Calculation of natural frequencies and damping coefficients of a multi-layered composite using homogenization theory ⁎ ⁎This work was supported by the Russian Science Foundation under grant 16-11-10343.
А. С. Шамаев, Владлена Валерьевна Шумилова · IFAC-PapersOnLine · 2018
In this paper a mathematical model that describes one-dimensional vibrations of a multi-layered composite with periodic structure is considered. It is supposed that the composite consists of a large number of alternating layers of an isotropic elastic material and an isotropic non-aging viscoelastic material with long-term memory, for which the regular part of the shear relaxation kernel is approximated by one exponential function. In addition, vibrations are assumed to be perpendicular to the layers. The homogenized model corresponding to the original composite model is also considered. It is proved that natural frequencies and damping coefficients of the multi-layered composite (respectively, homogenized material) are determined by using non-real roots of transcendental (respectively, cubic) equations. Moreover, it is proposed that non-real roots of the cubic equations can be used as initial approximations for solving the transcendental equations numerically. The accuracy of these approximations is investigated by calculating the natural frequencies and the damping coefficients of multi-layered composites with given numerical characteristics.