Dynamics of time-modulated, nonlinear phononic lattices

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 in both linear and nonlinear regimes. 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 linear theoretical predictions, wave-number band gaps emerge. The underlying instabilities associated to the wave-number band gaps 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, and results in a family of nonlinear time-periodic states. The bifurcation structure of the periodic states is studied. It is found the linear theory correctly predicts parameter values from which the time-periodic states bifurcate from the zero state. In the presence of an external drive, the parametric amplification induced by the wave-number band gap can lead to bounded and stable responses that are temporally quasiperiodic. 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.