Instability of vibrations of a mass that moves uniformly along a beam on a periodically inhomogeneous foundation

The stability of vibrations of a mass that moves uniformly along an Euler–Bernoulli beam on a periodically inhomogeneous continuous foundation is studied. The inhomogeneity of the foundation is caused by a slight periodical variation of the foundation stiffness. The moving mass and the beam are assumed to be always in contact. With the help of a perturbation analysis it is shown analytically that vibrations of the system may become unstable. The physical phenomenon that lies behind this instability is parametric resonance that occurs because of the periodic (in time) variation of the foundation stiffness under the moving mass. The first instability zone is found in the system parameters within the first approximation of the perturbation theory. The location of the zone is strongly dependent on the spatial period of the inhomogeneity and on the weight of the moving mass. The larger this period is and/or the smaller the mass, the higher the velocity is at which the instability occurs.