Low-frequency nonreciprocal flexural wave propagation via compact cascaded time-modulated resonators

Nonreciprocal mechanical devices are of great interest for directional elastic wave manipulation. In this Letter, we introduce a design of a compact low-frequency nonreciprocal metamaterial for flexural waves, whose dimension is less than 1 / 3 of the operating wavelength. This structure is made of two well-placed coil-cantilever-magnet resonators, where the electromagnetic forces can be temporally modulated, which enables time varying of the effective stiffness of the resonators. A phase shift is introduced between the stiffness modulations of these two resonators, which breaks the time-reversal symmetry and enables nonreciprocal wave propagation at the resonance frequency of the structure. A semi-analytical method based on harmonic wave decomposition is developed to describe the system, leading to results that match well with numerical predictions from a finite element method. We also experimentally demonstrate nonreciprocal flexural wave propagation with good agreement with the predictions made. Our system could inspire the design of compact nonreciprocal devices for flexural waves.