Autoparametric excitation of an elastic chain in a periodic potential

We describe the motions of an elastic chain submitted to an external periodic potential, without dissipation. Initially, the energy is injected in a single eigenmode of the chain. We show that this eigenmode acts as a parametric forcing of the other modes, and since there is no external energy source we speak of autoparametric forcing. If the chain is trapped in the potential well, then the motion of the center of mass is the lowest frequency mode. We show that this mode induces autoparametric resonances of the higher-frequency modes. We also exhibit specific conditions under which a mode of higher frequency induces autoparametric resonances of some other modes. Generically, a high-frequency mode in a trapped chain induces some coupling between other modes, with coupling coefficients that are periodic time functions. As a consequence, autoparametric internal resonances between one forcing mode and two forced modes take place under conditions on the relevant modes indices that we provide explicitly. When enough kinetic energy is injected into the motion of the center of mass for the chain to slide over the potential, we show that the basically translational motion of the chain may induce an autoparametric resonance of any vibrational mode of the chain. We also show that two modes can simultaneously display autoparametric resonance. Therefore, we generalize the notion of autoparametric resonance to systems of arbitrary size.