Deformation of a thin elastic plate with a fixed edge and attached rods. Part 2: The spectral problem

In the low-frequency range of the spectrum, we construct the asymptotics of the frequencies and modes of eigenvibrations of an isotropic and homogeneous elastic junction of thin cylindrical vertical rods and a horizontal plate. The junction surface is free of external loads everywhere except for the rigidly clamped edge of the plate. Several types of vibrations are identified, accompanied by bending deformations of the plate and/or the rods. The justification of the asymptotic formulas is based on an asymptotically sharp anisotropic weighted Korn inequality, a classical lemma on “almost eigenvalues,” and a convergence theorem for normalized eigenvalues.