  This work studies the acoustic band structure of tensegrity metamaterials,
and the possibility to tune the dispersion relation of such systems by playing
with local and global prestress variables. Building on established results of
the Bloch-Floquet theory, the paper first investigates the linearized response
of chains composed of tensegrity units and lumped masses, which undergo small
oscillations around an initial equilibrium state. The stiffness of the units in
such a state varies with an internal self-stress induced by prestretching the
cables forming the tensegrity units, and the global prestress induced by the
application of compression forces to the terminal bases. The given results show
that frequency band gaps of monoatomic and biatomic chains can be effectively
altered by the fine tuning of local and global prestress parameters, while
keeping material properties unchanged. Numerical results on the wave dynamics
of chains under moderately large displacements confirm the presence of
frequency band gaps of the examined systems in the elastically hardening
regime. Novel engineering uses of the examined metamaterials are discussed.
