Abstract
The article considers the spectrum of one-dimensional natural vibrations of a layered medium with a periodic structure consisting of an isotropic elastic material and a viscous incompressible fluid. It is established that the spectrum points are the roots of transcendental equations. In order to solve these equations numerically for multi-layered media, the roots of quadratic equations are proposed to use as initial approximations.
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REFERENCES
A. S. Shamaev and V. V. Shumilova, ‘‘Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin-Voigt viscoelastic materials,’’ Proc. Steklov Inst. Math. 295, 202–212 (2016). doi 10.1134/S0081543816080137
A. S. Shamaev and V. V. Shumilova, ‘‘Calculation of natural frequencies and damping coefficients of a multi-layered composite using homogenization theory,’’ IFAC-PapersOnLine 51 (2), 126–131 (2018). doi 10.1016/j.ifacol.2018.03.022
R. P. Gilbert and A. Mikelić, ‘‘Homogenizing the acoustic properties of the seabed: part I,’’ Nonlinear Anal. 40. 185–212 (2000). doi 10.1016/S0362-546X(00)85011-7
O. A. Oleinik, G. A. Iosif’yan, and A. S. Shamaev, Mathematical Problems in the Theory of Strongly Inhomogeneous Elastic Media (Moscow State Univ., Moscow, 1990).
Funding
This work was supported by the Russian Science Foundation (project no. 16–11–10343).
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Translated by L. Trubitsyna
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Shamaev, A.S., Shumilova, V.V. Spectrum of One-Dimensional Natural Vibrations of Layered Medium Consisting of Elastic Material and Viscous Incompressible Fluid. Moscow Univ. Math. Bull. 75, 172–176 (2020). https://doi.org/10.3103/S0027132220040063
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DOI: https://doi.org/10.3103/S0027132220040063