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

We construct asymptotics of natural oscillations of elastic junctions composed of a thin horizontal plate and several vertical rods attached to it. This construction is rigidly fixed along the plate edge and the exterior end faces of the rods, while the physical properties of its elements are chosen so that in the mid-frequency range of the spectrum the limit spectral problems consists of a self-adjoint operator obtained by extending differential operators, namely, a biharmonic 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.”