Vibration Characteristics of a Circular Cylindrical Ceramic Tube ...

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

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Correspondence Vibration Characteristics of a Circular Cylindrical Ceramic Tube Piezoelectric Transducer with Helical Electrodes Limei Xu, Feng Yang, Yuantai Hu, Hui Fan, and Jiashi Yang Abstract—We develop a 1-D mechanics model for coupled extensional and torsional vibrations of a circular cylindrical ceramic tube piezoelectric transducer with helical electrodes. A theoretical analysis of the electrically forced vibration of the tube is performed. Basic vibration characteristics of resonant frequencies, modes, and admittance are calculated and examined. The optimal orientation of the electrodes for exciting torsional modes is determined.

I. Introduction

cylindrical ceramic tube in a simple and effective manner still remains an interesting problem. An original idea for generating torsion was proposed by R. Adler [13]. In this patent, it was suggested that torsion in a circular cylindrical ceramic tube can be generated using helical electrodes as shown in Fig. 1. It was also pointed out in [13] that the same electrodes could be used for poling the ceramic tube and exciting torsional modes, and helical electrodes at 45° were recommended. In this paper, to further explore the idea in [13], we derive a theoretical model to quantitatively study the behavior of a circular cylindrical ceramic tube with helical electrodes. We are primarily interested in the optimal excitation of torsion. Although extensional motion often accompanies torsional motion in the ceramic tube we analyze, this paper is not a full parametric study with equal attention to torsion and extension.

T

orsional modes of circular cylindrical rods and tubes are widely used as the operating modes for acoustic wave devices. Examples are transformers [1], [2], angular rate sensors [3], fluid sensors [4], power harvesters [5], and various electromechanical transducers [6]–[10]. The extension of helical spring transducers [11], [12] is also closely related to torsion because the global extension of a spring is, in fact, the superposition of local torsions of differential spring elements. Because torsion at structural level is related to local shear, torsional modes offer the option of using the shear piezoelectric constant e15 instead of e33 for extensional modes. When a circular rod is in torsion, the surface displacement is tangential only. This is ideal for fluid sensing in which the fluid viscosity produces a drag force only, without generating compressional waves in the fluid. Usually, to excite torsional modes in a circular cylindrical ceramic tube, circumferential or tangential poling is needed [1]–[10]. With circumferential poling, torsion can be produced by an axial electric field. From a practical point of view, it is not easy to create circumferential poling in a circular cylindrical ceramic tube. In addition, applying a reasonably strong axial electric field to excite torsion in a circumferentially poled ceramic tube requires a relatively high voltage across the 2 ends of the tube. Therefore, how to excite torsional motion in a circular

Manuscript received October 10, 2008; accepted August 11, 2009. L. Xu is with the Institute of Astronautics and Aeronautics, University of Electronic Science and Technology, Chengdu, China. F. Yang, Y. Hu, and J. Yang are with the School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan, Hubei, China. J. Yang is also with the Department of Engineering Mechanics, University of Nebraska, Lincoln, NE (e-mail: [email protected]). H. Fan is with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Republic of Singapore. Digital Object Identifier 10.1109/TUFFC.2009.1347 0885–3010/$25.00

II. Structure Consider the circular cylindrical ceramic tube in Fig. 1 with length 2a, radius R, and thickness h. We assume a long and thin tube with a ≫ R ≫ h. There are 2 helical electrodes around the tube, shown by the solid and dotted lines. The width of the electrodes is very small and is negligible. The depth of the electrodes is the same as the tube thickness h. During one full turn of an electrode around the tube, the electrode advances a distance 2λ in the axial direction of the tube (the lead or pitch). A spiral electrode on a circular cylindrical surface is mathematically represented by

X = R cos g, Y = R sin g, 2l Z = g, 2p

(1)

where R is the radius of the circular cylindrical surface, and γ is an angular parameter. When γ varies, (1) describes a spiral curve. The exact range of γ will be determined later. The perpendicular distance between the 2 electrodes is d. On the 2 electrodes, electric potentials of ±V are applied, indicated by plus and minus signs in Fig. 1. The poling direction is represented by alternating arrows labeled with P. The direction of P is described by θ. We have tanθ = (2πR)/(2λ) and d = λsinθ. We consider a tube situated within −a < Z < a whose length is determined by m turns of an electrode and is given by 2a = m(2λ), where m is a positive integer. The range of the parameter γ is −mπ < γ < mπ. In solid mechanics [14], it is well known that in torsion, while a square differential element with its 4 sides along the “1” and “3” axes in Fig. 1 is in pure shear, a square

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S 1 = u ,1,

vol. 56, no. 11,

November

S 5 = Ry ,1.

2009

(2)

The electric field is dictated by the electrode configuration and the applied voltage. They are piecewise constant:

Fig. 1. A circular cylindrical ceramic tube with helical electrodes.

differential element at θ = 45° is in simultaneous and equal tension and compression in perpendicular directions. This roughly suggests that spiral electrodes at θ = 45° [13] may be effective in exciting torsion because they can produce tension (or compression) at θ = 45°. However, although the helical electrodes and the poling in Fig. 1 can produce tension (or compression) in the direction perpendicular to the electrodes through the piezoelectric constant in the poling direction (denoted by eˆ 33 in this paper), they cannot produce a simultaneous compression (or tension) exactly in the perpendicular direction through eˆ 31. Therefore, we expect that both torsion and extension will be excited in the tube under V, and the most effective excitation of torsion will not be by electrodes exactly at θ = 45°. One of the main goals of the present paper is to determine θ for the most effective excitation of torsion. III. One-Dimensional Equations We follow [15], [16] for the notation of the linear theory of piezoelectricity: u is the mechanical displacement vector; T is the stress tensor; S is the strain tensor; E is the electric field; D is the electric displacement; ϕ is the electric potential; ρ is the mass density; and cijkl, eijk, and εij are the elastic, piezoelectric, and dielectric constants, respectively. Under the compact matrix notation [15], [16], the material tensors cijkl, eijk, and εij can be represented by matrices. Consider polarized ceramics in 2 coordinate systems: (x, y, z) and (x1, x2, x3). The (x, y, z) system is the usual system in which the z axis is along the poling direction and the material constants in this system are denoted by cˆ pq, eˆip, and eˆij . The (x1, x2, x3) system is obtained by rotating the (x, y, z) system about the y axis by an angle θ, counterclockwise when viewed from the positive y direction. The material constants in the (x1, x2, x3) system are denoted by cpq, eip, and εij. The relations between the 2 sets of material constants are determined by tensor transformations and can be found in [17]. For a long and thin tube, 1-D models for extension and torsion can be constructed [14]. Let u(x1, t) be the extensional displacement and ψ(x1, t) be the angle of rotation for torsion [14]. Then the extensional strain and the shear strain related to torsion are given by [14]:

E1 = ±

2V sin q, d

E3 = ±

2V cos q. d

(3)

E1 and E3 change sign when crossing an electrode. Eq. (3) is under the assumption that d is not large. The electric field changes its direction when crossing an electrode. The piezoelectric constants do the same because of the alternating poling directions. Therefore, as a product of the electric field and the piezoelectric constants, the induced strain is uniform in the tube. The relevant constitutive relations for the 3 normal stresses related to extension and the shear stress T5 for torsion are T 1 = c 11S 1 + c 12S 2 + c 13S 3 + c 15S 5 - e 11E 1 - e 31E 3, (4a) T 2 = c 21S 1 + c 22S 2 + c 23S 3 + c 25S 5 - e 12E 1 - e 32E 3, (4b) T 3 = c 31S 1 + c 32S 2 + c 33S 3 + c 35S 5 - e 13E 1 - e 33E 3, (4c) T 5 = c 51S 1 + c 52S 2 + c 53S 3 + c 55S 5 - e 15E 1 - e 35E 3. (4d) The relevant electric displacement components are D 1 = e 11S 1 + e 12S 2 + e 13S 3 + e 15S 5 + e 11E 1 + e 13E 3, (5a) D 3 = e 31S 1 + e 32S 2 + e 33S 3 + e 35S 5 + e 31E 1 + e 33E 3. (5b) The electric displacement changes its direction or sign when crossing an electrode because the piezoelectric constants and the electric field do so. The major stress components are T1 for extension and T5 for torsion [14]. Therefore, we set T2 and T3 in (4b) and (4c) to zero (the usual stress relaxation to account for Poisson’s effect accompanying extension). This results in 2 equations from which we solve for S2 and S3 and substitute the resulting expressions into (4a), (4d), and (5). Thus, we obtain the following relaxed constitutive relations for extension and torsion of thin tubes:

T 1 = c 11S 1 + c 15S 5 - e 11E 1 - e 31E 3,

(6a)

T 5 = c 51S 1 + c 55S 5 - e 15E 1 - e 35E 3,

(6b)

D 1 = e 11S 1 + e 15S 5 + e 11E 1 + e 13E 3,

(6c)

D 3 = e 31S 1 + e 35S 5 + e 31E 1 + e 33E 3,

(6d)

where c pq are the relaxed elastic constants whose expressions can be obtained in a straightforward manner from

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2 é p ê m p(e E + e E ) sin q + m p(e E + e E ) cos q + m) + e 15R)C (m) sin(k (m)ml) sin q (e 11b (m 11 1 13 3 31 1 33 3 ê l êë m =1 ù p (e 31b (m) + e 35R)C (m) sin(k (m)ml) cos q úú . + l úû m =1 (16)

l2 Q e = 4h R 2 + 2 p 2

å

å

the above stress relaxation procedure, but they are somewhat lengthy and therefore are not provided here. The total extensional force N and the torque M over a cross section of the tube are calculated from N =

òAT 1dA = A(c 11u ,1 + c 15Ry ,1 - e 11E 1 - e 31E 3),

(7a)

M =

òAT 5RdA = AR(c 51u ,1 + c 55Ry ,1 - e 15E 1 - e 35E 3), (7b)

where A = 2πRh is the cross-sectional area of the tube. The equations of motion for extension and torsion are

N ,1 = rAu,

(8a)

M ,1 = rJ y,

(8b)

where J = 2πR3h = AR2 is the polar moment of inertia of the tube cross section about its center. Substitution of (7) into (8) gives the governing equations for u and ψ:

c 11u ,11 + c 15Ry ,11 = ru,

(9a)

c 51u ,11 + c 55Ry ,11 = rRy.

(9b)

For end conditions we may prescribe u or N, and ψ or M [14]. The total charge on an electrode is calculated from:

Qe =

ò L [D n ]hdL = ò L [D 1 sin q + D 3 cos q]hdL, (10) [(dX)2

where L is the length of the electrode; dL = + (dY)2 + (dZ)2]1/2, where dX, dY, and dZ are calculated from (1); Dn is the component of the electric displacement normal to the electrode; and [Dn] is the jump of Dn across the electrode.

IV. Electrically Forced Vibration Having derived the 1-D equations, we study the electrically forced vibration of a tube in this section. We consider the most common case when both ends of the tube are free. The boundary conditions are:

N (-a) = N (a) = 0,

(11a)

M (-a) = M (a) = 0.

(11b)

The applied voltage is V = V exp(iwt), where V and ω are constants; ω is the driving frequency and i is the imaginary unit. We use the usual complex notation. All fields are with the same time dependence and the exp(iωt) factor will be dropped. Because of the symmetry involved, u and ψ are odd functions of x1. Therefore, we let

u = B sin kx 1,

y = C sin kx 1,

(12)

where B, C, and k are undetermined constants. Substituting (12) into (9), we obtain 2 homogeneous equations for B and C:

(c 11k 2 - rw 2)B + c 15RCk 2 = 0,

(13a)

c 51Bk 2 + (c 55Rk 2 - rRw 2)C = 0.

(13b)

For nontrivial solutions of B and C, the determinant of the coefficient matrix of (13) must vanish. This yields a quadratic equation for k2. We denote the 2 roots of k2 by (k(1))2 and (k(2))2, and the corresponding amplitude ratios by

B c 15R(k (m)) 2 == b (m), C c 11(k (m)) 2 - rw 2

m = 1, 2.

(14)

Then the general solution with the desired symmetry can be constructed as 2

u =

å C (m)b (m) sin k (m)x 1,

m =1 2

y=

(15)

å C (m) sin k (m)x 1,

m =1

where C(1) and C(2) are undetermined constants. With (15), we apply the boundary conditions in (11). Because of symmetry, we only need to consider the 2 boundary conditions at x1 = a, which lead to 2 equations for C(1) and C(2). Once C(1) and C(2) are obtained, we can calculate D1 and D3 from (6c) and (6d). Then, from (10), we have the charge on an electrode as given by (16), see above. The current is given by I = Q e = iwQ e . The admittance Y of the transducer is calculated from Y = I/2V. We want to identify the nature of the modes excited, i.e., whether they are extensional or torsional and how much these 2 types of deformation are mixed. For this purpose, as a


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Fig. 2. (a) r versus frequency and θ. (b) 1/r versus frequency and θ.

measure, we calculate the ratio r between the average of the square of the extensional strain and the average of the square of the torsional shear strain

r =

1 2pRh(2a) 1 2pRh(2a)

a a

ò -a

a

2

ò -a S 1 2pRhdx 1 S 522pRhdx 1

=

ò -a (u ,1) a

ò -a

2

dx 1

(Ry ,1) 2dx 1

. (17)

The square of a strain component is proportional to the strain energy associated with the component. Therefore, (17) is proportional to the energy ratio between extension and torsion, which is sufficient for our purpose. V. Numerical Results As an example, consider a tube made from polarized ceramics PZT-5H [18]. Damping is introduced by allowing the elastic material constants to assume complex values, which can represent viscous damping in the material. In

Fig. 3. (a) Admittance versus frequency, (b) r versus frequency, and (c) 1/r versus frequency. All plotted for θ = 36°.

xu et al.: cylindrical ceramic tube piezoelectric transducer

our calculations, cˆ pq are replaced by cˆ pq(1 + iQ -1), where Q is a large, real, and positive number. For polarized ceramics, the value of Q is of the order of 102 to 103. We fix Q = 100. For geometric parameters, we choose a = 10 cm, R = 0.5 cm and h = 0.05 cm so that a ≫ R ≫ h is satisfied. We plot r and 1/r versus both the frequency f and θ in Figs. 2(a) and (b). It can be seen that r and 1/r are sensitive to both f and θ. The peaks in Fig. 2(a) or (b) show extension or torsion dominated resonances. Fig. 2(b) shows that θ = 45° does not have the largest 1/r for the strongest torsion. Numerical tests show that when θ = 36°, 1/r assumes its maximum with the largest torsional component. We plot this special case in Fig. 3. VI. Conclusion A 1-D model was established for a ceramic tube with helical electrodes. A theoretical analysis was performed. The results showed that a pair of helical electrodes can be used to pole a circular cylindrical ceramic tube into portions with alternating poling. When an alternating voltage is applied across the helical electrodes, the ceramic tube can be driven into coupled torsional and extensional motions. The ratio between the extensional component and the torsional component is sensitive to the poling direction θ. When θ is near 36° the torsional component is particularly strong. This offers a new way for exciting torsional modes effectively in a tube. Acknowledgment We thank the authors of [17] for its availability before it has been published.

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References [1] H. W. Schafft, “Piezoelectric transformer,” U.S. patent, 3 487 239, Dec. 30, 1969. [2] D. A. Berlincourt and L. S. Silker, “Piezoelectric transformer,” U.S. patent, 3 736 446, May 29, 1973. [3] J. S. Yang, H. Y. Fang, and Q. Jiang, “A vibrating piezoelectric ceramic shell as a rotation sensor,” Smart Mater. Struct., vol. 9, pp. 445–451, 2000. [4] J. S. Yang, Analysis of Piezoelectric Devices. Singapore: World Scientific, 2006. [5] Z. G. Chen, Y. T. Hu, and J. S. Yang, “Piezoelectric generator based on torsional modes for power harvesting from angular vibrations,” Appl. Math. Mech., vol. 28, no. 6, pp. 779–784, 2007. [6] R. A. Langevin, “The electro-acoustic sensitivity of cylindrical ceramic tubes,” J. Acoust. Soc. Am., vol. 26, no. 1, pp. 421–427, 1954. [7] C. A. Rosen, K. A. Fish, and H. C. Rothenberg, “Electromechanical transducer,” U.S. patent, 2 830 274, Apr. 8, 1958. [8] C. A. Rosen, “Electromechanical transducer,” U. S. patent, 2 974 296, Mar. 7, 1961. [9] S. Y. Lin, “Thickness shearing vibration of the tangentially polarized piezoelectric ceramic thin circular ring,” J. Acoust. Soc. Am., vol. 107, no. 5, pp. 2487–2492, 2000. [10] S. Y. Lin, “Torsional vibration of coaxially segmented, tangentially polarized piezoelectric ceramic tubes,” J. Acoust. Soc. Am., vol. 99, no. 6, pp. 3476–3480, 1996. [11] B. L. Jiao and J. D. Zhang, “Torsional modes in piezoelectric helical springs,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 46, no. 1, pp. 147–151, 1999. [12] B. L. Jiao, “Investigation on piezoelectric helix for use as a hydrophone,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 46, no. 6, pp. 1446–1449, 1999. [13] R. Adler, “Torsional ceramic transducer,” US patent, 3 900 748, Aug. 19, 1975. [14] S. Timoshenko and J. Gere, Mechanics of Materials. Boston, MA: PWS Publishing Co., 1997. [15] H. F. Tiersten, Linear Piezoelectric Plate Vibrations. New York: Plenum, 1969. [16] IEEE Standard on Piezoelectricity, ANSI/IEEE Std 176–1987, 1987. [17] H. F. Zhang, J. A. Turner, and J. A. Kosinski, “Analysis of thickness vibrations of c-axis inclined aluminum-nitrogen thin film resonators,” to be submitted. [18] B. A. Auld, Acoustic Fields and Waves in Solids, vol. 1. New York: Wiley, 1973, pp. 357–382.