Breaking reciprocity with spatiotemporal modulation provides an opportunity to design unprecedented optical, acoustic, and mechanical waveguides. A main challenge is to physically realize continuum-based metamaterials whose properties can be rapidly tuned in both space and time at the length and time scales of the propagated waves. We design a tunable elastic metamaterial by embedding in a beam a set of permanent magnets, and placing oscillating electrical coils coaxially adjacent to each magnet. By programming in space and time the ac input of the coils, the magnet-coil effective coupling stiffness is modulated along with the resonance frequency. Distinctly nonreciprocal flexural wave propagation is then experimentally observed. In addition, robust tunability of unidirectional band gaps and wave energy bias are quantitatively analyzed by applying different modulation current amplitudes, material damping coefficients, and modulation frequencies. Both simplified analytical and finite-element-method-based numerical models of the modulated metamaterial are suggested and analyzed in support of the experimental work. Specifically, unidirectional frequency conversions and band gaps due to the second-order mode interactions are discussed for the first time when the large modulation amplitude is implemented. The suggested prototype sheds light on nonreciprocal waveguiding, which could be applied in advanced wave diodes, phononic logic, energy localization, trapping, and harvesting.

• Received 14 March 2019
• Revised 20 May 2019

DOI:https://doi.org/10.1103/PhysRevApplied.11.064052