Bloch Oscillations in Mechanical Vibrations

We show the emergence of Bloch oscillations in mechanical vibrations. This is done studying numericaly the dynamics of a wave packet traveling in a beam in which a chirped structure is designed at one end. The torsional wave packet with central frequency $f_{C}$ is sent in the uniform part of the beam. Its evolution across the structured beam is numerically simulated using the transfer matrix method (TM). In order to avoid the superposition of undesirable reflected packets at the free end of the uniform part, the beam has a very long uniform part. The chirped structure is formed by $n$ coupled beams formed between grooves. The distance $\ell_{n}$ between grooves is given according to the rule $\ell_{n}$ = $\frac{\ell_{0}}{1+n\gamma}$ , where $\ell_{0}$ is a given distance and $\gamma$ is the parameter that mimics the constant electric field in the Bloch oscillations. The former rule implies that the distance between neighboring resonances is constant. For comparison purposes the case $\gamma$ = 0 is studied. It is shown that, for γ = 0, the wave packet is totally reflected or partially transmitted by the structure when its central frequency $f_{C}$ lies in the bandgap or in the passband, respectively. For larger values of the chirp parameter γ the mechanical Bloch oscillations emerge with a period given by the inverse of the level spacing at the corresponding band.