J.D. Yu

A new two-dimensional theory for vibrations of piezoelectric crystal plates with electroded faces

A system of two-dimensional (2-D) governing equations for piezoelectric plates with general crystal symmetry and with electroded faces is deduced from the three-dimensional (3-D) equations of linear piezoelectricity by expansion in series of trigonometric functions of thickness coordinate. The essential difference of the present derivation from the earlier studies by trigonometrical series expansion is that the antisymmetric in-plane displacements induced by gradients of the bending deflection (the zero-order component of transverse displacement) are expressed by the linear functions of the thickness coordinate, and the rest of displacements are expanded in cosine series of the thickness coordinate. For the electric potential, a sine-series expansion is used for it is well suited for satisfying the electrical conditions at the faces covered with conductive electrodes. A system of approximate first-order equations is extracted from the infinite system of 2-D equations. Dispersion curves for thickness shear...

Piezoelectric ceramic disks with thickness-graded material properties

A system of two-dimensional, first-order equations for electroded piezoelectric crystal plates with general symmetry and thickness-graded material properties recently was deduced from the three-dimensional equations of linear piezoelectricity. These equations are simplified for the two limiting cases of thickness-graded piezoelectric properties, i.e., the homogeneous plate and bimorph of piezoelectric ceramics. Closed-form solutions are obtained from these reduced equations for the flexural and thickness-shear vibrations and static response of bimorph disks as well as for the extensional and thickness-stretch vibrations of homogeneous disks. Frequency spectra and modes are computed and examined. Resonance frequencies for both homogeneous and bimorph disks of PZT-857 are computed and measured. The comparison of the results shows that the agreement is close.

Extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoceramic disks

A set of two-dimensional (2-D), second-order approximate equations for extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoelectric ceramic plates with electroded faces is extracted from the infinite system of 2-D equations deduced previously. The new truncation procedure developed recently is used for it improves the accuracy of calculated dispersion curves. Closed-form solutions are obtained for free vibrations of circular disks of barium titanate. Dispersion curves calculated from the present approximate 2-D equations are compared with those obtained from the 3-D equations, and the predicted resonance frequencies are compared with experimental data. Both comparisons show good agreement without any corrections. The frequencies of the edge modes calculated from the present 2-D equations are very close to the experimental data. Furthermore, mode shapes at various frequencies are calculated in order to identify the frequency segments of the spectrum at which one of the coupled modes-i.e., the radial extension (R), edge mode (Eg), thickness-stretch (TSt), and symmetric thickness-shear (s.TSh)-is predominant.

Governing equations of piezoelectric plates with graded properties across the thickness

Two-dimensional governing equations for electroded piezoelectric plates with graded material properties across the thickness are derived from the three-dimensional equations of linear piezoelectricity by extending Mindlins power series expansion to the mechanical displacement, electric potential and the thickness-graded properties, i.e., the elastic stiffnesses, piezoelectric coefficients, dielectric permittivity, and mass density. The couplings of the extensional and flexural motions due to the graded material properties, in addition to those due to the anisotropy of material properties, are clearly disclosed in these newly derived equations which reduce to Mindlins equations of homogeneous nonpiezoelectric plates when the graded material properties contain only the zeroth-order terms in the power series expansion. Dispersion curves for both the homogeneous piezoelectric plates and piezoelectric bimorphs are calculated from the 3-D equations and from the present 2-D first-order equations, and the comparison shows that the agreement is very close for frequencies up to and including the fundamental thickness-shear frequencies. An electrode-plated piezoelectric ceramic cantilever plate with thickness-graded piezoelectric coefficients is studied in detail. Resonance frequencies, modes of vibrations and static responses are computed and discussed.

Stress sensitivity of electromagnetic resonances in circular dielectric disks

A formula for predicting the stress effect on the electromagnetic resonances of dielectric resonators is obtained by applying a perturbation method to the three‐dimensional Maxwell’s equations in which the dielectric permittivity tensor is perturbed by the applied stress (or strain) field through the piezo‐optic effect. The dielectric resonator, which is surrounded by infinite free space, can be isotropic or anisotropic and of arbitrary shape. By using previously obtained two‐dimensional closed‐form solutions as the approximate unperturbed solutions, stress effect on the electromagnetic resonances in a dielectric circular disk is studied. Frequency changes of both the transverse electric and transverse magnetic modes are computed for disks of gallium arsenide and under three cases of loading: (1) a pair of diametral forces, (2) steady vertical acceleration, and (3) steady horizontal acceleration. In the latter two cases, the bottom face of the disk is supported by a rigid base.

Effect of stress on guided EM waves in anisotropic dielectric plates

The changes in the frequencies or velocities of guided EM (electromagnetic) waves in infinite and anisotropic dielectric plates affected by uniform and thickness-dependent stress fields are studied. For the uniformly applied stresses, frequency changes as a function of wavenumber are computed based on exact solutions of the 3-D Maxwell equations. For the thickness-dependent stress fields, such as those caused by the steady accelerations in plates attached to a rigid base, frequency changes are computed based on the variational techniques and the Rayleigh-Ritz method. For numerical computations, a dielectric plate of lithium niobate with thickness 2b = 3.27 mm is considered. It is found that the maximum frequency shift is about (1 /spl sim/ 2) /spl times/ 10/sup -5/ for the uniformly applied stress of 1MPa. For steady acceleration on plates attached to a rigid base, the maximum frequency shift is about (3 /spl sim/ 15) /spl times/ 10/sup -10//g for 2b = 3.27 mm.<<ETX>>

A new 2-D theory for vibrations of piezoelectric crystal plates with electroded faces

A system of two-dimensional governing equations for piezoelectric plates with general crystal symmetry and with electroded faces are deduced from the three-dimensional equations of linear piezoelectricity by expansion in series of trigonometric functions of thickness coordinate. In the cosine-series expansion for the mechanical displacements, the antisymmetric in-plane displacements induced by the gradients of deflection of plate is separated from the rest terms and is expressed by a linear function of thickness coordinate. For the electric potential, a sine-series expansion is used for it is well suited for satisfying the electrical conditions at the faces covered with conductive electrodes. A system of approximate first-order equations is extracted from the infinite system of 2-D equations. Dispersion curves for thickness-shear, flexure, and face-shear modes varying along x/sub 1/ and those for thickness-twist and face-shear varying along x/sub 3/ are calculated for AT-cut quartz plates and they are compared very closely with the corresponding ones computed from the 3-D equations, without introducing any corrections. Predicted frequency spectra by the present equations are shown to agree closely with the experimental data by Koga and Fukuyo and that by Nakazawa, Koriuchi, and Ito.

Guided electromagnetic waves in anisotropic dielectric plates

Guided electromagnetic waves in an infinite dielectric plate with general crystal symmetry surrounded by free space are studied in terms of the three‐dimensional Maxwell’s equations. To exhibit as how the crystal symmetry may affect the propagation, symmetry, and coupling of the waves, the study is divided into four cases: (I) β11,β22,β33≠0; (II) β11,β22,β33,β12≠0; (III) β11,β22,β33,β23≠0; (IV) all βij≠0; where βij is the impermeability tensor referred to the rectangular axes xi with the x2 axis normal to the plate faces. Closed‐form solutions are obtained and then the dispersion relations and modes are computed and studied for each case. It is found that in case I, solutions can be separated into the transverse‐electric or TE waves and the transverse‐magnetic or TM waves; TE and TM waves can be further separated into the symmetric and antisymmetric waves. In case II, the solutions for the TE waves remain the same as those in the case I; however, TM waves cannot be separated into symmetric and antisymmetr...

Guided EM waves in anisotropic dielectric plates

Guided EM waves in an infinite dielectric plate with general crystal symmetry surrounded by free space are studied in terms of the three-dimensional Maxwells equations. To exhibit how the crystal symmetry affects the propagation and coupling of the waves, the study is divided into four cases: (I) beta /sub 11/, beta /sub 22/, beta /sub 33/ not=0; (II) beta /sub 11/, beta /sub 22/, beta /sub 33/, beta /sub 12/ not=0; (III) beta /sub 11/, beta /sub 2(inf)/ beta /sub 33/, beta /sub 23/ not=0; (IV) all beta /sub ij/ not=0; where beta /sub ij/ is the impermeability tensor referred to the rectangular axes x/sub i/ with the x/sub 2/ axis normal to the plate faces. For each case, closed-form solutions are obtained, and dispersion relations and modes are computed and studied. It is found that in case 1, solutions can be separated into the TE and TM waves. TE and TM waves can be further separated into the symmetric and antisymmetric waves. In case II, the solutions for the TE waves remain the same as those in case I. However, TM waves can not be separated into symmetric and antisymmetric waves. In case III, solutions cannot be separated into the TE and TM waves, but they can still be separated into the symmetric and antisymmetric waves. In case IV, solutions can neither be separated into the TE and TM waves nor into the symmetric and antisymmetric waves.<<ETX>>

Extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoelectric ceramic disks

A set of two-dimensional second-order approximate equations for extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoelectric ceramic plates with electroded faces is extracted from the infinite system of 2-D equations deduced in a previous paper by Lee, Yu, and Lin. The new truncation procedure developed in a recent paper by Lee and Edwards is employed for it improves the accuracy of calculated dispersion curves. Closed form solutions are obtained for free vibrations of circular disks of barium titanate. Dispersion curves calculated from the present approximate 2-D equations are compared with those from the three-dimensional equations, and the predicted resonance frequencies are compared with the experimental data by Shaw. Both comparisons show good agreement without any corrections. The frequencies of the edge modes calculated from the present 2-D equations are very close to the experimental data. Furthermore, mode shapes at various frequencies are calculated in order to identify the frequency segments of the spectrum at which one of the coupled modes, i.e., the radial extension (R), edge mode (Eg), thickness-stretch (TSt), and symmetric thickness-shear (s.TSh), is predominant.