In this paper we consider the eigenvalue problem for fourth order ordinary differential equation that describes the bending vibrations of a homogeneous rod, in cross-sections of which the longitudinal force acts, the left end of which is fixed rigidly and on the right end an inertial mass is concentrated. We characterize the location of the eigenvalues on the real axis, we investigate the structure of root spaces and oscillation properties of eigenfunctions and their derivatives, we study the basis properties in the space L p , 1 < p < ∞, of the system of eigenfunctions of considered problem.