Parametric Instability of a Moving Particle on a Periodically Supported Infinitely Long String

A new method is proposed of theoretical analysis of the dynamic instability of a moving object on a periodically supported, infinitely long elastic structure. To demonstrate this method, a simple example is considered of a moving particle on an elastically supported string. The equations are obtained that govern the system parameters that correspond to the boundaries separating stability and instability in the parameter space. These equations are in the form of the determinant of an infinite matrix and are analogous to Hill’s infinite determinant. A parametric analysis of the instability zones is carried out in the plane of the normalized particle mass and particle velocity. The focus is placed on the effect of elasticity and viscosity of the supports. An analytical validation is presented of the numerically obtained instability zones. This is done using a simplified model of the string on the corresponding continuous foundation.