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Fig. 1. PZT-4 cylinder with short-circuit lateral surface (m = 1) Wave Propagation in Materials for Modern Applications

Figure 1 PZT-4 cylinder with short-circuit lateral surface (m = 1) Wave Propagation in Materials for Modern Applications

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Further we will assume that the outer cylindrical surface is free, hence, using (8a), (9a) and (10a) the boundary conditions are The Euler-Lagrange equation could be also presented in the standard form of the Navier equations of the linear dynamical elasticity (Redwood, 1960) and the Gauss law in the cylindrical coordinates:
Further we will assume that the outer cylindrical surface is free, hence, using (8a), (9a) and (10a) the boundary conditions are The Euler-Lagrange equation could be also presented in the standard form of the Navier equations of the linear dynamical elasticity (Redwood, 1960) and the Gauss law in the cylindrical coordinates:
We seek the solution of the problem (14) - (15) in terms of harmonic travelling waves along Z - axis in terms of several displacement and electric potentials first introduced by (Mirsky, 1964) and further used by (Winkel et al 1995) and (Shatalov et al., 2009) as follows:
We seek the solution of the problem (14) - (15) in terms of harmonic travelling waves along Z - axis in terms of several displacement and electric potentials first introduced by (Mirsky, 1964) and further used by (Winkel et al 1995) and (Shatalov et al., 2009) as follows:
In this case system of equations (14) is rewritten as follows: 4. Axisymmetric case
In this case system of equations (14) is rewritten as follows: 4. Axisymmetric case
Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material
Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material
Fig. 3. PZT-4 cylinder with reduced electro-mechanical coupling eee ee eee fe Eee ee ar 2 t follows from Fig. 1 - 5 that the first fundamental mode of the bending waves is no ensitive to the nature of the electric boundary conditions on the lateral cylindrical surface ‘urthermore it is practically not sensitive to the measure of electro-mechanical coupling o he material. The higher order modes are more sensitive to the nature of the electri oundary condition as well as to the measure of the electro-mechanical cross-coupling. Fig } - 5 shows that dispersion curves differ quite substantially from the curves in Fig. 1 and 2 This difference is explained mainly by the factor that the electro-mechanical couplin; efficients of PZT-7A are less than the corresponding factors of PZT-4. It is reflected in indulating behaviour of the propagating higher modes as well as the values of the cut-of requencies. The PZT-4 cylinder with short-circuit lateral surface demonstrates a negativ lope of the fourth branch in a quite broad range of wavenumbers (Fig. 1). Substantia
Fig. 3. PZT-4 cylinder with reduced electro-mechanical coupling eee ee eee fe Eee ee ar 2 t follows from Fig. 1 - 5 that the first fundamental mode of the bending waves is no ensitive to the nature of the electric boundary conditions on the lateral cylindrical surface ‘urthermore it is practically not sensitive to the measure of electro-mechanical coupling o he material. The higher order modes are more sensitive to the nature of the electri oundary condition as well as to the measure of the electro-mechanical cross-coupling. Fig } - 5 shows that dispersion curves differ quite substantially from the curves in Fig. 1 and 2 This difference is explained mainly by the factor that the electro-mechanical couplin; efficients of PZT-7A are less than the corresponding factors of PZT-4. It is reflected in indulating behaviour of the propagating higher modes as well as the values of the cut-of requencies. The PZT-4 cylinder with short-circuit lateral surface demonstrates a negativ lope of the fourth branch in a quite broad range of wavenumbers (Fig. 1). Substantia
Fig. 7. PZT4 cylinder with open-circuit lateral surface (m = 2)
Fig. 7. PZT4 cylinder with open-circuit lateral surface (m = 2)
Fig. 9. PZT7A cylinder with open-circuit lateral surface (m = 2)
Fig. 9. PZT7A cylinder with open-circuit lateral surface (m = 2)
Fig. 10. PZT-4 cylinder with short-circuit lateral surface (m = 3) Wave Propagation in Materials for Modern Applications
Fig. 10. PZT-4 cylinder with short-circuit lateral surface (m = 3) Wave Propagation in Materials for Modern Applications
Fig. 12. PZT-7A cylinder with short-circuit lateral surface (m = 3)
Fig. 12. PZT-7A cylinder with short-circuit lateral surface (m = 3)
Jispersion curves of axisymmetric waves in the cylinder made from PZT-4 and PZT-7A vith the short-circuit and free lateral (cylindrical) surface are depicted in Fig. 14 - 17 in the ame dimensionless coordinates (k-a+Q=o / (a-V.)), where V,=,/ci,/o, and a is the uter radius of the cylinder. The picture of the dispersion curves is obtained by a methoc lescribed above for real (propagating waves) and pure imaginary values of the vavenumber in the limits: Re(k-a),Im(k-a) €(0,8], Q=a/(a-V,) €(0,28] with resolutior 00 (250 pixels for real and 250 pixels for imaginary (k-a)) X 250 (Q=a@/(a-V,)) pixels.
Jispersion curves of axisymmetric waves in the cylinder made from PZT-4 and PZT-7A vith the short-circuit and free lateral (cylindrical) surface are depicted in Fig. 14 - 17 in the ame dimensionless coordinates (k-a+Q=o / (a-V.)), where V,=,/ci,/o, and a is the uter radius of the cylinder. The picture of the dispersion curves is obtained by a methoc lescribed above for real (propagating waves) and pure imaginary values of the vavenumber in the limits: Re(k-a),Im(k-a) €(0,8], Q=a/(a-V,) €(0,28] with resolutior 00 (250 pixels for real and 250 pixels for imaginary (k-a)) X 250 (Q=a@/(a-V,)) pixels.
Fig. 14. Axisymmetric dispesion curves in PZT-4 cylinder with short-circuit lateral surface
Fig. 14. Axisymmetric dispesion curves in PZT-4 cylinder with short-circuit lateral surface
Fig. 15. Axisymmetric dispesion curves in PZT-4 cylinder with open-circuit lateral surface Wave Propagation in Materials for Modern Applications
Fig. 15. Axisymmetric dispesion curves in PZT-4 cylinder with open-circuit lateral surface Wave Propagation in Materials for Modern Applications
Fig. 16. Axisymmetric dispesion curves in PZT-7A cylinder with short-circuit lateral surface
Fig. 16. Axisymmetric dispesion curves in PZT-7A cylinder with short-circuit lateral surface
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