An exact approach to the dynamics of locally-resonant beams

This paper presents an exact analytical approach to calculate the dynamic response of elastic beams with periodically-attached resonators, generally referred to as locally-resonant beams. Showing that a typical resonator is equivalent to an external constraint, whose reaction force on the beam depends on the deflection of the application point through a pertinent frequency-dependent stiffness, the beam-resonators coupled system is handled using only the beam motion equation, with Dirac's deltas modelling the shear-force discontinuities associated with the reaction forces of the resonators. This is the basis to tackle the dynamics of infinite as well as finite beams, the first by a transfer matrix method to calculate frequency band gaps, the second by a generalized function approach. The dynamics of the finite beam is studied in frequency and time domains deriving the exact frequency response and the exact modal response, including modal frequency and impulse response functions. The proposed approach is formulated for arbitrary number of resonators and loads, and applies for both non-proportional and proportional damping.