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.

Energy Growth of Infinite Harmonic Chain under Microscopic Random Influence

“…The effect of the total energy's growth in the case when only few degrees of freedom are influenced by a white noise was proved in [2] for finite linear Hamiltonian systems. Seminal physical papers [4,5] (and reference therein) are closely related to ours. In these articles authors have considered system with a discrete Laplacian at the right hand side of (1).…”

“…In the last equality we have used (13). Substituting ( 18) and (20) into the last expression we get (5). Formulas ( 2) and ( 6), for the expected value and variance of the energy H(t) immediately follows from (5).…”

“…In these articles authors have considered system with a discrete Laplacian at the right hand side of (1). In [5] instead of white noise there is a periodic force sin ωt acting on the fixed particle with index 0 and additional pinning term. The authors studied behavior of energies and solution as time t goes to infinity.…”

Equilibration of sinusoidal modulation of temperature in linear and nonlinear chains

“…If 0 < α < 1, we have anomalous thermal conductivity. Note that defect-free linear systems of any complexity always demonstrate ballistic heat propagation [32][33][34][35][36][37][38][39][40]. One might expect that in systems with weak anharmonicity, the linear theory can be very helpful [41].…”