Energy transfer to a harmonic chain under kinematic and force loadings: Exact and asymptotic solutions

Abstract

We consider dynamics of a one-dimensional harmonic chain with harmonic on-site potential subjected to kinematic and force loadings. Under kinematic loading, a particle in the chain is displaced according to sinusoidal law. Under force loading, a harmonic force is applied to a particle. Dependence of the total energy supplied to the chain on loading frequency is investigated. Exact and asymptotic expressions for the energy are derived. For loading frequencies inside the spectrum, the energy grows in time. The rate of energy growth depends on the group velocity corresponding to loading frequency. For non-zero group velocities, the energy grows linearly. If the group velocity vanishes, behavior of the system under kinematic and force loadings is qualitatively different. Under kinematic loading, the energy is bounded, while under force loading it grows in time as [Formula: see text]. Similar problem is solved in continuum formulation for a longitudinally vibrating elastic rod. It is shown that at large times, expressions for energies of the rod and the chain are identical, provided that sound speed and density are chosen properly. Generalization of results for the case of an arbitrary periodic excitation is discussed.