Bending-wave localization and interaction band gaps in quasiperiodic beams

Aperiodic metamaterials can host topological or localized wave modes, and gradually changing structures are known to induce conversion between elastic wave types. This work describes a family of one dimensional, aperiodic elastic structures that can host localized modes or achieve low-frequency attenuation by varying a single geometrical parameter, i.e., the ratio between the sizes of two interacting periodic structures. Two periodic sets of slits on the top and bottom of a slender beam scatter bending waves. The thickness profile and cross section of beams with a period ratio close to 1 varies slowly, resulting in localized wave modes. The localization happens because of spatially confined band gaps and is only triggered for certain excitation locations, unlike wave localization due to a local defect. If the periods differ more, broad band gaps appear at wave numbers equal to linear combinations of the individual Brillouin zone edges of each set of slits. We call these interaction band gaps, since they result from the interplay between two periodic sets of slits. Low-frequency wave attenuation is achieved while maintaining a high overall bending stiffness.