  We study longitudinal and transverse wave propagation in beams with elastic
properties that are periodically varying in space and time. Spatiotemporal
modulation of the elastic properties breaks mechanical reciprocity and induces
one-way propagation. We follow an analytic approach to characterize the
non-reciprocal behavior of the structures by analyzing the symmetry breaking of
the dispersion spectrum, which results in the formation of directional band
gaps and produces shifts of the First Brilloin Zone limits. This approach
allows us to relate position and width of the directional band gaps to the
modulation parameters. Moreover, we identify the critical values of the
modulation speed to maximize the non-reciprocal effect. We numerically verify
the theoretical predictions by using a finite element model of the modulated
beams to compute the transient response of the structure. We compute the
two-dimensional Fourier transform of the collected displacement fields to
calculate numerical band diagrams, showing excellent agreement between
theoretical and numerical dispersion diagrams.
