Dynamics of Time-Modulated, Nonlinear Phononic Lattices

Abstract

The propagation of acoustic and elastic waves in time-varying, spatially homogeneous media can exhibit different phenomena when compared to traditional spatially-varying, temporally-homogeneous media. In the present work, the response of a one-dimensional phononic lattice with time-periodic elastic properties is studied with experimental, numerical and theoretical approaches. The system consists of repelling magnetic masses with grounding stiffness controlled by electrical coils driven with electrical signals that vary periodically in time. For small amplitude excitation, in agreement with theoretical predictions, wavenumber bandgaps emerge. The underlying instabilities associated to the wavenumber bandgaps are investigated with Floquet theory and the resulting parametric amplification is observed in both theory and experiments. In contrast to genuinely linear systems, large amplitude responses are stabilized via the nonlinear nature of the magnetic interactions of the system. In particular, the parametric amplification induced by the wavenumber bandgap can lead to bounded and stable responses that are temporally quasi-periodic. Controlling the propagation of acoustic and elastic waves by balancing nonlinearity and external modulation offers a new dimension in the realization of advanced signal processing and telecommunication devices. For example, it could enable time-varying, cross-frequency operation, mode- and frequency-conversion, and signal-to-noise ratio enhancements.