Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a strongly nonlinear end attachment

We study the dynamics of a semi-infinite linear chain of particles that is weakly coupled to a strongly nonlinear oscillator at its free end. We analyze families of localized standing waves situated inside the lower or upper attenuation zones of the linear chain, corresponding to energy predominantly confined in the nonlinear oscillator. These families of standing waves are generated due to resonant interactions between the chain and the nonlinear attachment. In addition, we estimate the energy radiated from the nonlinear oscillator back to the chain, when the oscillator is excited under nonresonant conditions by wavepackets with dominant frequencies inside the propagation zone of the chain. We conclude that the system is capable of energy pumping, i.e., of one-way, irreversible energy transfer from the semi-infinite chain to the nonlinear oscillator. Such energy transfer closely depends on the excitation by the external forcing of a localized standing wave of the type studied in this work. A scenario for the realization of energy pumping phenomena in the system under consideration is discussed, and is confirmed by direct numerical simulations of the chain–attachment dynamic interaction.