Analysis of interaction between solids of revolution and a beam corresponding to the Kelvin model

Various formulations of the problem of a disk rolling on a plane have been studied by different researchers. In this paper, the effect of longitudinal oscillations of a beam, caused by the circular motion of a solid disk along the beam, on the mode of the disk motion is analyzed. Two versions of properties of a beam material are considered: an elastic beam and a viscoelastic beam corresponding to the Kelvin rheological model with relaxation and creep properties. The Fourier method is used as a method of separation of variables in the problem solving. When testing the beam and assuming its hereditary deformation, the rheological response force is introduced, which depends on longitudinal strains and their rate. The obtained result is presented as functions of time, which are adaptable for numerical integration. It is shown that beam oscillations arise from the disk motion and can be considered as self-oscillations.