Exact series model of Langevin transducers with internal losses P. A. Nishamol and D. D. Ebenezera) Naval Physical and Oceanographic Laboratory, Thrikkakara, Kochi 682021, India

(Received 11 July 2013; revised 9 January 2014; accepted 24 January 2014) An exact series method is presented to analyze classical Langevin transducers with arbitrary boundary conditions. The transducers consist of an axially polarized piezoelectric solid cylinder sandwiched between two elastic solid cylinders. All three cylinders are of the same diameter. The length to diameter ratio is arbitrary. Complex piezoelectric and elastic coefficients are used to model internal losses. Solutions to the exact linearized governing equations for each cylinder include four series. Each term in each series is an exact solution to the governing equations. Bessel and trigonometric functions that form complete and orthogonal sets in the radial and axial directions, respectively, are used in the series. Asymmetric transducers and boundary conditions are modeled by using axially symmetric and anti-symmetric sets of functions. All interface and boundary conditions are satisfied in a weighted-average sense. The computed input electrical admittance, displacement, and stress in transducers are presented in tables and figures, and are in very good agreement with those obtained using ATILA—a finite element package for the analysis of sonar transducers. For all the transducers considered in the analysis, the maximum difference between the first three resonance frequencies calculated using the present method and ATILA is less than 0.03%. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4864469] V PACS number(s): 43.38.Ar, 43.38.Fx, 43.20.Bi, 43.40.At [MRB]

I. INTRODUCTION

A classical Langevin transducer1 is comprised of an axially polarized piezoelectric ceramic cylinder sandwiched between two elastic cylinders. In the original Langevin transducer, the ratio of the thicknesses of the transducer to the lateral dimensions is very small. However, sandwich transducers with larger ratios are now called Langevin-type transducers. In this paper, a method is presented to analyze a Langevin-type transducer with three cylinders, all having the same radius, a, as shown in Fig. 1, and any length. The bottom and top elastic cylinders are known as the tail mass and head mass, respectively, of the transducer. An exact infinite series method is presented to determine the displacement and stress fields and the electrical admittance when the transducer is electrically excited. Analytical, dynamic, axisymmetric models of finite, solid, elastic, and piezoelectric, circular cylinders used in conjunction with continuity conditions at the interfaces between the components of Langevin transducers yield models of the transducers. Several approximate models are available in literature. The effects of the approximations on the error in the computed transducer parameters of interest are often only qualitatively discussed. The errors can be quantitatively determined by comparing the model results with exact results. In this paper, results obtained by using the analytical model are compared with those obtained using ATILA (Ref. 2)—a finite element package for the analysis of underwater transducers. The classical model of longitudinal vibration of long, thin, elastic rods, developed during the 18th century, is the a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

J. Acoust. Soc. Am. 135 (3), March 2014

Pages: 1159–1170

simplest model of cylinders. In this model, it is assumed that only the longitudinal stress is non-zero. Pochhammer, and Chree,3 independently4 present a model for free vibration of an infinite circular cylinder, governed by exact equations of elasticity, and with zero normal and shear stress on the curved surface. When either model is used to analyze transducers, continuity of only average stress and average displacement along the axis can be ensured at the plane interfaces between components. Filon5 and Purser6 present infinite series models of static, finite, elastic cylinders. Each term in the series is an exact solution to the exact biharmonic7 equation of elasticity. Filon uses complete sets of trigonometric functions and satisfies arbitrary boundary conditions only on the curved surface. Purser uses complete sets of trigonometric and Bessel functions and satisfies all the boundary conditions on all the surfaces exactly or in a weighted-average sense. Purser’s approach is extended by other investigators. Hutchinson analyzes free vibration of finite elastic cylinders and presents resonance frequencies and mode shapes of cylinders with zero displacement8 and zero stress9,10 on the boundaries. Grinchenko11 analyzes hollow static cylinders of finite length with specified non-zero normal stress on the outer and inner curved surfaces. Grinchenko and Meleshko12 analyze a finite elastic cylinder subjected to uniform normal dynamic stress on the curved surface. Meleshko13 revisits Filon’s static shear problem and analyzes it using two infinite series. Sburlati14 analyzes static cylinders by expressing the Love function in the form of a Fourier-Bessel series. She uses two auxiliary terms that are dependent on the axial coordinate. Shear stress is zero on all the surfaces in Refs. 6 and 8–13. Ebenezer and co-workers15 study free and forced vibration of finite elastic cylinders. They show16 that the same

0001-4966/2014/135(3)/1159/12/$30.00

C 2014 Acoustical Society of America V

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FIG. 1. (Color online) Schematic of a Langevin transducer.

numerical results are obtained by using different complete sets of functions and present alternative forms of the solution that are to be used at the frequencies at which one of the functions in the complete set becomes zero. They compute the dynamic response of a steel cylinder to local stress on the boundary and show good agreement with that obtained using ATILA.2 A few models of axially polarized piezoelectric cylinders used in Langevin transducers are available. In the simplest one found in several books, it is assumed that only the longitudinal stress exists—just as in the case of elastic rods. Paul17 presents the frequency equation for axially polarized, infinite, hollow cylinders with zero normal and shear stress on the curved surfaces. Brissaud18,19 uses one exact and one approximate solution to the exact axisymmetric governing equations to approximately satisfy boundary conditions on a finite cylinder. Ebenezer and Ramesh20 use two exact solutions to approximately analyze electrically excited axially polarized finite solid cylinders with zero stress on the surface. Series solutions are also used to analyze axially polarized finite piezoelectric cylinders. Each term in the series used by Holland and Eer Nisse21 is not a solution to the governing equations. However, the sum of the infinite series satisfies both the governing equation and the boundary conditions of a free cylinder. Others use exact series solutions where each term in the series is an exact solution to the governing equation. Paul and Natarajan22 use complete sets to satisfy the boundary conditions of a stress-free cylinder with electrodes on all surfaces. Ebenezer et al.23 analyze axially polarized piezoelectric cylinders with internal losses that are simultaneously excited electrically and mechanically. They use exact series solutions to the governing equations to satisfy arbitrary boundary conditions. They present numerical values of complex input electrical admittance and axial displacement. Analytical and numerical models of Langevin transducers are available. The classical theory of longitudinal vibration of elastic rods is used together with the 1160

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

corresponding model for longitudinal piezoelectric vibrators, in the one-dimensional analysis of Tonpilz transducers.24–26 Iula et al.27 use Brissaud’s model18 to present an approximate analytical model of an axisymmetric Langevin transducer. Iula et al.28 also use finite element three-dimensional analysis to study the vibrational behavior. Shuyu29 neglects the torsional and shear stresses and analyzes sandwich transducers with a large cross section area or high resonance frequency. Fu et al.30 study constrained multi-objective optimization of Langevin transducers modeled using rod theory. Adachi and co-workers31,32 use the finite element method to analyze bolt-clamped Langevin transducers. In this paper, an exact series method is presented to analyze Langevin transducers with internal losses. Internal losses give rise to hysteresis loops and internal heating. The effect of internal losses is modeled using complex dielectric, elastic, and piezoelectric coefficients. Expressions for all field variables in each cylinder include four exact series solutions. One pair is axially symmetric with respect to the mid-plane of the cylinder and another pair is anti-symmetric. In each pair, one series contains terms that form a complete set in the radial direction and the other contains terms that form a complete set in the axial direction. Therefore, arbitrary boundary and continuity conditions are satisfied on the flat and curved surfaces and at the interfaces. The coefficients in the series are determined by satisfying the boundary and interface conditions in a weighted-average sense by using the orthogonal property of the functions in the complete sets. Numerical results are presented for different Langevin transducers to illustrate the good agreement between the values computed using the present method and ATILA.2 II. GOVERNING EQUATIONS AND SOLUTIONS

Consider a Langevin transducer shown in Fig. 1. Cylindrical coordinates (r, h, z) are used in the analysis. Local coordinates are used for each cylinder and the origin is at the center of each cylinder. The radius of each cylinder is a. The three cylinders in the transducer are numbered 1, 2, and 3 starting from the bottom, elastic cylinder. Superscripts denote the number of the cylinder. The length of the cth cylinder is L(c), c ¼ 1, 2, 3. The top and bottom flat surfaces of the piezoceramic cylinder are fully electroded. The applied potentials on the top and bottom surfaces are U0 and zero, respectively. The Langevin transducer, the boundary conditions, and the electrical excitation are axisymmetric. Therefore, axisymmetric, linearized, governing differential equations for the elastic and piezoceramic cylinders are used here. The electrical excitation and the response of the transducer have an exp(þjxt) variation in time that is suppressed everywhere for convenience. Values of the imaginary parts of the coefficients of the piezoelectric cylinder satisfy the condition that energy is dissipated15,33 and not created. For the piezoceramic cylinder, the exact axisymmetric equations of dynamic equilibrium, equations of state, straindisplacement relations, and the Gauss electrostatic condition used in Refs. 17 and 34 are used here with a change in the P. A. Nishamol and D. D. Ebenezer.: Langevin transducer

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notation. Here, U ð pÞ and W ð pÞ are the axial and radial displacements, respectively, in the piezoelectric cylinder; whereas U and W are used in Ref. 34. Here and in Ref. 34, U is the electric potential, cE11 ; cE12 ; cE13 ; cE33 ; and cE44 are the elastic stiffness coefficients, e31 ; e33 ; and e15 are the piezoelectric stress coefficients, and es11 and es33 are dielectric permittivity coefficients. The solution to Eq. (5) in Ref. 34 is expressed here as the sum of five independent exact solutions20,23 ½ U ð pÞ

W ð pÞ

T



T

U  ¼ U 1 ð p Þ W1 ð p Þ U 1  T þ U2 ð pÞ W 2 ð pÞ U2  T þ U3 ð pÞ W 3 ð pÞ U3  T þ U4 ð pÞ W 4 ð pÞ U4  T þ U 5 ð p Þ W5 ð p Þ U 5 ;

and (1a)

(1b)

where Kr0 ¼ x½q=cE11 0:5 ; J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

(1f)

ð pÞ

(1c)

where Kz0 ¼ x½q=ðcE33 þ e233 =eS33 Þ0:5 ;

m¼1 s¼1

8 9 8 9 ð pÞ > Qð pÞ J0 ðKr0 rÞ = < U5 > = < ð pÞ 0 ¼  ; > W5 ; > : ð pÞ  : Q e15 =es11 J0 ðKr0 rÞ U5 9 8 Mq X 3  p  > > X > > p p ð Þ ð Þ ð Þ > > > Qms J0 ðKrms rÞcos Kzm z > > > > > > > > > m¼1 s¼1 > > > > M = > > > m¼1 s¼1 > > > > > Mq X 3 > >  p > X > > pÞ pÞ ð ð ð Þ > > > Qms #ms J0 ðKrms rÞcos Kzm z > > > ; : m¼1 s¼1

m¼1 s¼1

9 8 8 9 ð pÞ > 0 = < < U3 > = ð pÞ ¼ Bð pÞ J1 ðKr0 rÞ W > ; ; : : 3 > 0 U3 9 8 Mz X 3 > >   X > > ð pÞ ð pÞ ð pÞ > > > Bms J0 ðkrms rÞsin kzm z > > > > > > > > > m¼1 s¼1 > > > > Mz X = > > > m¼1 s¼1 > > > > > > M 3 z X   X > > > > p p p ð Þ ð Þ ð Þ > > > > B c J ðk rÞsin k z zm ms ms 0 rms > > ; :

(1e)

m¼1 s¼1

where 9 8 8 9 ð pÞ > = < 0 = < U1 > ð pÞ W1 > ¼ : 0 ;; > ; : Dz þ E U1 9 9 8 8 ð pÞ sinðK zÞ > > ð pÞ > A > > > z0 > > = = > < < U2 > 0 ð pÞ ¼ W 2 > > > ð pÞ e33 > > > sinðKz0 zÞ > > ; > >A : ; : U2 es33 9 8 Mr X 3 > > X > > p p ð Þ ð Þ > > > > ð Þsinðk A J k r zÞ > > ms zms 0 rm > > > > > > m¼1 s¼1 > > > > > > = > > > m¼1 s¼1 > > > > > > > > Mr X 3 X > > > > p p ð Þ ð Þ > > > Ams vms J0 ðkrm rÞsinðkzms zÞ > > > ; :

9 8 Pð pÞ cosðKz0 zÞ > > > > > > = < 0 ð pÞ ¼ W4 > > > > e33 > > > > > ; ; > : Pð pÞ s cosðKz0 zÞ > : U > e33 4 9 8 Mr X 3 > > X > > ð pÞ ð pÞ > > > Pms J0 ðkrm rÞcosðkzms zÞ > > > > > > > > > m¼1 s¼1 > > > > > > > > Mr X = > > > m¼1 s¼1 > > > > > > > > > > M 3 r X X > > > > ð pÞ ð pÞ > > > > ð Þcosðk P n J k r zÞ ms zms 0 rm ms > > ; : 9 8 ð pÞ > > > U4 > > > =
> > > > = ; þ Mr X 2 > > X > > > > ðeÞ ðeÞ ðeÞ > > Ams ams J1 ðkrm r Þcosðkzms zÞ > > ; : 8 Mr X 2 X > > ðeÞ > AðeÞ J ðk r Þsinðkzms zÞ > > < m¼1 s¼1 ms 0 rm

(2b)

m¼1 s¼1

8 9 < U ðeÞ = 2

: W ðeÞ ;

( ¼

2

)

0 ðeÞ

BðeÞ J1 ðK1 rÞ 8 Mz X 2 X > > ðeÞ > BðeÞ J ðkðeÞ rÞsinðkzm zÞ > > < m¼1 s¼1 ms 0 rms

9 > > > > > = ; þ Mz X 2 > > X > > > > e ðeÞ ðeÞ > > BðmseÞ bðmsÞ J1 ðkrms rÞcosðkzm zÞ > > ; :

(2c)

3

: W ðeÞ ;

¼

3

ðeÞ

; 0 8 Mr X 2 X > > ðeÞ > PðeÞ J ðk r Þcosðkzms zÞ > > < m¼1 s¼1 ms 0 rm

ðeÞ ; W4

¼

9 > > > > > =

þ ;

ðpÞ ðpÞ ðpÞ þðcE33 þ e33 cms Þkzm gJ0 ðkrms rÞcosðkzm zÞ

m¼1 s¼1

1162

þ

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

Mr X 3 X E PðpÞ ms fc13  ms krm m¼1 s¼1

ðpÞ ðpÞ ðcE33 þ e33 nms Þkzms gJ0 ðkrm rÞsinðkzms zÞ

;

9 > > > > > > = ; þ > > Mq X 2 > > X > > > eÞ ðeÞ ðeÞ ðeÞ > > Qðms gms J1 ðKrms ÞsinðKzm zÞ > > > ; :

Mz X 3 X E ðpÞ BðpÞ ms fc13 /ms krms m¼1 s¼1

(2d)

2

8 Mq X 2 > X > > ðeÞ ðeÞ > QðeÞ J0 ðKrms rÞcosðKzm zÞ > > < m¼1 s¼1 ms

Mr X 3 X E AðpÞ ms fc13 wms krm

ðpÞ ðpÞ þðcE33 þ e33 vms Þkzms gJ0 ðkrm rÞcosðkzms zÞ

8 9 < QðeÞ J0 ðK ðeÞ rÞ = 0

ðeÞ

m¼1 s¼1

Mr X 2 > > X > > > > ðeÞ > PðmseÞ 1ðmseÞ J1 ðkrm r Þsinðkzms zÞ > > > ; :

:

ðeÞ

ðeÞ

1

:

ðeÞ

1ms ; and gms are presented in Refs. 15 and 16. In Eqs. (2c) and (2e), the axial displacement is expressed in terms of functions that are axially symmetric and anti-symmetric, respectively, with respect to the mid-plane of the cylinder. In this analysis, krm in Eqs. (1c), (1e), (2b), and (2d) are chosen such that krm a are the roots of J1 ðkrm aÞ ¼ 0 and are approximately equal to 0, 3.83, 7.02,… for m ¼ 0,1,2,…, respectively. For Mr ¼ 1; Jv ðkrm aÞ form a point-wise complete set of functions when  ¼ 0 and norm-wise complete ðcÞ ; c ¼ 1; 2; or 3, in sets of functions when  ¼ 1. Further, kzm ðcÞ ðcÞ L =2 ¼ mp, Eqs. (1d) and (2c) are chosen such that kzm ðcÞ m ¼ 0; 1; 2; :::; Mz ; and Kzm in Eqs. (1f) and (2c) are chosen ðcÞ ðcÞ such that Kzm L ¼ ð2m  1Þp; m ¼ 1; 2; :::; Mz . In both sets, for Mz ¼ 1; sinðÞ and cosðÞ are complete sets of functions. The components of stress and charge density are expressed in terms of the components of displacement and electric potential by using the stress-strain and straindisplacement relations for elastic and piezoelectric materials. Then, they are expressed in terms of complete sets of functions by using Eqs. (1) and (2). For example, the normal stress along the axial direction of a piezoceramic cylinder is E 2 s expressed, by using cD 33 ¼ c33 þ e33 =e33 , as

m¼1 s¼1

:

  ðeÞ2 ¼ kðeÞ þ 2lðeÞ =qðeÞ ; and c2

 PðpÞ cD 33 Kz0 sinðKz0 zÞ þ

8 9 < PðeÞ cosðK ðeÞ zÞ =

þ

8 9 < U4ðeÞ =

ðeÞ2

ðpÞ E TzzðpÞ ¼ AðpÞ cD 33 Kz0 cosðKz0 zÞ þ B c13 Kr0 J0 ðKr0 rÞ þ De33

m¼1 s¼1

8 9 < U ðeÞ =

ðeÞ

¼ x=cs ; s ¼ 1; 2; c1

¼ lðeÞ =qðeÞ : Here; AðeÞ ; BðeÞ ; PðeÞ ; QðeÞ Ams ; Bms ; Pms ; and Qms are coefficients and are determined by using the boundary and continuity conditions and the excitation. Equations (2b)–(2e) are made exact solutions to the governing ðeÞ ðeÞ equations for arbitrary values of krm ; kzm ; and Kzm by using the procedure presented above for piezoelectric cylðeÞ ðeÞ ðeÞ ðeÞ inders. Explicit expressions for kzms ; krms ; Krms ; ams ; bðmseÞ ; ðeÞ

ðeÞ T W2 

where 8 9 < U ðeÞ =

ðeÞ

Ks

þ

Mq X 3 X E ðpÞ QðpÞ ms fc13 tms krms m¼1 s¼1

(2e)

ðpÞ ðpÞ ðpÞ  ðcE33 þ e33 #ms Þkzm gJ0 ðkrms rÞsinðKzm zÞ:

(3)

The current in the piezoceramic cylinder is expressed as P. A. Nishamol and D. D. Ebenezer.: Langevin transducer

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I¼

ða jxDz 2prdr 0

Mz X ðpÞ zÞ ¼ jpxfa½BðpÞ 2e31 J1 ðKr0 aÞ  Daes33   2 cosðkzm

" 

m¼1 3 X

ðpÞ BðpÞ ms fe31 /ms krms

s¼1

þ ðe33 

ðpÞ ðpÞ ðpÞ es33 cms Þkzm gaJ1 ðkrms aÞ=krms



Mq 3 X X ðpÞ ðpÞ zÞ QðpÞ  2 sinðKzm ms fe31 tms Krms m¼1

 ðe33 

s¼1 s ðpÞ ðpÞ ðpÞ e33 #ms ÞKzm gaJ1 ðKrms aÞ=Krms g:

(4)

The current in Eq. (4) is independent of z, as seen from the Gauss zero-divergence condition.34 Hence, the sum of the series in Eq. (4) should be zero—and this is used as a check for the calculations. Finally, the complex input electrical admittance, Y ¼ G þ jB, where G is the conductance and B is the susceptance, is expressed as Y ¼ G þ jB ¼ I=U0

(5)

and is obtained by using Eq. (4) and the specified applied potential difference, U0. It is seen from Eqs. (1) and (2) that the displacement, potential, stress, and charge density are expressed in terms of complete sets of functions in the axial and radial directions ðcÞ ðcÞ , and kzm ; and in terms because of the chosen values of krm , kzm of functions that are axially symmetric and anti-symmetric with respect to the mid-plane of each cylinder. Therefore, the response of the transducer to any16,23 specified distribution of displacement, stress, electric potential, and charge density on the boundary can be determined by using this method. When internal losses are present, the conductance is non-zero and energy is dissipated in the transducer. Therefore, if the transducer is given an initial displacement and released, the free vibrations will be damped and gradually reduce to zero. In general, when losses are present, normal modes do not exist, but the present method can be used. The analysis permits the use of complex values for all piezoelectric and elastic coefficients. Therefore, the present method can be used to determine the power dissipation density36 and the total power, U20 G, dissipated in the transducer. ðcÞ ðcÞ It is noted that krm , kzm , and kzm are real even when there are losses and that other functions and coefficients are complex.

ðcÞ respectively, on the flat surfaces of the cylinders; and T^rr ðcÞ and T^rz are normal and shear stresses on the curved surfaces, respectively. All boundary and continuity conditions are satisfied in a weighted-average sense. Each condition is multiplied by weights and integrated over the surface. An infinite set of weights that form a complete set of orthogonal functions is used. In some cases, the functions form only a norm-wise complete set and the error at the rim may not be small. In such cases, alternative sets of functions16 can be used. An infinite set of equations is obtained for each condition because the solutions in Eqs. (1) and (2) are the sums of four infinite series. A finite set of equations is obtained by truncating the series. (In some cases, the solution can be found without truncation.11,13) The number of complex equations is equal to the number of complex coefficients if the series are truncated: Mr should be the same for each cylinder but Mz and Mq can be different for different cylinders and can be chosen based on the relative lengths of the cylinders. The orthogonal property of the functions results in equations for coefficients with no coupling to other coefficients in the same series.

A. Electrical conditions

The electrical boundary conditions on the flat electrodes of the piezoceramic cylinder are: U ¼ 0 at z ¼ LðpÞ =2 and U ¼ U0 at z ¼ LðpÞ =2. These conditions are satisfied by using the orthogonal property of J0 ðkrn rÞ. Equating the specified values of potential to the expression in Eq. (1), multiplying both sides by rJ0 ðkrn rÞ, and integrating over r yields one equation for each value of n. For the top flat surface (z ¼ L(2)/2, 0  r  a), using n ¼ 0 yields DLðpÞ =2 þ E þ ðe33 =eS33 Þ½AðpÞ sinðKz0 LðpÞ =2Þ þ PðpÞ cos ðKz0 LðpÞ =2Þ þ QðpÞ 2e15 =ðKr0 aes11 Þ þ ð2=aÞ 

Mq X 3 X ðpÞ ðpÞ QðpÞ ms #ms cos ðKzm L =2Þ

m¼1 s¼1 ðpÞ ðpÞ J1 ðKrms aÞ=Krms

¼ U0 ;

(6a)

and using n ¼ 1,2,…,Mr yields 2 1 QðpÞ 2e15 Kr0 ðaeS11 Þ1 J0 ðkrn aÞJ1 ðKr0 aÞðKr0 2  krn Þ " 3 X ðpÞ ðpÞ þ a2 J02 ðkrn aÞ AðpÞ ns vns sinðkzns L =2Þ s¼1

þ

3 X

PðpÞ ns nns

#

ðpÞ ðpÞ cosðkzns L =2Þ

s¼1

III. BOUNDARY AND CONTINUITY CONDITIONS

The specified electric conditions, the boundary conditions, and the continuity conditions are used to determine the coefficients in Eqs. (1) and (2). Here, only zero stress on all the exposed surfaces of the elastic and piezoelectric cylinders is considered. However, arbitrary boundary conditions can be satisfied because all the fields of interest are expressed in terms of complete sets of functions. The symbols  and Ù denote that the function is to be evaluated on a flat surface and a curved surface, respectively. ðcÞ ðcÞ For example, Tzz and Trz are the normal and shear stresses, J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

Mq X 3 X ðpÞ ðpÞ ðpÞ QðpÞ þ 2a ms #ms cosðKzm L =2ÞKrms J0 ðkrm aÞ m¼1 s¼1 2

ðpÞ ðpÞ 2 1 aÞðKrms  krn Þ ¼ 0:  J1 ðKrms

(6b)

Similar equations are obtained by using the zero potential boundary condition on the bottom flat surface. On the curved surface of the piezoceramic cylinder, i.e., on r ¼ a, jzj  LðpÞ =2; the electric field displacement in the radial direction, D^ r , is zero. Equating the expression for D^ r on r ¼ a to ðpÞ zÞ, and zero, multiplying both sides by the weights 1, sinðkzn P. A. Nishamol and D. D. Ebenezer.: Langevin transducer

1163

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ðpÞ cosðKzn zÞ, and integrating over the length of the cylinder, and using the orthogonal properties of the functions yields ð LðpÞ =2 Mq X 3 X ðpÞ1 ^ QðpÞ D r dz ¼ ms Kzm LðpÞ =2

m¼1 s¼1 ðpÞ ðpÞ ðpÞ  feS11 #ms Krms  e15 ðKrms  tms Kzm Þg ðpÞ ðpÞ ðpÞ  J1 ðKrms aÞ sinðKrms L =2Þ ¼ 0;

(7a)

for n ¼ 1,2,…,Mq. B. Stress boundary conditions

The boundary of the transducer is comprised of two flat surfaces and three curved surfaces. At the flat ends of the transducer, the following boundary conditions are specified for r  a:

for n ¼ 0, ð LðpÞ =2 3 X BðpÞ D^ r sinðkðpÞ zÞdz ¼ 0:5LðpÞ zn

LðpÞ =2

 

ns s¼1 ðpÞ ðpÞ feS11 cns krns  e15 ðkrns ðpÞ J1 ðkrns aÞ ¼ 0

LðpÞ =2

at

z ¼ Lð1Þ =2

(8a)

ð3Þ

ð3Þ Trz gT ¼ f0

0gT

at

z ¼ Lð3Þ =2:

(8b)

fTzz

(7b)

ðeÞ

ðpÞ zÞdz ¼ 0:5LðpÞ D^ r cosðKzn

3 X

QðpÞ ns

s¼1

ðpÞ aÞ ¼ 0  J1 ðKrns

0

0gT

and

ðpÞ ðpÞ ðpÞ  fe15 ðtns Kzn  Krns Þ þ eS11 #ns Krns g

ða

ð1Þ Trz gT ¼ f0

ðpÞ þ /ns kzn Þg

for n ¼ 1,2,…,Mz, and ð LðpÞ =2

ð1Þ

fTzz

(7c)

ðeÞ ðeÞ Trz rdr ¼ QðeÞ lðeÞ ½1  J0 ðK2 aÞ  lðeÞ

Mr X 2 X

Multiplying both sides of the boundary conditions on Tzz by the weights rJ0 ðkrn rÞ, n ¼ 0,1,2,…,Mr, and integrating over r yields one equation for each n on each flat surface. These are not presented here for the sake of brevity. Following the same procedure for the boundary ðeÞ conditions on Trz and using the weights r and rJ1 ðkrn rÞ yields

ðeÞ ðeÞ AðeÞ ms ½krm þ ams kzms 

m¼1 s¼1 ðeÞ ðeÞ L =2Þ½1  J0 ðkrm aÞ=krm  lðeÞ  sinðkzms

Mz X 2 X ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ BðeÞ ms ½krms þbms kzm sinðkzm L =2Þ½1  J0 ðkrms aÞ=krms m¼1 s¼1

þ lðeÞ

Mr X 2 X

ðeÞ ðeÞ ðeÞ ðeÞ PðeÞ ms ½krm þ 1ms kzms cosðkzms L =2Þ½1  J0 ðkrm aÞ=krm

m¼1 s¼1 Mz X 2 X ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ þ lðeÞ QðeÞ ms ½Krms þ gms Kzm cosðKzm L =2Þ½1  J0 ðKrms aÞ=Krms ¼ 0;

(9a)

m¼1 s¼1

for n ¼ 0 and ða 0

2

ðeÞ ðeÞ ðeÞ ðeÞ 2 1 Þ Trz rJ1 ðkrn rÞdr ¼ QðeÞ lðeÞ K2 akrn J0 ðkrn aÞJ1 ðK2 aÞðK2  krn

 0:5a2 lðeÞ J02 ðkrn aÞ

( ) 2 X ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ Ans ½krn þans kzns sinðkzns L =2Þ þ Pns ½krn  1ns kzns cosðkzns L =2Þ s¼1

ðeÞ ðeÞ J0 ðkrn aÞ þ alðeÞ krn

Mz X 2 X ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ2 2 1 QðeÞ ¼ 0; ms ½gms Kzm  Krms cosðKzm L =2ÞJ1 ðKrms aÞðKrms  krn Þ

(9b)

m¼1 s¼1

for n ¼ 1, 2,...,Mr. On the curved surface of each cylinder, the following conditions are specified for jzj  LðcÞ =2: ðcÞ

fT^rr 1164

ðcÞ T^rz gT ¼ f0 0gT

on

r ¼ a:

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

(10)

They are satisfied by using the orthogonal properties of ðcÞ ðcÞ cosðkzn zÞ and sinðKzn zÞ, where c ¼ 1; 2; or 3. For examðpÞ ðpÞ ple, on using cosðkzn zÞ, the boundary condition on T^rr yields P. A. Nishamol and D. D. Ebenezer.: Langevin transducer

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ð LðpÞ =2 LðpÞ =2

ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ 2 1 T^rr cosðkzn zÞdz ¼ AðpÞ 2cD 13 ½kzn sinðkzn L =2ÞcosðKz0 L =2Þ  Kz0 cosðkzn L =2ÞsinðKz0 L =2Þðkzn  Kz0 Þ 2

þ

Mr X 3 X E E ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ AðpÞ ms fc11 wms krm þ c13 þ e31 vms Þkzms gJ0 ðkrm aÞ2kzms ½sinðkzn L =2Þcosðkzms L =2Þ m¼1 s¼1 2

2

ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ 1 L =2Þsinðkzms L =2Þðkzn  kzms Þ þ 0:5LðpÞ  cosðkzn

3 X E ðpÞ BðpÞ ns ½c11 /ns krns s¼1

þ

ðcE13

þ

ðpÞ ðpÞ e31 cns Þkzn J0 ðkrns aÞ

1

þa

ðcE12



ðpÞ cE11 Þ/ns J1 ðkrns aÞ

¼ 0;

(11)

1

E S for n ¼ 1,2,…,Mz, where cD 13 ¼ c13 þ e31 e33 e33 . Similar equations are obtained by using n ¼ 0, and by using the weights ðcÞ sinðKzn zÞ.

C. Continuity conditions

The Langevin transducer has two flat interfaces between cylinders. The continuity of the axial and radial displacements and the normal and shear components of stress at these interfaces are used to build the model of the transducer. At the interface between the bottom elastic and the piezoceramic cylinder, the following conditions are satisfied for 0  r  a: fU

ð1Þ

ð1Þ T^zz

 ð1Þ W

ð1Þ ð2Þ T^rz gz¼Lð1Þ =2 ¼ fU

ð2Þ T^zz

 ð2Þ W

ð2Þ T^rz gz¼Lð2Þ =2 :

(12a)

Similarly, at the interface between the piezoceramic and top elastic cylinders, the following conditions are satisfied: fU

ð2Þ

ð2Þ T^zz

 ð2Þ W

ð2Þ ð3Þ T^rz gz¼Lð2Þ =2 ¼ fU

ð3Þ T^zz

 ð3Þ W

ð3Þ T^rz gz¼Lð3Þ =2 :

(12b)

Multiplying both sides of the continuity conditions on U, at the interface between the bottom elastic cylinder and the piezoðcÞ electric cylinder, by the weights rJ0 ðkrn rÞ, and integrating over r yields ð1Þ

ð1Þ

ð1Þ

ð1Þ

Að1Þ sinðK1 Lð1Þ =2Þ þ Pð1Þ cosðK1 Lð1Þ =2Þ þ Qð1Þ 2J1 ðK2 aÞ=ðK2 aÞ þ 2a1

Mq X 2 X ð1Þ ð1Þ ð1Þ ð1Þ1 Qð1Þ ms cosðKzm L =2ÞJ1 ðKrms aÞKrms m¼1 s¼1

¼ Að2Þ sinðKz0 Lð2Þ =2Þ þ Pð2Þ cosðKz0 Lð2Þ =2Þ þ Qð2Þ 2J1 ðKr0 aÞ=ðKr0 aÞ þ 2a1

Mq X 3 X ð2Þ ð2Þ ðpÞ ðpÞ1 Qð2Þ ms cosðKzm L =2ÞJ1 ðKrms aÞKrms m¼1 s¼1

(13)  Tzz , and Trz yield similar equations. for n ¼ 0, and similar equations for n ¼ 1,2,…,Mr. The continuity conditions for W, D. Matrix form

In order to compute numerical results, all the equations based on boundary and continuity conditions are truncated, combined, and expressed in a matrix form as ½FfXg ¼ fGg;

(14a) ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

f XgT ¼ fAð1Þ ; Bð1Þ ; Pð1Þ ; Qð1Þ ; A11 ; A12 ; A21 ; A22 ; …; AMr 1 ; AMr 2 ; B11 ; B12 ; B21 ; B22 ; :::; ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð2Þ

ð2Þ

ð2Þ

ð1Þ

ð1Þ

BMz 1 ; BMz 2 ; P11 ; P12 ; P21 ; P22 ; :::; PMr 1 ; PMr 2 ; Q11 ; Q12 ; Q21 ; Q22 ; :::; QMq 1 ; ð1Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

QMq 2 ; Að2Þ ; Bð2Þ ; D; E; Pð2Þ ; Qð2Þ ; A11 ; A12 ; A13 ; A21 ; A22 ; A23 ; :::AMr 1 ; AMr 2 ; AMr 3 ; ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

B11 ; B12 ; B13 ; B21 ; B22 ; B23 ; :::; BMz 1 ; BMz 2 ; BMz 3 ; P11 ; P12 ; P13 ; P21 ; P22 ; P23 ; ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

…; PMr 1 ; PMr 2 ; PMr 3 ; Q11 ; Q12 ; Q13 ; Q21 ; Q22 ; Q23 ; …QMq 1 ; QMq 1 ; QMq 3 ; Að3Þ ; Bð3Þ ; Pð3Þ ; ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

Qð3Þ ; A11 ; A12 ; A21 ; A22 ; …; AMr 1 ; AMr 2 ; B11 ; B12 ; B21 ; B22 ; …; BMz 1 ; BMz 2 ; P11 ; P12 ; ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

P21 ; P22 ; :::; PMr 1 ; PMr 2 ; Q11 ; Q12 ; Q21 ; Q22 ; :::; QMq 1 ; QMq 1 g

(14b)

where fXg is the matrix with 14Mr þ 7Mz þ 7Mq þ 14 coefficients to be determined, ½F is a square matrix. The elements of the column matrix fGg are non-zero only when the corresponding boundary conditions on the surfaces are non-zero. In the special case mentioned here, the only non-zero element of fGg is J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

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GMr þ2 ¼

ða

U0 rdr ¼ 0:5U0 a2 :

(15)

0

Solving Eq. (14) for fXg yields the coefficients and the fields of interest are then determined. It is noted that the functions and coefficients in each term in the series solution are frequency dependent. It is seen from the form of the solutions and the numerical results that the number of terms required to get accurate results depends primarily on the spatial distribution of the excitation15,16 and not on the frequency. At zero frequency, the cubic equations for ðpÞ2

ðpÞ2

ðeÞ2

ðeÞ2

kzms and krms , and the quadratic equations for kzms , krms , and ðeÞ2

Krms reduce to linear equations and the static form13,14 of the solutions to the governing equations is to be used. When the ðpÞ frequency is low and the value of m is high, the values of kzms , ðpÞ ðeÞ ðeÞ ðeÞ krms , kzms , krms , and Krms are nearly independent of s. Therefore, the ½F matrix is ill-conditioned. However, in most cases, it is sufficient15 to use small values of Mr, Mz, and Mq to obtain accurate results. Each row of the ½F matrix is normalized and Gauss elimination is used to solve the equations. Alternative complete sets of functions16 are used if necessary.

respectively, and the Lame’s constants are k ffi 51.813 GPa and l ffi 26.692 GPa. In all the transducers, both elastic and piezoceramic cylinders are of radius 5 mm; and the bottom steel cylinder and the piezoceramic cylinder with losses are of length 10 mm. In Transducer 1, the piezoelectric cylinder is sandwiched between two elastic steel cylinders. All three cylinders have the same length, 10 mm. In Transducers 2 and 3, the lengths of the aluminum head are 10 and 5 mm, respectively. Detailed analyses of the cylinders in Transducer 1 are presented in Refs. 15, 16, and 23. For each transducer, the critical frequencies and the admittances at these frequencies are computed in the neighborhoods of the first three resonances. The frequency fs at which the conductance G reaches a local maximum; the value of G at this frequency, Gmax; the frequencies f1/2s and f1/2s at which the susceptance B reaches a local maximum and minimum, respectively; and the values of B, Bmax, and Bmin, at these frequencies, are computed using ATILA and the present method, and shown in tables. The percentage error, computed by treating the ATILA value as the reference, is also shown. The conductance and susceptance are computed at intervals of 1 kHz and shown in the figures. Values computed using the present method are shown using dots and those computed using ATILA are shown using a solid line.

IV. RESULTS AND DISCUSSIONS

Numerical results are presented for three different Langevin transducers and are compared with those computed using ATILA to illustrate the accuracy of the analytical model. Values of certain critical frequencies, the input electrical admittance, displacement, and stresses are presented. In ATILA, second order, axisymmetric, rectangular, equi-sized elements are used. The results are computed using (I, J) elements in the transducer: I elements in the radial direction and J elements in the axial direction. The number of finite elements that is necessary for accurate analysis depends on the frequency and spatial distribution of the excitation and is determined through convergence studies. Convergence is illustrated for one of the transducers. All the ATILA results presented are computed with I ¼ 20 and J ¼ 120 unless specified. The properties of the PZT4 piezoceramic cylinder with internal losses used in the present analysis are presented in Table III of Ref. 23. The Young’s modulus Y, Poisson’s ratio, and density q, of the steel cylinder are 200 GPa, 0.3, and 7800 kg/m3, respectively, and the Lame’s constants are k ffi 115.38 GPa and l ffi 76.923 GPa. Corresponding values for the aluminum cylinder are 71 GPa, 0.33, and 2700 kg/m3,

A. Transducer 1

All analytical results are computed using Mr ¼ Mz ¼ Mq ¼ 10. The critical frequencies and admittances at these frequencies are shown in Table I. The critical frequencies in the neighborhoods of the first three resonances are in good agreement with those computed using ATILA. The maximum absolute percentage error in the critical frequencies is 0.02%. The maximum error of 1.9% in the admittance occurs at the third resonance frequency. At the first resonance frequency of 55.65 kHz, the displacement in the transducer is primarily along the axial direction. The magnitude of the axial displacement has a nodal plane at the center of the transducer because the transducer is symmetric. The second resonance frequency occurs at approximately (and not exactly) three times the first resonance frequency because the transducer is not homogeneous and the overall length to diameter ratio is three. The displacement is primarily axial at this frequency also. The magnitude of the axial displacement has one nodal plane at the center and two other nodal planes within the head and tail sections. At the third

TABLE I. Critical frequencies, G, and B of Transducer 1. Resonance First

Method

fs (kHz)

Gmax (mS)

f-1/2s (kHz)

Bmax (mS)

f-1/2s (kHz)

Bmin (mS)

ATILA

55.65 55.66 0.02 205.61 205.65 0.02 269.68 269.71 0.01

2.88 2.88 0 5.04 5.02 0.39 0.0162 0.0159 1.9

55.54 55.54 0 205.49 205.52 0.01 269.26 269.29 0.01

1.45 1.45 0 2.58 2.57 0.39 0.0789 0.0788 0.13

55.76 55.77 0.02 205.74 205.77 0.01 270.08 270.11 0.01

1.43 1.43 0 2.46 2.45 0.41 0.0634 0.0636 0.32

Analytical % Error Second

ATILA

Analytical % Error Third

ATILA

Analytical % Error

1166

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

P. A. Nishamol and D. D. Ebenezer.: Langevin transducer

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FIG. 2. (Color online) (a) Conductance and (b) susceptance of Transducer 1. Solid line: ATILA; dots: Present method.

resonance frequency, there are five nodal regions that are approximately parallel to the flat ends of the transducer. Three of them are within the piezoceramic (PZT) cylinder and two are in the steel cylinders. In addition, the axial displacement is nearly zero at r ¼ 5 mm near the interfaces between the cylinders and at r ¼ 4 mm near the center of the transducer. The conductance and susceptance are shown in Fig. 2 up to 500 kHz. The agreement between the results obtained using the present method and ATILA is very good in the entire band. It can also be seen that the present method predicts all the peaks in G. Very narrow resonances, if present, will not be seen in Fig. 2 because the frequency resolution is 1 kHz. In the tables, the frequency resolution is 10 Hz. Therefore, the maxima and minima in the admittance in Table I are not the same as those in Fig. 2. B. Transducer 2

Transducer 2 is the same as Transducer 1 in size. However, it has an aluminum head. All analytical results are computed using Mr ¼ Mz ¼ Mq ¼ 10. The critical frequencies, conductance, and susceptance for this transducer are shown in Table II for the first three resonances. The maximum absolute percentage error in the critical frequencies is 0.02%. The maximum error of 0.87% in the admittance occurs at the second resonance frequency. The second and third resonance frequencies are approximately equal to two and three times, respectively, the first resonance frequency. At the first, second, and third resonance frequencies, there are one, two, and three nodal planes that are

very nearly parallel to the flat ends. The displacement is primarily axial at all three resonances. The conductance and susceptance are shown in Fig. 3 up to 500 kHz. The agreement between the results obtained using the present method and ATILA is very good in the entire band. The convergence of results obtained using ATILA and the present method is shown in Table III. The conductance and susceptance are computed using ATILA at a few frequencies (100,200,…,500 kHz) for different mesh densities and are compared with those computed analytically for different Mr, Mz, and Mq values. ATILA results are shown for (I, J) ¼ (10, 60), (20, 120), and (40, 240). The values of G and B are shown up to four significant digits. In ATILA, on comparing results for (I, J) ¼ (20, 120) and (I, J) ¼ (40, 240), it is seen that G and B have converged to three significant digits. In the present method, in all cases, increasing the value of Mr ¼ Mz ¼ Mq results in monotonic convergence. It is seen from Table III that the values of G and B obtained using (I, J) ¼ (40, 240) and Mr ¼ Mz ¼ Mq ¼ 15 are the same up to three significant digits in several cases. The real and imaginary parts of U and Tzz, at the first resonance frequency, computed using ATILA and the present method, are shown in Fig. 4. The analytical results are computed using Mr ¼ Mz ¼ Mq ¼ 5. The axial displacement, U, is shown at the center of the piezoceramic cylinder, as a function of the radial coordinate, in Figs. 4(a) and 4(b). The corresponding normal component of stress, Tzz, is shown in Figs. 4(c) and 4(d). The real and imaginary parts of U and Tzz are nearly uniform at this frequency, and the same small variations are seen in the results obtained using the two methods. The

TABLE II. Critical frequencies, G, and B of Transducer 2. Resonance First

Method

fs (kHz)

Gmax (mS)

f1/2s (kHz)

Bmax (mS)

fþ1/2s (kHz)

Bmin (mS)

ATILA

67.00 67.00 0 146.53 146.54 0.01 198.67 198.71 0.02

3.51 3.51 0 0.230 0.228 0.87 5.03 5.02 0.2

66.86 66.87 0.01 146.46 146.47 0.01 198.53 198.57 0.02

1.77 1.77 0 0.165 0.164 0.61 2.57 2.57 0

67.13 67.13 0 146.61 146.62 0.01 198.82 198.86 0.02

1.74 1.74 0 0.0647 0.0639 1.2 2.46 2.45 0.41

Analytical % Error Second

ATILA

Analytical % Error Third

ATILA

Analytical % Error

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

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FIG. 3. (Color online) (a) Conductance and (b) susceptance of Transducer 2. Solid line: ATILA; dots: Present method. TABLE III. Convergence studies for Transducer 2. Present method

ATILA

G (lS) (I, J) ¼ (10,60)

(I, J) ¼ (20,120)

(I, J) ¼ (40,240)

Mr ¼ Mz ¼ Mq ¼ 5

Mr ¼ Mz ¼ Mq ¼ 10

Mr ¼ Mz ¼ Mq ¼ 15

100 200 300 400 500

1.148  101 6.186  101 1.258 4.907  101 1.940

1.148  101 6.177  101 1.248 4.909  101 1.997

1.148  101 6.175  101 1.248 4.911  101 2.004

1.148  101 7.479  101 3.039 4.909  101 1.667

1.148  101 6.553  101 1.499 4.907  101 1.876

1.148  101 6.449  101 1.355 4.907  101 1.907

100 200 300 400 500

0.01946  0.4952 0.07812 0.1099 0.1616

0.01947 0.4948 0.07816 0.1099 0.1619

0.01947 0.4948 0.0782 0.1099 0.1619

0.01946 0.5106 0.07768 0.1099 0.1612

0.01947 0.5063 0.07782 0.1099 0.1614

Frequency (kHz)

B (mS) 0.01946 0.5481 0.07632 0.1099 0.1601

FIG. 4. (Color online) Axial displacement (U) and normal stress along the axis (Tzz) at different locations of Transducer 2 at its first resonance. Solid line: ATILA; dots: Present method. 1168

J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

P. A. Nishamol and D. D. Ebenezer.: Langevin transducer

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TABLE IV. Critical frequencies, G, and B of Transducer 3. Resonance First

Method

fs (kHz)

Gmax (mS)

f1/2s (kHz)

Bmax (mS)

fþ1/2s (kHz)

Bmin (mS)

ATILA

76.65 76.66 0.01 179.08 179.08 0 228.21 228.14 0.03

3.96 3.96 0 2.52 2.51 0.4 0.459 0.483 5.2

76.49 76.51 0.03 178.92 178.93 0.01 227.95 227.88 0.03

2.00 2.00 0 1.31 1.31 0 0.287 0.299 4.2

76.81 76.82 0.01 179.23 179.23 0 228.48 228.41 0.03

1.96 1.96 0 1.21 1.20 0.8 0.171 0.183 7.0

Analytical % Error Second

ATILA

Analytical % Error Third

ATILA

Analytical % Error

normal axial stress, on the axis of the transducer, is shown in Figs. 4(e) and 4(f). It is, as expected, zero on the flat ends. C. Transducer 3

In Transducer 3, the piezoceramic cylinder is placed in between an aluminum head of 5 mm length and a steel tail of 10 mm length. The analytical results are computed with Mr ¼ Mz ¼ Mq ¼ 5. The critical frequencies of this transducer around the first three resonances are shown in Table IV. The maximum percentage error in the critical frequencies is 0.03% and that in the associated G and B values is less than 7%. The displacement is primarily axial at all three resonances. The distribution of the magnitude of the axial displacement is studied. At the first resonance frequency it has a nodal plane, parallel

to the ends of the transducer, near the center of the transducer even though the transducer is not axially symmetric. On the top flat surface (see Fig. 1), jUj is twice that at the bottom flat surface. At the second resonance frequency it has two nodal planes that are parallel to the flat ends. One is in the PZT cylinder near the PZTAl interface and the other is in the steel cylinder. At the third resonance frequency it has three nodal regions that are nearly parallel to the flat ends. The first is near the PZT-Al interface, the second is near the center of the transducer, and the third is in the steel cylinder. In addition, it is nearly zero close to the curved surface of the part of the PZT cylinder that is close to the Al cylinder. The conductance and susceptance are shown in Fig. 5 up to 250 kHz with 1 kHz resolution. The axial displacement, U, at r ¼ 0 and z ¼ Lð3Þ =2, is shown in Fig. 6. The analytical

FIG. 5. (Color online) (a) Conductance and (b) Susceptance of Transducer 3. Solid line: ATILA; dots: Present method.

FIG. 6. (Color online) (a) Real and (b) imaginary parts of the axial displacement at r ¼ 0 at the top interface of Transducer 3. Solid line: ATILA and dots: Present method. J. Acoust. Soc. Am., Vol. 135, No. 3, March 2014

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results are in good agreement with those computed using ATILA in all the cases. V. CONCLUSIONS

An analytical method is presented to analyze the response of a Langevin transducer with arbitrary length to diameter ratio to arbitrary mechanical and electrical excitations. Series solutions are used and each term in the series is an exact solution to the exact governing equations. The displacement, potential, stress, and electric displacement fields are expressed in terms of complete sets of functions. Arbitrary boundary conditions and internal losses are included in the model and these are significant advantages. Numerical values of the complex input electrical admittance and the complex displacement are computed and compared with finite element results. The error is less than 1% in most of the cases. The frequency, fs, at which the conductance G reaches a local maximum Gmax, and the frequencies f1/2s and f1/2s at which the susceptance B reaches a local maximum Bmax and minimum Bmin, respectively, are all in excellent agreement with ATILA. These frequencies are of interest in the design of transducers and in their characterization. Numerical results are presented for transducers with three cylinders but can also be obtained for transducers with any number of cylinders. When the number of terms is large, the ½F matrix may be ill-conditioned even if each row is normalized. However, increasing the number of terms in the series results in monotonic convergence. Results obtained with a different number of terms in the series can be used to estimate whether convergence is taking place and the results need not be compared with those obtained using some other method. However, as illustrated, a few terms are enough for some practical applications. The method can be extended to analyze transducers with a stack of piezoceramic rings. It can further be extended to analyze Tonpilz transducers with conical heads. Further extensions are required to include the effect of fluid loading and to analyze an array of Tonpilz transducers. The method can also be adapted to analyze transformers, ultrasonic motors, and other devices. ACKNOWLEDGMENT

The facilities provided by the Director, NPOL are gratefully acknowledged. 1

C. H. Sherman and J. L. Butler, Transducers and Arrays for Underwater Sound (Springer, New York, 2007), pp. 5–6, 517. 2 ATILA user’s manual, Version 5.1.1 (Acoustics Laboratory, ISEN, Lille Cedex, France, 1997). 3 C. Chree, “Longitudinal vibrations of a circular bar,” Q. J. Pure Appl. Math. 21, 287–298 (1886). 4 A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 2nd ed. (Cambridge University Press, London, 1906), Art 201, pp. 276–278. 5 L. N. G. Filon, “On the elastic equilibrium of circular cylinders under certain practical systems of load,” Proc. R. Soc. London, Ser. A 198, 147–233 (1902). 6 F. Purser, “On the application of Bessel’s functions to the elastic equilibrium of a homogeneous isotropic cylinder,” Trans. R. Irish Acad. A 32, 31–60 (1902). 7 V. V. Meleshko, “Selected topics in the history of the two-dimensional biharmonic problem,” Appl. Mech. Rev. 56(1), 33–85 (2003).

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P. A. Nishamol and D. D. Ebenezer.: Langevin transducer

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