  In this work, we investigate non-Hermitian elastic waveguides with
periodically applied proportional feedback efforts, implemented through
piezoelectric sensors and actuators. Using onedimensional spectral models for
longitudinal motion, it is shown that dispersion diagrams of this family of
structures exhibit non-reciprocal imaginary frequency components, manifesting
as wave attenuation or amplification along opposite directions for all pass
bands. The effects of positive and negative proportional feedback, as well as
local and non-local actuation are investigated. Overall, switching the sign of
the feedback effort inverts the amplification direction, while increasing the
degree of non-locality produces splitting of the pass bands into multiple bands
with interchanging non-reciprocal behavior. Furthermore, skin modes localized
at the boundaries of finite domains are investigated and successfully predicted
by the winding number of the complex dispersion bands. These results contribute
to recent efforts in designing metamaterials with novel properties associated
with the physics of non-Hermitian systems, which may find fruitful
technological applications relying on vibration and noise control, wave
localization, filtering and multiplexing.
