Research output: Contribution to journal › Article › peer-review
The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section. / Nazarov, S. A.
In: Sbornik Mathematics, Vol. 209, No. 9, 01.09.2018, p. 1287-1336.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 40973348