Asymptotic formulas for eigenvalues and eigenfunctions of some ordinary differential operators of fourth order

In this paper we consider a spectral problem for a fourth order ordinary differential equations with a spectral parameter fractionally linearly entering in the boundary condition. This problem, in particular, describes the bending vibrations of a homogeneous rod, in cross-sections of which a longitudinal force acts, the left end of which is fixed and a mass is concentrated at the right end. We obtain refined asymptotic formulas for the natural frequencies and harmonics of this problem.