  Elastic metamaterials are man-made structures with properties that transcend
naturally occurring materials. One predominant feature of elastic metamaterials
is locally resonant bandgaps, i.e., frequency ranges at which wave propagation
is blocked. Locally resonant bandgaps appear at relatively low frequency and
arise from the existence of periodically placed mechanical local resonators.
Typically, elastic metamaterials exhibit both locally resonant and
Bragg-scattering bandgaps, which can generally be different in width and
frequency ranges. This paper proposes two designs of active elastic
metamaterials that only exhibit locally resonant bandgaps, which are infinite
in number, evenly spaced in the frequency spectrum, and identical in width. The
mathematical model is established using the transfer matrix method and
synthesis of locally resonant bandgaps is achieved via an active elastic
support with carefully designed frequency-dependent stiffness. A single unit
cell of each proposed metamaterials is thoroughly studied, and its dispersion
relation is derived analytically, along with the periodically repeating bandgap
limits and widths. Following the dispersion analysis and bandgap parametric
studies, finite arrays of the proposed metamaterials are considered, and their
frequency response is calculated to verify the analytical predictions from
dispersion analyses.
