Assessment of the dynamic characteristics (eigenfrequencies, modes and decrement of vibration) of rolling stock wheelsets is an urgent task. This study is devoted to the evaluation of the dynamic characteristics of wheelsets together with the rail taking into account the inelastic properties of the shaft material, wheel sliding, and rolling stock speed. A mathematical model for assessing the dynamic characteristics of wheelsets, taking into account the viscoelastic properties of the material of the system was developed using the Boltzmann-Volterra hereditary theory. A method and algorithm were developed to reduce the problem of natural oscillations of the system to an algebraic eigenvalue problem. Dynamic characteristics of the wheelsets were determined under various parameters of the viscoelasticity of the material. A mathematical model was developed to evaluate the dynamic characteristics of wheelsets with a rail, while considering dissipative processes and using a linear combination of mass and stiffness matrices of the system. Eigenfrequencies, modes and decrement of vibrations of rolling stock wheelsets with a rail were determined taking into account wheel sliding and rolling stock speed. Some mechanical effects were revealed.