薄壁圆管换能器的耦合振动分析

Coupled vibration analysis of the thin-walled cylindrical rs ance ivers acc g ax s pre 0–22 5 �1 s is the con nd t The pol alid ly a of c 21/1 0.E built from the thin-walled short rings, for which the assump- mathematical treatment of the problem and the specific as- tion of one-dimensional nature of vibrations in the circum- ferential direction is valid. With increasing of the height-to- diameter aspect ratio of the comprising piezoelements the one-dimensional approximation fails, and vibrations of the cylindrical piezoelements have to be considered as two- dimensional coupled vibrations in the circumferential and axial directions. The first treatment of vibration of piezoelectric ceramic thickness poled shells in the two-dimensional approximation was carried out by Haskins and Walsh1 in the framework of the “membrane” theory of shells following the treatment of vibration of the passive thin isotropic elastic tubes described by Love.2 Afterwards a number of papers were published, related to different aspects of the problem of coupled vibra- tions in piezoceramic tubes.3–9 Only a few of these references4,9 considered calculation of properties of under- water electroacoustic transducers, made from the tubes, for which the analysis of the coupled vibrations was applied. The common feature of all of the references is that they treat the problem by means of solving the partial differential equations of motion of the tubes as piezoelectric elastic me- dium under certain electrical and mechanical boundary con- ditions. In the case that the tubes are treated as electroacous- sumptions made regarding the thickness of the tubes, namely, whether the tubes are considered as thin-walled �the “mem- brane theory” approximation�3–5 or as finite thickness.6–8 Be- cause of complexity of the problem, solutions in all of the referenced cases are obtained by numerical computation, and the most of the papers are concerned with improving of the mathematical models rather than with revealing the trends in the electromechanical and electroacoustic characteristics of transducer behavior as functions of aspect ratio of the com- prising piezoceramic tubes. The dependence of parameters of the thickness poled air-backed cylindrical transducers from aspect ratios were considered by means of experimental stud- ies and discussed in two recent papers.10,11 However, a reasonably accurate theoretical prediction of performance of the transducers, employing vibrations of variously poled tubes mechanically and acoustically loaded as a function of their aspect ratio, has not been previously presented. The objective of this paper is to establish an analytical solution to the coupled electromechanical and mechanoa- coustical problem that extends the physical insight and pro- vides a foundation for calculating the cylindrical electroa- coustic transducer parameters for various polarization states and aspect ratios of the comprising piezoceramic tubes. The solution is based on employing the energy method12 and in-a�Electronic mail: baronov@comcast.net piezoelectric ceramic transduce Boris Aronova� BTech Acoustic LLC, Acoustics Research Laboratory, Adv Department of Electrical and Computer Engineering, Un 151 Martine Street, Fall River, Massachusetts 02723 �Received 21 June 2008; revised 1 December 2008; Analysis of electromechanical transducers employin ceramic thin-walled tubes of arbitrary aspect ratio i method �B. S. Aronov, J. Acoust. Soc. Am. 117, 21 and Blechshmidt �Ann. Physik, Ser. 5 18, 417–48 vibration as a coupled vibration of two partial system piezoceramic tube vibration, and the energy of extensional vibration of a tube is also taken into describing the transducer vibrations are derived, a introduced that conveniently represents the solution. coefficients, and velocity distributions for differently height-to-diameter aspect ratio are calculated. The v the case that a piezoceramic tube is mechanical acoustically loaded on its outer surface. The results results of experimental investigations. © 2009 Acoustical Society of America. �DOI: 10.11 PACS number�s�: 43.38.Ar, 43.38.Fx, 43.40.At, 43.4 I. INTRODUCTION The cylindrical piezoelectric ceramic transducers are widely used for underwater applications. Calculation of their parameters is well known in the case that the transducers are J. Acoust. Soc. Am. 125 �2�, February 2009 0001-4966/2009/125�2 Downloaded 16 Nov 2011 to 166.111.132.53. Redistribution subject to ASA licens d Technology and Manufacturing Center, and ity of Massachusetts Dartmouth, epted 4 December 2008� ially symmetric vibrations of piezoelectric sented based on application of the energy 0 �2005��. The suggestion made by Giebe 933�� regarding representation of a tube used to choose the assumed modes of the flexural deformations accompanying the sideration. The Lagrange type equations he equivalent electromechanical circuit is resonance frequencies, effective coupling ed piezoceramic tubes as functions of their ity of the equivalent circuit is extended to nd acoustically loaded on the ends and alculations are in good agreement with the .3056560� y �AJZ� Pages: 803–818 tic transducers, the boundary conditions on their radiating surface involve a reaction of acoustical loading, and the problem becomes a rather complicated coupled piezoelectric-elastic and coupled elastoacoustic problem. The main differences between the cited references consist in the © 2009 Acoustical Society of America 803�/803/16/$25.00 e or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp volves application of equivalent electromechanical circuits for modeling the transducers. The method permits separation 2a t of the radiation problem from the treatment of a transducer as an electromechanical system and thus simplifies the over- all procedure for obtaining and interpreting a solution. In the application of the energy method it is crucial to appropriately choose the assumed modes of vibration for the mechanical system of a transducer. In the current paper vibration of a thin-walled tube is considered as the dynamical interaction of two coupled partial mechanical systems, namely, of a thin ring undergoing radial vibrations and of a thin longitudinally vibrating bar, which was first proposed by Giebe and Blechshmidt.13 This is an alternative to the partial differential equations of motion treatment of the problem, which was introduced by Love.2 The Giebe and Blechshmidt’s solution13 was derived in the form of Lagrange’s equations and gave exactly the same results for the resonance frequen- cies of the finite length elastic tubes as obtained by Love’s approach.2 A common prediction of both of these analyses was the existence of a so called “dead zone,” i.e., some fre- quency range, in which no resonance vibrations of a tube may occur. This physically improbable result is rooted in the membrane theory approximation, in which case the thickness of a tube is assumed to be small to the extent that the energy of flexural deformations can be neglected in comparison with the energy of extensional deformations at any aspect ratio. It was shown by Junger and Rosato14 that such an assumption is especially wrong for the range of aspect ratios around the point of strongest coupling. In order to correct this shortcom- ing, the energy related to flexural vibration of the bar as one of the partial systems is taken into account in the current paper. The structure of the paper is as follows. In Sec. II as- sumptions are made on the modes of deformation in the thin- walled elastic tubes of arbitrary aspect ratio, following the idea to represent the vibration of a tube as the coupled vibra- tions of two partial systems. In distinction from the mem- brane theory approach the flexural deformations accompany- ing the extensional vibrations of a tube are included. In Secs. III and IV the energy method12 is applied to derive Lagrange’s equations describing vibrations of thickness poled piezoelectric tubes, and an electromechanical circuit equivalent to the equations of motion is introduced for the most practical range of aspect ratios pertaining to the first region of strong coupling. The results of calculating the reso- nance frequencies, modes of vibration, and effective cou- pling coefficients of the tubes and their experimental verifi- cation are presented. In Sec. V the analysis is extended for the tubes that are polarized in circumferential and axial di- rections. Section VI deals with including the acoustic and mechanical effects on a transducer in the equivalent electro- mechanical circuit. Examples of the transmit and receive re- sponses of air-backed transducers built from the tubes having different aspect ratios are presented. It has to be noted that a terminology vagueness exists regarding a naming of the thin-walled cylindrical piezoele- ments that are considered in the course of treating their two- dimensional vibration. So far as the passive elastic cylindri- cal elements are concerned, it is common to refer to elements 804 J. Acoust. Soc. Am., Vol. 125, No. 2, February 2009 B Downloaded 16 Nov 2011 to 166.111.132.53. Redistribution subject to ASA licens as “rings” at aspect ratios h /2a�1 and as the “tubes” at h /2a�1. But it is not clear when a ring transitions into a tube. Probably by this reason the piezoelectric elements con- sidered in terms of two-dimensional �coupled� vibration are interchangeably referred to as cylindrical shells,1 cylindrical tubes,3,4 piezoelectric shells of finite length,5 piezoceramic cylinders of finite size,7 and ceramic cylinders.8 At the same time, a convention exists among suppliers of piezoceramic parts to specify the thin-walled cylindrical parts with elec- trodes applied to the side surfaces as tubes regardless of their height-to-diameter ratio. Following this convention, we will name the objects of this treatment as the piezoceramic tubes �or just tubes, when it is clear from context that they are made from piezoelectric ceramics�. II. ASSUMPTIONS ON THE DISTRIBUTION OF DEFORMATIONS IN THE THIN-WALLED ELASTIC TUBES Consider extensional axially symmetric vibrations of a thin-walled isotropic elastic tube shown in Fig. 1�a�. In this treatment we will represent the vibration in the tube as the coupled vibrations of the two partial one-dimensional elastic systems, namely, of the radial vibrating ring �Fig. 1�b�� and of the longitudinally vibrating thin bar �Fig. 1�c��. The common assumption for the thickness, t, of the “thin-walled” tube is that t�2a. In other words this means that the resonance frequency of vibration of the ring in the direction of thickness, f t �f t�c /2t, where c=�Y /� is the sound speed in a thin bar, Y is Young’s modulus, and � is the density of the tube material�, and the resonance frequency of the radial vibration of the ring, fr �fr=c /2�a�, are very far apart. Thus, these vibrations can be considered as indepen- dent, and at frequencies close to the radial resonance of the ring the thickness is very small compared with the exten- sional wavelength. Therefore the stresses in radial direction, Tr, are practically constant, and being zero on the ring sur- faces they remain negligible inside of its volume, i.e., Tr=0, which allows the problem to be treated as two dimensional. Consider the extreme cases for the height, h, of a tube, in which h /2a�1 and h /2a→�. In the case that h /2a�1, the tube reduces to ring Fig. 1�b�, the first resonance fre- r x (a) (b) (c) h t ϕ FIG. 1. A thin-walled elastic tube �a� and its partial subsystems: the radially vibrating short ring �b� and the longitudinally vibrating thin bar �c�. oris S. Aronov: Coupled vibration analysis of cylindrical transducers e or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp quency of the axial vibrations, fh1�c /2h, is much higher than the resonance frequency of the radial vibration, f , and x h/2 r by the reasons discussed for the thickness of a ring it can be assumed that the axial stress Tx=0 in the volume of the ring. Thus, the problem becomes one dimensional with a well- known solution. In the case that h /2a→�, the tube becomes very long, the resonance frequency of the axial vibrations becomes much lower than for the radial vibration, and the vibrations in the vicinity of the radial resonance frequency may be considered as one dimensional with the mechanical conditions Tr=0 and Sx=0, where Sx is the strain in the axial direction. The latter condition is valid, strictly speaking, for an infinitely long tube because of the symmetry consider- ations, but it may be assumed to be applicable to a tube long in comparison with the extensional wavelength in a fre- quency range of interest. For this case the resonance fre- quency of the radial vibration, fr�, is known to be fr� =�Y /��1−�2�, where � is Poisson’s ratio. Both of these one-dimensional approximations are valid practically to a broad extent of aspect ratios so far as the separation between the resonance frequencies of vibration in the axial and radial directions is sufficiently large. But it remains to be estimated what is large enough in terms of an acceptable accuracy of calculations based on these approxi- mations. From a general theory of coupled vibrations it fol- lows that the strongest coupling between the partial systems should take place under the condition that the resonance fre- quencies of the partial systems are equal. In our case this condition takes place at first in the vicinity of the aspect ratio h /2a=� /2, at which point fr= fh1, and then repeatedly at the aspect ratios related to the harmonics of the axial resonance frequency, fhi. Thus, it can be expected that the one- dimensional ring approximation may be valid for the tube with the aspect ratios h /2a�� /2. It is not clear what the lowest acceptable value of the aspect ratio is for the one- dimensional long tube approximation to be valid. This has to be determined. When considering vibrations of a tube as being two di- mensional, an assumption regarding the distribution of dis- placements over the tube surface can be made based on rep- resentation of the vibrations, as a result of interaction of the partial system vibrations. At first consider deformations of a ring �Fig. 1�b��. Denote the radial displacement of the ring surface as �0, then the strain S� in the circumferential direc- tion may be presented as S� = 2��a + �a� − 2�a 2�a = �0 a , �1� and the strain Sx in the axial direction will be determined as Sx=−�S�=−��0 /a. Correspondingly, the displacement in the axial direction �xr generated in the tube by the radial dis- placement will be �xr = − �0 �x a . �2� Consider now deformations of a thin bar �Fig. 1�c��. Dis- placement in the axial direction, �xx, may be represented as an expansion in the series J. Acoust. Soc. Am., Vol. 125, No. 2, February 2009 Boris S. Downloaded 16 Nov 2011 to 166.111.132.53. Redistribution subject to ASA licens �xx = � i=1 2L−1 �xi sin�i�x/h� , �3� where L is a number of modes taken into account for an approximation, which may be considered as acceptable. Ex- pression for the strain in the axial direction is Sx=��xx /�x, and the strain in the lateral direction, S�, will be found as S�=−�Sx=−���xx /�x. The deformation corresponding to this strain, which is produced in the circumferential direction of the ring, should cause a displacement of the ring surface that is proportional to deformation S� according to formula �1�. Thus, the deformation of the bar in axial direction gen- erates the radial displacement of the tube surface, �rx, which can be defined as �rx � ���xx/�x� �4� and can be expressed with reference to Eq. �3� in the general form of �rx = � i=1 2L−1 �ri cos�i�x/h� . �5� Summarizing the assumptions made above, the displacement distribution in an axial symmetrically vibrating thin-walled tube can be represented as follows: �x = �xx + �xr = �− �x/a��0 + � i=1 2L−1 �xi sin�i�x/h� , �6� �r = �0 + �rx = �0 + � i=1 2L−1 �ri cos�i�x/h� . �7� Distribution of the displacements in a tube is shown qualita- tively in Fig. 2, as a superposition of displacements gener- ated by vibration of the partial systems �only the fundamen- tal mode of a bar vibration is illustrated for simplicity�. So far as the displacement distribution is defined by Eqs. �6� and �7�, the axial symmetric strain distribution in the body of the tube can be represented in the cylindrical coordinates x and � as follows: (a) (b) z -t/2 t/2 ξ rx ξ0 -h/2 ξ xx ξ0 FIG. 2. Distribution of displacements in a tube: �a� in the extensional vibra- tions and �b� in the flexural vibrations. Aronov: Coupled vibration analysis of cylindrical transducers 805 e or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp S = ��x − z �2�r = − � � + 2L−1 � i� cos�i�x/h� 10 x �x �x2 a 0 � i=1 xi h − z � i=1 2L−1 �ri� i�h 2 cos�i�x/h� , �8� S� = �r a = �0 a + 1 a � i=1 2L−1 �ri cos�i�x/h� . �9� The term �−z�2�r /�x2� in Eq. �8� accounts for the strains due to the flexural deformations of the wall of the tube �the co- ordinate axis z goes in the radial direction and has its origin on the mean circumferential surface of the tube, as it is shown in Fig. 2�b��. This term is a matter of principle in this treatment. It takes into consideration the energy of the flex- ural deformations of the wall of a tube having a finite thick- ness, and makes the proposed approach to the problem dif- ferent from that, which was used in the framework of the membrane theory. The stresses in the tube will be found as follows15 �re- member that Tr=0�: Tx = Y 1 − �2 �Sx + �S�� , �10� T� = Y 1 − �2 ��Sx + S�� . �11� By substituting the strains Sx and S�, defined by Eqs. �8� and �9�, into Eq. �10� it is easy to make certain that the boundary conditions on the free ends of a tube, Tx� h /2�=0, are met. With distribution of strain in the tube known, the equations of motion of an isotropic passive tube can be derived as Lagrange’s equations. A note has to be made regarding a number of terms in series �3� to be taken into account for practical calculations in order to achieve an acceptable level of accuracy for the results. This depends on the range of aspect ratios under consideration. For the range below and around the first re- gion of strong coupling, namely, 0 h /2a 3, to the first approximation it should be sufficient to retain only the first term of the series, which corresponds to the fundamental mode of the axial vibration. For the higher range of aspect ratios, but below and around the second region of a strong coupling, that is, for 0 h /2a 6, the next term has to be included, which corresponds to the third harmonic of the axial vibration. One more note has to be made regarding representation of the flexural term in Eq. �8�. The flexural term is repre- sented based on the elementary theory of bending. This can be considered as sufficiently accurate until the half length of flexure-to-thickness ratio for a bar as a partial system is much larger than unity, i.e., h / it�1, where i is a number of half waves of flexure on the length of the bar. Otherwise, corrections accounted for the effects of rotary inertia and shearing deformations on the kinetic and potential energies related to the flexural vibrations must be taken into consideration.15 It is shown in Ref. 16 that with these correc- 806 J. Acoust. Soc. Am., Vol. 125, No. 2, February 2009 B Downloaded 16 Nov 2011 to 166.111.132.53. Redistribution subject to ASA licens tions for rectangular bars a good agreement between calcu- lated and measured resonance frequencies takes place at h / it�5, even this condition can not be met at small aspect ratios, especially for a high order flexural vibration. Thus, for example, for the tubes having the mean diameter 2a =35.8 mm and thickness t=3.2 mm, which were used for experimental investigat