Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, analytical study

We study targeted energy transfer in a two degree-of-freedom damped system under the condition of 1:1 transient resonance capture. The system consists of a linear oscillator strongly coupled to an essentially nonlinear attachment or nonlinear energy sink. In a companion paper [Quinn et al., Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: part I, Journal of Sound and Vibration 311 (2008) 1228–1248] we studied the underlying structure of the Hamiltonian dynamics of this system, and showed that for sufficiently small values of viscous damping, nonlinear damped transitions are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the Hamiltonian system. In this work direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed through slow/fast partitions of the transient responses, in order to investigate analytically the parameter region of optimal targeted energy transfer. To this end, we determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator. Then, we prove that optimal targeted energy transfer is realized for initial energies close to the neighborhood of a homoclinic orbit of the underlying Hamiltonian system. We study analytically transient orbits resulting as perturbations of the homoclinic orbit in the weakly damped system, and show that this yields an additional slow-time scale in the averaged dynamics, and leads to optimal targeted energy transfer from the linear oscillator to the nonlinear energy sink in a single “super-slow” half-cycle. We show that at higher energies, this “super-slow” half-cycle is replaced by strong nonlinear beats, which lead to significant but suboptimal targeted energy transfer efficiency. Finally, we investigate numerically targeted energy transfer efficiency in this system over a wide range of system parameters and verify the analytical predictions.