Tailoring band gaps for pantographic beams modelled by discrete elements

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Abstract
Determining the band gaps of metamaterials based on well-defined unit cells is a crucial line of investigation, as it provides fundamental design principles for tailored mechanical metamaterials with targeted properties. Practical applications notably include frequency-filtering devices, which are especially advantageous for structural vibration mitigation and acoustic insulation. For elementary unit cell geometries, dispersion relations can be derived analytically, yielding direct estimates of band-gap boundaries and actionable insights into parameter tuning. However, as unit-cell topologies become increasingly intricate -- a distinctive feature of modern metamaterials -- analytical approaches become intractable, necessitating numerical computational strategies. In this study, we investigate the dynamic behaviour of planar pantographic beams using a discrete element model composed exclusively of translational and rotational springs. Through numerical experiments, we delineate the parameter space governing band-gap formation, providing concise, practical guidelines for optimal unit-cell design. Ultimately, the framework presented herein offers a generalizable methodology for analysing more complex and topologically distinct unit-cell configurations.

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