Energy exchange between coupled mechanical oscillators: linear regimes

Nonlinear mechanisms are frequently invoked to explain unexpected observations during processes of energy exchange in mechanical systems. However, whether the same phenomena could be observed in a purely linear system is seldom considered. In this paper, we revisit the problem of two linearly coupled, damped, harmonic oscillators, with emphasis on the dynamics of their mutual exchange of energy. A novel criterion is established to discern between two well-differentiated regimes of energy exchange under ringdown conditions, when all external excitations are turned off and the oscillatory motion is left to decay by dissipation. The two regimes are induced by different sets of initial conditions, and correspond to extremal values of the total net energy transferred during ringdown. Although the problem is in principle fully solvable, its algebraic involvedness requires in practice to resort to approximated expressions for oscillation frequencies and decay rates. We explicitly provide such approximations in the limit of high quality factors and weak coupling. This limit is relevant to current experiments on micro- and nanomechanical oscillators, whose physical behavior is usually explained by extensive allusion to energy exchange. Our results should help to discern between observations that are genuinely due to nonlinear effects, and those that can be explained in terms of linear mechanisms only. (paper)