactance and acoustical compliance reactance of the cavity resonates, Zr1 will appear a minimum and the electrical input admittance Y will appear a maximum, and it is called the cavity mode. However, for the complicated structure and vibration of the transducer, it is difficult to determine the analytic expressions of the equivalent impedances of Zs, Zr1 and Zr2. Therefore, it is hard to find resonant frequency equation with an analytic method. Alternatively, Finite Element Method provides an effective solution of computing complicated structure vibration and fluid–structure interaction problem. 2.3. Finite element analysis

ð5Þ

Here the symbol of ‘‘//’’ is an operator which means computing he impedance of components connected in parallel. Eq. (4) is the impedance of the longitudinal vibration branch, and Eq. (5) is the impedance of the radial vibration branch. From Eq. (3), when the frequencies satisfy the equation:

  Im MN  Z 2c ¼ ImðMNÞ ¼ 0

Material properties

ð6Þ

the electrical input admittance Y will appear a maximum, and we call it ‘‘a vibrating mode’’, where ‘‘Im’’ is the operator of the imaginary part of a complex number. As to the longitudinal vibration

Finite Element Method is used to calculate the resonant frequencies of the three modes. The properties of the transducer materials are listed in Table 1. The resonant frequencies of the three modes both in air and water are listed in Table 2. Table 2 Computed resonant frequencies of the transducer. Modes Cavity Longitudinal Radial

Frequencies in air/Hz

Frequencies in water/Hz

6664 8822

2200 6100 7900

Y. Pan et al. / Applied Acoustics 72 (2011) 836–840

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Fig. 5. Prototype of the transducer. (a) components, (b) without boot, and (c) with boot.

Fig. 8. Measured TVR (Ref. 1 l PaV1 at 1 m) at 90° direction.

Fig. 6. Experimental setup for transmitting voltage response.

Fig. 7. Measured admittance in water.

In order to expand the working bandwidth, these resonances should be arranged reasonably. The cavity mode works at lower frequency, followed by the longitudinal vibration mode, and then the radial vibration mode of the tube at higher frequency.

on one end of a 2 m steel bar together with the standard hydrophone on the other end. The steel bar was positioned horizontally under 10 m lake water. The transmitting voltage response (TVR) was measured using free field calibration method. The experimental setup is shown in Fig. 6. The devices used in the experiment are Function/Arbitrary Waveform Generator with type of Agilent 33120A, Power Amplifier with type of INSTRUMENT Model L10, Dual Channel Programmable Filter with type of NF 3628 and Oscilloscope with type of Tektronix TSD2024. The admittance was measured by HP 4192A Impedance Analyzer. And the measured admittance and TVR are shown in Figs. 7 and 8. From these figures it can be seen that there are three resonances, which are the cavity, the longitudinal and the radial with the corresponding frequencies of 2400 Hz, 5600 Hz and 7600 Hz. The measured cavity resonant frequency is about 8.3% higher than computation. That is probably caused by the reduced inner radius of the cylindrical cavity after covering polyurethane boot. Similarly, the mass of the polyurethane on the ends of the transducer makes the measured longitudinal resonant frequency about 8.2% lower than computation. The transmitting voltage response (TVR) at the cavity resonant frequency of 2400 Hz is 124.3 dB. In the frequency range from 2200 Hz to 9000 Hz, the TVR fluctuates less than 10 dB, with the minimum TVR 123.3 dB at about 3100 Hz and the maximum TVR 132.5 dB at about 5800 Hz near the longitudinal resonance. The peak value of TVR at the cavity resonance frequency is not very clear, probably because the vibration volume displacement of the cavity mode is not large enough.

4. Summary 3. Results and discussion A prototype of the transducer is fabricated as shown in Fig. 5. After covering a boot with polyurethane, the transducer was fixed

The new design of flextensional transducer consists of a tube stacked by axially polarized piezoelectric ceramic rings as the driven element, two annulus steel end plates and a dual convex

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aluminum shell. By reasonably arranging the cavity, the longitudinal and the radial resonances, the broadband working performance is enhanced. The transmitting voltage response (TVR) of the transducer fluctuates less than 10 dB from 2200 Hz to 9000 Hz. Acknowledgements This work was supported by grants from the National Natural Science Foundation of China No. 11074276. The authors are grateful to the editors and the reviewers for their valuable comments. References [1] Rolt KD. History of the flextensional electroacoustics transducer. J Acoust Soc Am 1990;87(3):1340–7. [2] Merchant HC. Underwater transducer apparatus. US patent: 3258738, June; 1966. [3] Hayes HC. Sound generating and directing apparatus. US patent: 2064911, December; 1936.

[4] McMahon GW, Jones DF. Barrel stave projector. US patent: 4922470, May; 1990. [5] Bayliss C. Application and development of finite element techniques for transducer design and analysis. Birmingham: University of Birmingham; 1998. [6] Butler Stephen C. High frequency multi-resonant broadband transducer development at NUWC. Undersea Defence Technol 2002. 6A-2. [7] Jones DF, Chirstopher DA. A broadband omnidirectional barrel-stave flextensioanl transducer. J Acoust Soc Am 1999;106(2):L13–7. [8] McMahon GW. Performance of open ferroelectric ceramic cylinders in underwater transducers. J Acoust Soc Am 1964;36(3):528–33. [9] Lin S. The radial composite piezoelectric ceramic transducer. Sensor Actuat A 2008;141(1):136–43. [10] Lin S. Study on the equivalent circuit and coupled vibration for the longitudinally polarized piezoelectric ceramic hollow cylinders. J Sound Vib 2004;275(3–5):859–75. [11] Feng F, Shen J, Deng J. A 2D equivalent circuit of piezoelectric ceramic ring for transducer design. Ultrasonics 2006;44(Suppl. 1):e723–6. [12] Henriquez TA, Young AM. The Helmholtz resonator as a high-power deepsubmergence source for frequencies below 500 Hz. J Acoust Soc Am 1980;67(5):1555–8. [13] Woollett RS. Underwater helmholtz – resonator transducers: general design principles. Naval underwater systems centre technical report number 5633; 1977.