Kirchhoff plate with the Winkler–Steklov conditions at small parts of the edge

We construct asymptotics of eigenvalues and eigenfunctions of the bi-harmonic equation with the Neumann conditions perturbed by the spectral Winkler–Steklov conditions at small parts of plate's edge. The null eigenvalue has multiplicity three and the corresponding eigenfunctions are linear. Asymptotic expansions of positive eigenvalues differ sufficiently in the mid- and high-frequency ranges of the spectrum. In particular, eigenfunctions at low and mid frequencies are distributed along the whole domain while eigenfunctions at high frequencies are concentrated in the vicinity of the edge perturbations.