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The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section
S. A. Nazarovab
a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg
b St. Petersburg State University, Mathematics and Mechanics Faculty
After scaling, a long Kirchhoff plate with rigidly clamped ends and free lateral sides is described by a mixed boundary-value problem for the biharmonic operator in a thin domain with weakly curved boundary. Based on a general procedure for constructing asymptotic formulae for solutions of elliptic problems in thin domains, asymptotic expansions are derived for the eigenvalues and eigenfunctions of this problem with respect to a small parameter equal to the relative width of the plate and are also justified. In the low-frequency range of the spectrum the limiting problem is a Dirichlet problem for a fourth-order ordinary differential equation with variable coefficients, and in the mid-frequency range it is (quite unexpectedly) a Dirichlet problem for a second-order equation. The phenomenon of a boundary layer in a neighbourhood of the ends of the plate is investigated. This makes it possible to construct infinite formal asymptotic series for simple eigenvalues and the corresponding eigenfunctions, and to develop a model with enhanced precision. Asymptotic constructions for plates with periodic rapidly oscillating boundaries or for other sets of boundary conditions corresponding to mechanically reasonable ways to fix the ends of the plate are discussed.
Bibliography: 46 titles.
Kirchhoff plate, eigenvalues and eigenfunctions, asymptotic behaviour, dimension reduction, boundary layer, one-dimensional model.
|Russian Science Foundation
|This work was supported by the Russian Science Foundation under grant no. 17-11-01003.
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Sbornik: Mathematics, 2018, 209:9, 1287–1336
MSC: Primary 35P20, 74B04; Secondary 76N05
Received: 29.05.2017 and 07.08.2017
S. A. Nazarov, “The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section”, Mat. Sb., 209:9 (2018), 35–86; Sb. Math., 209:9 (2018), 1287–1336
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\paper The asymptotics of natural oscillations of a~long two-dimensional Kirchhoff plate with variable cross-section
\jour Mat. Sb.
\jour Sb. Math.
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F. L. Bakharev, S. A. Nazarov, “Asimptotika sobstvennykh chisel dlinnykh plastin Kirkhgofa s zaschemlennymi krayami”, Matem. sb., 210:4 (2019), 3–26
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