TY - JOUR

T1 - The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section

AU - Nazarov, S. A.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - After scaling, a long Kirchhoff plate with rigidly clamped ends Keywords: Kirchhoff plate, eigenvalues and eigenfunctions, behaviour, dimension reduction, boundary layer, one-dimensional model. 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.

AB - After scaling, a long Kirchhoff plate with rigidly clamped ends Keywords: Kirchhoff plate, eigenvalues and eigenfunctions, behaviour, dimension reduction, boundary layer, one-dimensional model. 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.

UR - http://www.scopus.com/inward/record.url?scp=85057566810&partnerID=8YFLogxK

U2 - 10.1070/SM8974

DO - 10.1070/SM8974

M3 - Article

AN - SCOPUS:85057566810

VL - 209

SP - 1287

EP - 1336

JO - Sbornik Mathematics

JF - Sbornik Mathematics

SN - 1064-5616

IS - 9

ER -