Spectral energy scattering and targeted energy transfer in phononic lattices with local vibroimpact nonlinearities

We propose a method for manipulating wave propagation in phononic lattices by employing local vibroimpact (VI) nonlinearities to scatter energy across the underlying linear band structure of the lattice, and transfer energy from lower to higher optical bands. First, a one-dimensional, two-band phononic lattice with embedded VI unit cells is computationally studied to demonstrate that energy is scattered in the wave number domain, and this nonlinear scattering mechanism depends on the energy of the propagating wave. Next, a four-band lattice is studied with a similar technique to demonstrate the concept of nonresonant interband targeted energy transfer (IBTET) and to establish analogous scaling relations with respect to energy. Both phononic lattices are shown to exhibit a maximum energy transfer at moderate input energies, followed by a power-law decay of relative energy transfer either to the wave number domain or between bands on input energy. Last, the nonlinear normal modes (NNMs) of a reduced order model (ROM) of a VI unit cell are computed with the method of numerical continuation to provide a physical interpretation of the IBTET scaling with respect to energy. We show that the slope of the ROM's frequency-energy evolution for 1:1 resonance matches well with IBTET scaling in the full lattice. Moreover, the phase-space trajectories of the NNM solutions elucidate how the power-law scaling is related to the nonlinear dynamics of the VI unit cell.