Resonance energy transport and exchange in oscillator arrays

It is well known that complete energy transfer between two weakly coupled linear oscillators occurs only at resonance. If the oscillators are nonlinear, the amplitude dependence of their frequencies may destroy, in general, any eventual resonance. This means that no substantial energy transfer may occur unless, exceptionally, resonance persists during the transfer. In this paper, the self-sustained resonance is considered for an oscillator array consisting of n coupled linear oscillators (a primary system) initially excited by impulse loading and connected to an essentially nonlinear attachment (NLA). Under the condition of resonance, initial energy is transferred to the NLA and then travels back and forth between the linear and nonlinear oscillators. It is shown that the general mechanism of the energy transport is similar to that in the previously studied system of two coupled oscillators but, in contrast to the two degree-of-freedom case, the multidimensional system requires a proper tuning not only for the NLA but for the entire array. In this work, we develop an order-reduction procedure, which allows the separated dynamical analysis for the pair of nonlinearly coupled oscillators and the remaining (n-1) linear oscillators. Using simplifications based on the low-order reduced model, we detect an admissible domain of parameters ensuring resonance interaction and then derive a closed-form approximate solution adequately describing the transient processes in the entire system. We obtain explicit approximate solutions for both conservative (complete energy exchange) and dissipative (irreversible energy transfer) systems and then illustrate the theoretical results by an example of the four degree-of-freedom system. Analytical results are confirmed by numerical simulations.