Wave Propagation 2011 Part 13 pdf

Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material 345 Hence in (17) the electric and mechanical parts are separated and the determining equation () () 2 22 45 0 jj bb ξξ ++= describes the wave dynamics in the passive transversely isotropic medium. This equation coincides with the corresponding equation (Berliner & Solecki, 1996) and hence, all the presented results are converted into the well known results (Mirsky, 1964) and (Berliner & Solecki, 1996) in the limiting case of small electro-mechanical coupling coefficients. Additional dispersion line, corresponding to ( ) 22 11 33 0 j k ξε ε − = could be considered as an artefacts in this case. Furthermore if we suppose that 15 31 0ee = = in the first expression (31) we obtain that ( ) ( ) 22 2 11 44 13 44 EE j j EE ckc i kcc ρω ζ η −− = + (40) which also coincides with the known result (Berliner & Solecki, 1996). Hence, the results obtained contain the classical results of investigation of a passive transversely isotropic material as a particular limiting case. 4. Axisymmetric case Axisymmetric vibrations of the piezoelectric cylinder could be considered as a particular case of the general problem with 0 m = , 0v = , 2 0E = , 46 0TT = = and all variables are θ - independent. In this case system of equations (14) is rewritten as follows: () 222 13 11 13 44 44 15 31 22 2 1 EEEE uuu w u E E cccceeu rrrr rz z z r ρ ⎛⎞ ∂∂ ∂ ∂ ∂ ∂ +−++ + − − = ⎜⎟ ∂∂ ∂∂∂ ∂ ∂ ⎝⎠  , () 22 2 11 3 13 44 44 33 15 33 22 11 EE E E uu ww w EE E cc c c e e w rz rz r rr z r r z ρ ⎛⎞ ⎛⎞ ∂∂ ∂ ∂ ∂ ∂ ∂ ⎛⎞ +++++−+−= ⎜⎟ ⎜⎟ ⎜⎟ ∂∂∂ ∂ ∂ ∂ ∂ ∂ ⎝⎠ ⎝⎠ ⎝⎠  , 22 2 11 3 15 31 11 33 2 111 0 SS uuww uv EE E ee rz r r r r r r z rz r r z εε ⎛⎞⎛⎞ ∂∂∂∂ ∂∂ ∂ ∂ ⎛⎞ + ++ + + + ++ = ⎜⎟⎜⎟ ⎜⎟ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ⎝⎠ ⎝⎠⎝⎠ (41) Boundary conitions (15) are: 111 1213 31 0 EEE ra ra uw u Tc c c e rr z z φ = = ∂∂∂ ⎡⎤ ⎛⎞ ⎛ ⎞ ⎛⎞ ⎛⎞ = ++ + = ⎜⎟ ⎜⎟ ⎜ ⎟ ⎜⎟ ⎢⎥ ∂∂∂ ⎝⎠ ⎝⎠ ⎝ ⎠ ⎝⎠ ⎣⎦ , 544 15 0 E ra ra uw Tc e zr r φ = = ∂∂ ∂ ⎡⎤ ⎛⎞ = ++ = ⎜⎟ ⎢⎥ ∂∂ ∂ ⎝⎠ ⎣⎦ , (42) 11115 0 S ra ra uw De rzr φ ε = = ∂∂∂ ⎡⎤ ⎛⎞ = −+ + = ⎜⎟ ⎢⎥ ∂∂∂ ⎝⎠ ⎣⎦ or φ = = 0 ra The variables are changed as follows: Wave Propagation in Materials for Modern Applications 346 () ( ) () ,, itkr r uurzt e r ω ϕ + ∂ ⎡⎤ == ⎢⎥ ∂ ⎣⎦ , ()() ( ) ,, itkr wwrzt re ω χ + == , ()() ( ) ,, itkr rzt re ω φφ τ + == (43) and system of equations (20) is obtained. All relationships (21) – (32) are true for the axisymmetric case and the final representation of solutions of system (41) is: () ( ) ( ) ( ) () 123 ,, itkr rrr uurzt e rrr ω ϕϕϕ + ∂∂∂ ⎡⎤ ==++ ⎢⎥ ∂∂∂ ⎣⎦ , ( ) () () () ( ) 11 22 33 ,, itkr wwrzt r r re ω ηϕ ηϕ ηϕ + ==++ ⎡⎤ ⎣⎦ , (44) ( ) () () () ( ) 11 22 33 ,, itkr rzt r r r e ω φφ μϕ μϕ μϕ + ==++ ⎡⎤ ⎣⎦ Solution of the Helmholtz equations (28): ϕζϕ ∇ += 22 0 jjj , ( ) = 1, 2, 3j is: () ( ) jj jmj rAW r ϕϕ ζ == (45) where j A - arbitrary constants, representing the circumferential wave number, () () mj mj WrJr ζ ζ = is the Bessel function of the first kind if j ζ is real or complex and ( ) ( ) mj m j WrI r ζζ = is the Bessel function of the second kind if j ζ is pure imaginary number. Boundary conditions are obtained from determinant (37) by means of elimination of third row and fourth colimn. 5. Numerical results and discussion In this section we present the dispersion curves for non-axisymmetric waves with circumferential wavenumbers 1,2,3m = and axisymmetric waves resulting from the characteristic equation (38). Two piezoelectric materials are chosen, PZT-4 and PZT-7A, to illustrate the influence of electro-mechanical coupling coefficients on the configuration of the dispersion curves. The relevant material parameters for PZT-4 and PZT-7A are given in the Table 1. In order to obtain the dispersion curves we made use of a method similar to the novel method (Honarvar et al., 2008) where the dispersion curves are not produced as a result of solving of the dispersion equation by a traditional iterative find-root algorithm but are obtained by a zero-level cut in the velocity-frequency plane. In this chapter we modify this approach calculating the logarithm of modulus of determinant (37) on the mesh () 12 1 2 ,,11,,;21,, jj kjNj N ω ==…… (Shatalov et al., 2009). In those points where the real and imaginary parts of determinant (37) are close to zero substantial negative spikes occurwhich are displayed on a surface plot and give a picture of the configuration of the dispersion curves. The main advantage of this approach is that the local minima of the [N 1 ×N 2 ] - matrix of the logarithms are the proper guess values of the dispersion equation’s Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material 347 Geometric and Material Constants PZT-4 PZT-7A SI Units Radius ( ) a 1 1 m () ρ 3 7.5 10 ⋅ 3 7.6 10 ⋅ 3 /kg m − 11 E c 10 13.9 10⋅ 10 14.8 10⋅ 2 /Nm − 12 E c 10 7.78 10⋅ 10 7.62 10⋅ 2 /Nm − 13 E c 10 7.43 10⋅ 10 7.42 10⋅ 2 /Nm − 33 E c 10 11.5 10⋅ 10 13.1 10⋅ 2 /Nm − 44 E c 10 2.56 10⋅ 10 2.53 10⋅ 2 /Nm − 15 e 12.7 9.2 2 /Cm − 31 e -5.2 -2.1 2 /Cm − 33 e 15.1 9.5 2 /Cm − 11 S ε 0 730 ε ⋅ 0 460 ε ⋅ 22 1 Cm N −− 33 S ε 0 635 ε ⋅ 0 235 ε ⋅ 22 1 Cm N −− Table. Material constants and geometric parameters for PZT-4 and PZT-7A ( ) 12 2 2 1 0 8.85 10 Cm N ε −−− =⋅ roots. Hence all roots of equation (37) could be found for values of wavenumbers ( ) 1j k on the real axis and on or near the imaginary axis as a function of frequency ( ) 2j ω . Another advantage of this method is that it is much faster than the traditional root finding methods and as fast as the Honarvar method (Honarvar et al., 2008). The main disadvantage of this approach is that the roots of characteristic arguments () ( ) , 0,1,2,3 j j ξ = are also displayed on the surface plots as obvious artefacts. An elaborate discussion of these artefacts is given by Yenwong-Fai, (Yenwong-Fai, 2008). These artefacts could be simply detected and eliminated from the dispersion plots by program tools.Our algorithm, as it has been implemented, does not search for branches of the dispersion relation well away from the real and imaginary axes for k. It would be relatively straightforward in principle to locate these additional branches. Dispersion curves of bending waves ( ) 1m = in the cylinder made from PZT-4 with the short-circuit lateral (cylindrical) surface are depicted in Fig. 1 in dimensionless coordinates () () / s ka aV ω ⋅÷Ω= ⋅ , where 44 / E s Vc ρ = , and a is the outer radius of the cylinder. The picture of the dispersion curves is obtained by a method described above for real (propagating waves) and pure imaginary values of the wavenumber in the limits: ( ) ( ) ( ] Re , Im 0,8ka ka⋅⋅∈, ( ) ( ] /0,14 s aV ω Ω= ⋅ ∈ with resolution 500 (250 pixels for real and 250 pixels for imaginary ( ) ka ⋅ ) × 250 ( ( ) / s aV ω Ω =⋅) pixels. The same resolution is used for Fig. 2 – 17. The first dispersion curve of the propagating waves (real values of the wavenumber) tends to an asymptote of the surface wave propagation. It is joined to the second curve which tends to the asymptote of the shear waves through the domain of the evanescent waves. Wave Propagation in Materials for Modern Applications 348 Fig. 1. PZT-4 cylinder with short-circuit lateral surface (m = 1) Fig. 2. PZT-4 cylinder with open-circuit lateral surface (m = 1) Dispersion curves of bending waves ( ) 1m = in the cylinder made from PZT-4 with the open-circuit lateral surface are demonstrated in Fig. 2. It is obvious that the electric boundary conditions substantially influence both propagating and evanescent waves. For example, in the case of the open-circuit lateral surface the dispersion curves are even steeper than the corresponding curves in the case of the short-circuit lateral surface. In Fig. 3 a conceptual case of reduced electro-mechanical coupling coefficients () ( ) 3 15 31 33 33 4 0, 10 PZT ee e e − − == = is shown. In Fig. 4 and 5 the dispersion curves of PZT-7A material are presented for the open- and short-circuit lateral surfaces respectively. In comparison with PZT-4 this material has lower values of the electro-mechanical coupling coefficients but practically the same elastic coefficients and mass density. Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material 349 Fig. 3. PZT-4 cylinder with reduced electro-mechanical coupling () ( ) 3 15 31 33 33 4 1, 0, 10 PZT mee e e − − ==== Fig. 4. PZT-7A cylinder with short-circuit lateral surface (m = 1) It follows from Fig. 1 - 5 that the first fundamental mode of the bending waves is not sensitive to the nature of the electric boundary conditions on the lateral cylindrical surface. Furthermore it is practically not sensitive to the measure of electro-mechanical coupling of the material. The higher order modes are more sensitive to the nature of the electric boundary condition as well as to the measure of the electro-mechanical cross-coupling. Fig. 3 - 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 coupling coefficients of PZT-7A are less than the corresponding factors of PZT-4. It is reflected in undulating behaviour of the propagating higher modes as well as the values of the cut-off frequencies. The PZT-4 cylinder with short-circuit lateral surface demonstrates a negative slope of the fourth branch in a quite broad range of wavenumbers (Fig. 1). Substantial Wave Propagation in Materials for Modern Applications 350 dependence of dispersion curves on electric boundary conditions is obvious from the behaviour of the curves for the evanescent waves. Fig. 5. PZT-7A cylinder with open-circuit lateral surface (m = 1) Dispersion curves of non-axisymmetric waves with the circumferential wavenumber 2m = in the cylinders made from PZT-4 and PZT-7A with the open- and close-circuit lateral surface are depicted in Fig. 6 - 9. Again as for the case 1m = the substantial difference in the dispersion curves behaviour is explained by different types of the electric boundary conditions as well as by the difference in the electro-mechanic coupling coefficients. Dispersion curves of non-axisymmetric waves with the circumferential wavenumber 3m = in the cylinder made from PZT-4 and PZT-7A with the open- and close-circuit lateral surface are depicted in Fig. 10-13. The dispersion curves of higher circumferential wavenumbers (m = 2, 3) are sensitive to the nature of the electric boundary condition as well as to the measure of the electro-mechanical cross-coupling for both propagating and evanescent waves. These dispersions curves obtained from the exact solution of the problem could be used as references data for developing of reliable finite elements for approximate solution of the problems of wave propagation in piezoelectric structures. Fig. 6. PZT4 cylinder with short-circuit lateral surface (m = 2) Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material 351 Fig. 7. PZT4 cylinder with open-circuit lateral surface (m = 2) Fig. 8. PZT7A cylinder with short-circuit lateral surface (m = 2) Fig. 9. PZT7A cylinder with open-circuit lateral surface (m = 2) Wave Propagation in Materials for Modern Applications 352 Fig. 10. PZT-4 cylinder with short-circuit lateral surface (m = 3) Fig. 11. PZT-4 cylinder with open-circuit lateral surface (m = 3) Fig. 12. PZT-7A cylinder with short-circuit lateral surface (m = 3) Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material 353 Fig. 13. PZT-7A cylinder with open-circuit lateral surface (m = 3) Dispersion curves of axisymmetric waves in the cylinder made from PZT-4 and PZT-7A with the short-circuit and free lateral (cylindrical) surface are depicted in Fig. 14 - 17 in the same dimensionless coordinates ( ) ( ) / s ka aV ω ⋅÷Ω= ⋅ , where 44 / E s Vc ρ = , and a is the outer radius of the cylinder. The picture of the dispersion curves is obtained by a method described above for real (propagating waves) and pure imaginary values of the wavenumber in the limits: ( ) ( ) ( ] Re , Im 0,8ka ka⋅⋅∈, ( ) ( ] /0,28 s aV ω Ω= ⋅ ∈ with resolution 500 (250 pixels for real and 250 pixels for imaginary ( ) ka ⋅ ) × 250 ( ( ) / s aV ω Ω =⋅) pixels. Fig. 14. Axisymmetric dispesion curves in PZT-4 cylinder with short-circuit lateral surface Wave Propagation in Materials for Modern Applications 354 Fig. 15. Axisymmetric dispesion curves in PZT-4 cylinder with open-circuit lateral surface Fig. 16. Axisymmetric dispesion curves in PZT-7A cylinder with short-circuit lateral surface Fig. 17. Axisymmetric dispesion curves in PZT-7A cylinder with open-circuit lateral surface [...]... short waves with a large wave number Z are more sensitive to m than long waves with a small Z For a very small m=0.0001, the effect on short waves is already significant The gaps raise the frequencies of short waves with a large Z To examine the effect of the gaps on long waves with a small Z, we plot the case of m=0.1 in Fig 7 for anti-symmetric and symmetric waves together The anti-symmetric waves... thickness-twist wave due to fluid viscosity 5 Conclusion In this Chapter, we study the propagation of thickness-twist waves in an infinite piezoelectric ceramic plate Three cases are taken into account: 1) Propagation of thicknesstwist waves in a piezoelectric ceramic plate with attached electrodes; 2) Propagation of thickness-twist waves in a piezoelectric ceramic plate with unattached electrodes; 3) Propagation. .. 2c E E E J m + 1 (ξ 2 a ) + ⎢i k (η 2 c 13 + μ 2 e13 ) − ξ 22 c13 + 266 m ( m − 1 ) ⎥ J m (ξ 2 a ) , a a ⎣ ⎦ m ⎡ ⎤ E a22 = − ⎡(η 2 + i k ) c 44 + μ 2 e15 ⎤ ⎢ξ 2 J m + 1 (ξ 2 a ) − J m (ξ 2 a ) ⎥ , ⎣ ⎦⎣ a ⎦ a32 = a13 = E 2 mc 66 a m−1 ⎡ ⎤ ⎢ξ 2 J m + 1 (ξ 2 a ) − a J m (ξ 2 a ) ⎥ , ⎣ ⎦ E ⎡ ⎤ 2ξ 3c 66 2c E E E J m + 1 (ξ 3 a ) + ⎢i k (η 3c 13 + μ 3 e13 ) − ξ 32 c13 + 266 m ( m − 1 ) ⎥ J m (ξ 3 a ) , a... this part Then we analyze the propagation of thickness-twist waves in an infinite piezoelectric plate with air gaps between the plate surfaces and two electrodes Dispersion relations of the waves are obtained and plotted Results show that the wave frequency or speed is sensitive to the air gap thickness This effect can be used to manipulate the behavior of the waves and has implications in acoustic wave. .. case of symmetric gaps the waves separate into symmetric and antisymmetric ones The wave frequency is sensitive to the gap thickness, especially for short waves In the examples calculated the gaps raise the wave frequencies The dependence of frequency on air gaps provides a new design factor for acoustic wave devices Equations determining the Propagation of Thickness-Twist Waves in an Infinite Piezoelectric... Enjilela, E.; Sinclair, A & Minerzami, S (2007) Wave propagation in transversely isotropic cylinders, International Journal of Solids and Structures, 44, 5236-5246 Mirsky, I (1964) Wave propagation in transversely isotropic circular cylinders Part 1: Theory, J Acoust Soc Am 36, 2106-2122 Nayfeh, A.; Abdelrahman, W & Nagy, P (2000) Analysys of axisymmetric waves in layered piezoelectric rods and their... in the imaginary part Fig 5 Effect of air gaps on short anti-symmetric waves Fig 6 Effect of air gaps on short symmetric waves 370 Wave Propagation in Materials for Modern Applications Fig 7 Effect of air gaps on long waves of Fig 6 Again the gaps raise the frequencies In Yang et al (2009) it was found that for pure thickness-shear waves with Z=0 in quartz plates the gaps also raise the frequencies This... for a given ξ1, we have denoted the wave frequency when the viscosity is neglected by ω(0) and (ξ ) = ( ω ) (0) 2 2 (0) 2 2 / vT − ξ 12 (43) 371 Propagation of Thickness-Twist Waves in an Infinite Piezoelectric Plate Given a wave number ξ1, Eq (42) determines a series of frequencies ω(0)(ξ1) for guided waves in the plate When the fluid viscosity is considered, these wave frequencies are perturbed and... 2c E 2 E E I m + 1 ( ξ1 a ) + ⎢i k (η1c13 + μ1 e13 ) + ξ 1 c13 + 266 m ( m − 1 ) ⎥ I m ( ξ1 a ) , a ⎣ ⎦ m ⎡ ⎤ E a21 = ⎡(η1 + i k ) c 44 + μ1 e15 ⎤ ⎢ ξ 1 I m + 1 ( ξ 1 a ) + I m ( ξ1 a ) ⎥ , ⎣ ⎦⎣ a ⎦ a31 = − a12 = − E 2 ξ 2 c 66 a E 2mc66 a m−1 ⎡ ⎤ ⎢ ξ1 I m + 1 ( ξ1 a ) + a I m ( ξ1 a ) ⎥ , ⎣ ⎦ E ⎡ ⎤ 2c66 2 E E I m + 1 ( ξ 2 a ) + ⎢i k (η 2c 13 + μ 2 e13 ) + ξ 2 c 13 + 2 m ( m − 1 ) ⎥ I m ( ξ 2 a ) ,... A1 through A4 are undetermined constants, ω is the wave frequency, and ξ1 and ξ2 are waves numbers in the x1 and x2 directions Eq (13) satisfies Eq (5) when ⎛ν 2 ⎞ − 1⎟ , ⎝ν ⎠ ξ 22 = ξ12 ⎜ 2 T (14) 2 where the wave speed v is given by ξ 12 = ω 2 /ν 2 and ν T = c / ρ is the speed of plane shear waves propagating in the x1 direction Substitution of Eq (13) into the boundary conditions at x2 = ± h in Eq . propagating waves (real values of the wavenumber) tends to an asymptote of the surface wave propagation. It is joined to the second curve which tends to the asymptote of the shear waves through. () 66 32 2 1 2 2 21 E mm mc m aIaIa aa ξξ ξ + − ⎡ ⎤ =− + ⎢ ⎥ ⎣ ⎦ , () () () () 2 366 66 13 13 313 313 313 3 2 2 2 1 E E EE m m c c aIaikcecmmIa aa ξ ξημξ ξ + ⎡⎤ =− + + + + − ⎢⎥ ⎣⎦ , () () () 23. 6335–6350 Wave Propagation in Materials for Modern Applications 358 Berliner, M. & Solecki, R. (1996). Wave propagation in fluid-loaded transversely isotropic cylinders. Part 1. Analytical