  Locally resonant metamaterials are characterized by bandgaps at wavelengths
that are much larger than the lattice size, enabling low-frequency vibration
attenuation. Typically, bandgap analyses and predictions rely on the assumption
of traveling waves in an infinite medium, and do not take advantage of modal
representations typically used for the analysis of the dynamic behavior of
finite structures. Recently, we developed a method for understanding the
locally resonant bandgap in uniform finite metamaterial beams using modal
analysis. Here we extend that framework to general locally resonant
metastructures with specified boundary conditions using a general operator
formulation. Using this approach, along with the assumption of an infinite
number of resonators tuned to the same frequency, the frequency range of the
locally resonant bandgap is easily derived in closed form. Furthermore, the
bandgap expression is shown to be the same regardless of the type of vibration
problem under consideration, depending only on the added mass ratio and target
frequency. It is shown that the number of resonators required for the bandgap
to appear increases with the target frequency range, i.e. respective modal
neighborhood. Furthermore, it is observed that there is an optimal, finite
number of resonators which gives a bandgap that is wider than the
infinite-absorber bandgap, and that the optimal number of resonators increases
with target frequency and added mass ratio. As the number of resonators becomes
sufficiently large, the bandgap converges to the derived infinite-absorber
bandgap. The derived bandgap edge frequencies are shown to agree with results
from dispersion analysis using the plane wave expansion method. Numerical and
experimental investigations are performed regarding the effects of mass ratio,
non-uniform spacing of resonators, and parameter variations among the
resonators.
