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Abstract <p> We construct asymptotics of natural oscillations of elastic junctions composed of a thinhorizontal plate and several vertical rods attached to it. This construction is rigidly fixed alongthe plate edge and the exterior end faces of the rods, while the physical properties of its elementsare chosen so that in the mid-frequency range of the spectrum the limit spectral problems consistsof a self-adjoint operator obtained by extending differential operators, namely, a biharmonic one inplate’s longitudinal section and ordinary second-order differential operators at rod’s axes. Thelow-frequency range of the spectrum is formed by eigenvalues of the Dirichlet problem for ordinaryforth-order differential operators describing transverse oscillations of rods with fixed ends.Justification of asymptotic formulas is performed by means of anisotropic Korn’s inequality andthe classical lemma on “almost eigenvalues.”</p>

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Elastic Junctions of a Plate with Rods and Self-Adjoint Extensions of Differential Operators

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • S. A. Nazarov

摘要

Abstract

We construct asymptotics of natural oscillations of elastic junctions composed of a thinhorizontal plate and several vertical rods attached to it. This construction is rigidly fixed alongthe plate edge and the exterior end faces of the rods, while the physical properties of its elementsare chosen so that in the mid-frequency range of the spectrum the limit spectral problems consistsof a self-adjoint operator obtained by extending differential operators, namely, a biharmonic one inplate’s longitudinal section and ordinary second-order differential operators at rod’s axes. Thelow-frequency range of the spectrum is formed by eigenvalues of the Dirichlet problem for ordinaryforth-order differential operators describing transverse oscillations of rods with fixed ends.Justification of asymptotic formulas is performed by means of anisotropic Korn’s inequality andthe classical lemma on “almost eigenvalues.”