tailieunhanh - Wave Propagation 2011 Part 13

Tham khảo tài liệu 'wave propagation 2011 part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 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 b4 b5 0 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 2 eu - k2e33 0 could be considered as an artefacts in this case. Furthermore if we suppose that e15 e31 0 in the first expression 31 we obtain that i_ V z Cn - k24 4 n k C cE4 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 m 0 v 0 E2 0 T4 T6 0 and all variables are 0 -independent. In this case system of equations 14 is rewritten as follows E d2u 1 du u V E C11I 2 --------2 I C13 ỡr r dr r E d w E d u dE C44 T- Ce 2 - e1YY dr dz dz2 dz ỠE3 e31 3 pu 31 dr E E Y1 du d2u V E d2w 1 dw Y E d2w CE CE. CE. 1 4 r dz dr dz 4 dr2 r dr 3 dz2 e15 ỔE3 eY p dz d2u 1 du d2w 1 dw I 1 du d2v S dE-. E-. Y S ỔE3 e151 . . --1- - - -- - I e311 I eS I I S33-H3 0 15 31 11 33 I dr dz r dr dr r dr I r dz dr dz dr r dz 41 Boundary conitions 15 are T1 r a 0 r a T5 r a E d u dw 1 dỏ C44 Yr zd I e15 dz dr dr 0 42 D1 r a dE1 E1 dr r The variables are changed as follows 346 Wave Propagation in Materials for Modern Applications u u r z t dọ r dr e wt kr w w r z i x r e kr ộ ộ r z t T r e r 43 and system of equations 20 is obtained. All relationships 21 -