Gaps in the Spectrum of a Periodic Family of Bodies Connected by Thin Elastic Rods

We investigate the low-frequency range of the spectrum of periodic spatial isotropic and homogeneous elastic waveguide composed of identical massive bodies connected by thin cylindrical rods. For a model problem on the periodicity cell, asymptotics of eigenvalues dependent on the Floquet variable are constructed by applying dimension reduction for the rods and analyzing the interaction of singular fields and rigid motions in the body. Additionally, the sizes and positions of spectral bands (wave passing zones) inside the low-frequency range are determined and open spectral gaps (wave stopping zones) are detected. We also formulate open questions, in particular, about the existence of open gaps in the spectrum of an analogous planar elastic waveguide.