  We demonstrate, both experimentally and numerically, the mechanical rainbow
trapping effect and the mechanical Bloch oscillations for torsional waves
propagating in chirped mechanical beams. After extensive simulations, three
quasi-one-dimensional chirped structures were designed, constructed and
experimentally characterized by Doppler spectroscopy. When the chirp intensity
vanishes a perfect periodic system, with bands and gaps, is obtained. The
mechanical rainbow trapping effect occurs for small values of the chirp
intensity. The wave packet traveling along the beam is progressively slowing
down and is reflected back at a certain depth, which depends on its central
frequency. In this case a new kind of oscillation, here named "{\em rectified
rainbow-Bloch oscillation}", appears since the wave packet is reflected at one
side by the interface between the structure and the uniform rod and by the
minigap at the opposite side. For larger values of the chirp parameter the
rainbow trapping yields the penetration length where the mechanical Bloch
oscillations emerge. Numerical simulations based on the transfer matrix method
are in agreement with experimental data.
