Analyzing the influence of the disk motion on longitudinal oscillations of a beam with rheological properties

Abstract

The paper gives the analysis of the influence of longitudinal oscillations of a beam, which is caused by a circular hard disk motion along the beam, on the mode of disk motion. The problem of rolling the disk on the plane in its various formulations has been investigated by different researchers as well as by the authors of this paper. The paper considers two variants of properties of the beam material: an elastic beam and a viscoelastic beam, which fits the Kelvin rheological model and has relaxation and creep properties. When solving the problem, the Fourier method is used as a method of separation of variables. When examining the beam, and assuming it as hereditary deformable, a rheological force of the reaction is introduced, this depends on both on longitudinal strains and on the rate of these strains The result is presented as functions of time which are adaptable for numerical integration. It is shown that oscillations of a rheological beam are produced by a moving disk and can be considered as self-oscillations.