Flexural wave propagation in locally-resonant beams with uncoupled/coupled bending-torsion beam-like resonators

A theoretical framework is presented to compute the dispersive properties of a locally-resonant Euler–Bernoulli beam hosting a periodic array of beam-like resonators, the bending-torsion response of which may be either uncoupled or coupled depending on the cross-section geometry. It is shown that, in the frequency domain, the reaction of every beam-like resonator can be expressed in an exact analytical form depending only on deflection and rotation of the host beam at the connection points. In this way, the beam-like resonators can be substituted by equivalent fictitious external supports with frequency dependent stiffness. On this basis: (a) band gaps of the infinite locally-resonant beam are easily computed employing a transfer matrix approach; (b) the frequency response, as well as the transmittance of the finite locally-resonant beam, are computed in closed analytical forms extending an efficient formulation recently proposed by the authors, based on generalized functions. In addition to presenting the novel theoretical framework, the main purpose of the paper is promoting coupled bending-torsion beam-like resonators as a novel concept for locally-resonant structures, which may prove even more efficient than conventional uncoupled bending-torsion beam-like resonators. Indeed, it is shown the former offer more and better opportunities to open band gaps, taking advantage of the inherent coupling between bending and torsional vibrations.