Analytical Solution for Free Vibration of Multilayered Circular Plate

Abstract:

An analytical solution is deduced for a three-dimensional transversely isotropic axisymmetric multilayered circular plate made of piezoelectric (PE) and piezomagnetic (PM) under simply supported boundary condition. The state space vector, finite Hankel transform and propagation matrix methods are utilized together to obtain the full-field solutions for the multilayered circle plate. Numerical examples for five-layered PE/PM composites with dimensionless frequencies of the multilayered plate under simple-supported lateral boundary conditions are presented. The frequencies increase with the ratio of the thickness to radius.

• Periodical: Advanced Engineering Forum (Volumes 2-3)
• Pages: 948-953
• DOI: https://doi.org/10.4028/www.scientific.net/AEF.2-3.948
• Online since: December 2011
• Authors: Hong Jun Ren, Xu Yuan Song, Qing Kai Han
• Keywords: Finite Hankel Transform, Multilayered Circular Plate, Propagation Matrix Method

References

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[7] H. Ding, R. Xu, F. Lin, Exact solution for axisymmetric transversely isotropic piezoelectric circular plate-II. Exact solution and numerical example for the elastic circular plate (in Chinese), Science in China (Series E). 29 (1999) 298-305. Appendix , , , , , , , , , , , , , , , , , , , Table 1. Material properties of BaTiO3 and CoFe2O4 [1] (Cij: elastic constants in GPa; eij: piezoelectric coefficients in N/(V·m); qij: piezomagnetic coefficients in N/(A·m); εij: permittivity coefficients in 10-9 C/(V·m); and μij: permeability coefficients in 10-6 Wb/(A·m). BaTiO3 C11 C12 C13 C22 C23 C33 C44 C55 C66 166 77 78 166 78 162 43 43 44.5 e13 e23 e33 e42 e51 ε11 ε22 ε33 -4.4 -4.4 18.6 11.6 11.6 11.2 11.2 12.6 CoFe2O4 C11 C12 C13 C22 C23 C33 C44 C55 C66 286 173 170.5 286 170.5 269.5 45.3 4 5.3 56.5 q13 q23 q33 q42 q51 μ11 μ22 μ33 580.3 580.3 699.7 550 550 590 590 157 Table 2. BFBFB dimensionless frequencies of the magnetoelectroelastic plate () s k1=2.40483 k2=5.52008 k3=8.65373 Ω1 Ω2 Ω3 Ω1 Ω2 Ω3 Ω1 Ω2 Ω3 0.1 0.01562 0.23272 1.71047 0.07888 0.53281 1.79070 0.18151 0.83130 1.91611 0.2 0.06061 0.46460 1.76793 0.27844 1.05485 2.03336 0.57808 1.61408 2.39830 0.3 0.13031 0.69477 1.85390 0.54010 1.55037 2.35170 1.03827 2.24727 2.96779 0.4 0.21912 0.92223 1.96164 0.82860 1.99147 2.70765 1.51217 2.63614 3.54627 0.5 0.32189 1.14572 2.08594 1.12765 2.33879 3.07842 1.98629 2.90999 4.08241 0.6 0.43452 1.36361 2.22270 1.43014 2.58258 3.44812 2.45745 3.18860 4.54287 0.7 0.55395 1.57381 2.36869 1.73299 2.76742 3.80358 2.92541 3.49932 4.93070 0.8 0.67803 1.77356 2.52136 2.03501 2.93759 4.13367 3.39072 3.84164 5.27807 0.9 0.80524 1.95947 2.67870 2.33572 3.11363 4.43144 3.85403 4.20973 5.61486 1.0 0.93449 2.12786 2.83905 2.63509 3.30232 4.69688 4.31590 4.59787 5.95722 Table 3. FBFBF dimensionless frequencies of the magnetoelectroelastic plate () s k1=2.40483 k2=5.52008 k3=8.65373 Ω1 Ω2 Ω3 Ω1 Ω2 Ω3 Ω1 Ω2 Ω3 0.1 0.01673 0.23826 1.74117 0.08421 0.54523 1.82013 0.19293 0.84984 1.94877 0.2 0.06476 0.47551 1.79730 0.29488 1.07714 2.07089 0.60698 1.63950 2.45178 0.3 0.13880 0.71065 1.88451 0.56761 1.57610 2.40330 1.08091 2.25530 3.03636 0.4 0.23256 0.94240 1.99607 0.86555 2.00972 2.77142 1.56539 2.63715 3.60805 0.5 0.34039 1.16923 2.12583 1.17249 2.34294 3.14772 2.04818 2.92896 4.11292 0.6 0.45793 1.38910 2.26877 1.48170 2.58234 3.51301 2.52665 3.22994 4.53701 0.7 0.58198 1.59946 2.42098 1.79045 2.77525 3.85316 3.00067 3.56007 4.90375 0.8 0.71036 1.79716 2.57944 2.09771 2.95885 4.16019 3.47054 3.91168 5.24441 0.9 0.84151 1.97873 2.74170 2.40315 3.14932 4.43400 3.93660 4.29381 5.57986 1.0 0.97442 2.14104 2.90575 2.70674 3.35153 4.68087 4.39903 4.68552 5.92080