Extracting torsional band gaps and transient waves in phononic crystal beams: Method and validation

Abstract

Due to an absence of proper measuring techniques, torsional waves/vibrations in one-dimensional phononic crystals (PCs) or elastic metamaterials are usually theoretically or numerically studied without experimental validations. This work proposes a measurement method capable of extracting torsional band gaps and transient torsional waves from the total responses of PC and metamaterial beams. Specifically, a pair of point-wise self-demodulated fiber Bragg grating (SFBG) displacement sensors, which can be arranged at points very close to each other on a thin PC beam, is utilized along with the proposed extraction method. The SFBGs are symmetrically set up at two locations about the central line on the surface of the PC beam. The torsional band gaps and the transient torsional waves are extracted from the differential responses of the two SFBGs, in which the common bending waves are cancelled out. In addition to the torsional behaviors, the bending waves can also be extracted from the summation responses of the two SFBGs. We first examine the dynamic linearity of the SFBGs. Then, steady-state detection of the torsional and bending band gaps of a PC beam is performed. Finally, a cantilever PC beam subjected to an eccentric steel ball impact, whose loading history is recorded by a pair of polyvinylidene fluoride (PVDF) films, is investigated to demonstrate the extraction of the torsional/bending bandgaps and waves from the early short time transient responses. The feasibility of the proposed torsional (or bending) extraction method is justified by the method of reverberation-ray matrix (MRRM), which is capable of providing “pure” torsional (or bending) behaviors of the PC beam. With the proposed extraction method, we also observe surface localized modes in the transmission of the PC beam, which are strongly related to the arrangement of the constituent components with different impedances in the unit cells. We believe that the proposed extraction method for torsional wave, along with the SFBG sensing technique and the MRRM, can provide future researchers with measurement and analysis method to validate their designs of torsional phononic crystals or metamaterials.