ELASTIC JUNCTIONS OF A PLATE WITH RODS AND SELF-ADJOINT EXTENSIONS OF DIFFERENTIAL OPERATORS

We construct asymptotics of natural oscillations of elastic junctions composed of a thin horizontal plate and several vertical rods joined to it. This construction is rigidly fixed along the plate edge and the exterior end-faces of the rods while physical properties of its elements are chosen such that in the mid-frequency range of the spectrum the limiting spectral problems consists of a self-adjoint operator obtained as extensions of differential operators, namely, a bi-harmonic one in plate’s longitudinal section and ordinary second-order differential operators at rod’s axes. The low-frequency range of the spectrum is formed by eigenvalues of the Dirichlet problem for ordinary forth-order differential operators describing transverse oscillations of rods with fixed ends. Justification of asymptotic formulas is performed by means of anisotropic Korn’s inequality and the classical lemma on “almost eigenvalues”.