Influence of Winkler--Steklov conditions on natural oscillations of elastic weighty body

S. A. Nazarov

Ufa mathematical journal, Tome 16 (2024) no. 1, pp. 53-79

Résumé

We consider the spectral problem for the spacial system of equations of the elasticity theory. Small parts of the body surface are supplied with the Winkler–Steklov conditions, which model spring mount, while the remaining part of the boundary is traction-free. In several cases (the relative stiffness of springs and their positions are varied) we construct asymptotics for eigenfrequencies of the body and for corresponding eigenmodes. The limiting problems are ones for the body (spectral or stationary in some case) and problems of the elasticity theory for the half-spaces with the Winkler–Steklov conditions on flat sets (separated or joined into a single spectral theory in some cases). The discreteness of the spectrum of the problem in the half-space is ensured by a polynomial property of the system of equations of the elasticity theory. We study particular cases, formulate open questions and discuss patological situations, in which the spectrum loses usual properties. We construct asymptotic models of the problem, which provide two-terms asymptotics for the eigenpairs of the initial problem and which use the technique of self-adjoint extensions of differential operators or Hilbers spaces with separated asymptotics.

Détail

Keywords: elastic body, Winkler–Steklov conditions of spring mount, asymptotics of eigenfrequencies.
Mots-clés : singular perturbation