The Bloch oscillations (BO) and the rainbow trapping (RT) are two apparently unrelated phenomena, the former arising in solid state physics and the latter in metamaterials. A Bloch oscillation, on the one hand, is a counter-intuitive effect in which electrons start to oscillate in a crystalline structure when a static electric field is applied. This effect has been observed not only in solid state physics but also in optical and acoustical structured systems since a static electric field can be mimicked by a chirped structure. The RT, on the other hand, is a phenomenon in which the speed of a wave packet is slowed down in a dielectric structure; different colors then arrive to different depths within the structure thus separating the colors also in time. Here we show experimentally the emergence of both phenomena studying the propagation of torsional waves in chirped metallic beams. Experiments are performed in three aluminum beams in which different structures were machined: one periodic and two chirped. For the smaller value of the chirping parameter the wave packets, with different central frequencies, are back-scattered at different positions inside the corrugated beam; the packets with higher central frequencies being the ones with larger penetration depths. This behavior represents the mechanical analogue of the rainbow trapping effect. This phenomenon is the precursor of the mechanical Bloch oscillations, which are here demonstrated for a larger value of the chirping parameter. It is observed that the oscillatory behavior observed at small values of the chirp parameter is rectified according to the penetration length of the wave packet.

Theoretical and experimental results for in-plane vibrations of a uniform rectangular plate with free boundary conditions are obtained. The experimental setup uses electromagnetic-acoustic transducers and a vector network analyzer. The theoretical calculations were obtained using the plane wave expansion method applied to the in-plane thin plate vibration theory. The agreement between theory and experiment is excellent for the lower 95 modes covering a very wide frequency range from DC to 20 kHz. Some measured normal-mode wave amplitudes were compared with the theoretical predictions; very good agreement was observed. The excellent agreement of the classical theory of in-plane vibrations confirms its reliability up to very high frequencies Keywords: rectangular plate, in-plane vibrations, plane wave expansion method * Corresponding author, Tel. +52 55 562 27788; Fax: +52 55 562 27775. Email addresses: arreolaarturo@gmail.com (A. Arreola-Lucas), jofravil@fis.unam.mx (J. A. Franco-Villafañe), gbaez@correo.azc.uam.mx (G. Báez), mendez@fis.unam.mx (R. A. Méndez-Sánchez)

The mechanical rainbow trapping effect and the mechanical Bloch oscillations for torsional waves propagating in chirped mechanical structures are here experimentally demonstrated. 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 is experimentally characterized for small values of the chirp intensity. The wave packet traveling along the structure is progressively slowing down and is reflected back at a certain depth, which depends on its central frequency. For larger values of the chirping parameter the rainbow trapping yields the penetration length where the mechanical Bloch oscillations emerge. Numerical simulations based on the transfer matrix method show an excellent agreement with experimental data.

In this work, the Poincaré map numerical method was successfully developed to solve the fourth-order differential equation that describes the flexural vibrations of a beam, within the Timoshenko beam theory. The Euler-Bernoulli continuity conditions were considered, which are valid for frequencies smaller than the critical frequency. As an example, this method was used to design a complex elastic structure, characterized by a flexural frequency spectrum with a broad band gap. Such structure consists of two coupled phononic crystals, which were designed with filling factor values in such a way that in their bending frequency spectra, an allowed band of the first part, overlaps with a band gap of the second one and vice versa. The resulting composed system has a much wider effective gap than its original components, between 4 and 10.5 kHz. This system works as an elastic bending wave filter. Finally, these three structured elastic systems were constructed, and characterized by the acoustic resonance spectroscopy technique. The natural flexural frequencies as well as the corresponding wave amplitudes of each structured beam were measured. The experimental measurements show excellent agreement with the numerical simulation.

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