Phased arrays have been a cornerstone of non-destructive evaluation, structural health monitoring and medical imaging for years due to their unique beam steering and focusing capabilities. Despite the recent advances in parallel beamforming and nonlinear imaging, such arrays are bounded by reciprocal symmetry which significantly limits the scope of their operation and applicability. Unlike band gap structures where nonreciprocity is often associated with a unidirectional diode-like behavior, a breakage of reciprocity in phased arrays manifests itself in the form different and independently tunable wave transmission (TX) and reception (RX) patterns. In this work, we present a combined analytical and experimental realization of an elastic phased array which operates within multiple frequency channels and is capable of simultaneous steering of multiple beams. To achieve this, we devise a class of phase shifters which augment a dynamic phase modulation on top of the conventional static phase gradient along the array transducers. As a result, the emergent array exhibits non-identical TX and RX profiles. The system’s performance is fully demonstrated via scanner laser vibrometer measurements of the displacement field and confirms the array’s ability to guide incident waves within frequency channels which are commensurate with the modulation rate and along the intended directional channels.
Non-reciprocal wave propagation in elastic structures has received considerable attention lately. A common mechanism to break elastic wave reciprocity is the use of phononic materials with traveling-wave-like properties. Among the popular methods to study wave dispersion in periodic media are the plane wave expansion and transfer matrix method (TMM). However, owing to the time-variant nature of such non-reciprocal systems, the implementation of both methods requires the truncation of harmonic terms. In this work, we adopt the TMM to extract the dispersion patterns of a moving phononic material with a prescribed velocity. In the presence of a temporal modulation of material properties in one direction accompanied by physical motion in an opposing direction, both effects cancel out and the problem becomes effectively time-invariant. This facilitates the analysis and yields interesting results. Subsequently, we exploit the well-established relationship between the momentumenergy spaces of moving and stationary elastic media to reconstruct exact dispersion diagrams of a stationary space-time-periodic system. The proposed approach provides a platform to incorporate the TMM in the analysis of non-reciprocal time-variant materials. Finally, given the lack of harmonic truncation, the accuracy of the new method does not diminish as the modulating speed increases.
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