Effects of the local resonance in bending on the longitudinal vibrations of reticulated beams

This work investigates the dynamic behaviour of reticulated beams obtained by repeating a unit cell made up of interconnected beams or plates forming an unbraced frame. As beams are much stiffer in tension–compression than in bending, the longitudinal modes of such structures (governed by tension–compression at the macroscopic scale) can appear in the same frequency range as the bending modes of the elements. The condition of scale separation being respected for compression, the homogenization method of periodic discrete media is used to rigorously derive the macroscopic behaviour at the leading order. In the absence of bending resonance, the longitudinal vibrations of the structure are described at the macroscopic scale by the usual equation for beams in tension–compression. When there is resonance, the form of the equation is unchanged but the real mass of the structure is replaced by an effective mass which depends on the frequency. This induces an abnormal response in the neighbourhood of the natural frequencies of the resonating elements. This paper focuses on the consequences on the modal properties and the transfer function of the reticulated structure. The same macroscopic mode shape can be associated with several natural frequencies of the structure (but the deformation of the elements at the local scale is different). Moreover the vibrations are not transmitted when the effective mass is negative. These phenomena are first evidenced theoretically and then illustrated with numerical simulations.