New low-order continuum models for the dynamics of a Timoshenko beam lattice with next-nearest interactions

Abstract

In this paper, the dynamic behaviour of a novel Timoshenko beam lattice with long-range interactions, accounting for both bending and shear deformations, is investigated. Several new non-classical continuum models are developed with the aim of capturing its dispersive behaviour with a lower computational cost. For this, innovative continualization procedures are used, comparing them with techniques commonly used in lattices continualization, as well as with advanced ones. Moreover, low-order continuum governing equations are pursued, thus avoiding the need for extra boundary conditions, whose physical meaning is unclear. A comprehensive analysis of the transition frequency, which initiates the shear propagation spectrum, has been performed here for the first time for this lattice and the corresponding continuum models. The capability of these continuum models in capturing the behaviour of the lattice is assessed by conducting both dispersion and natural frequency analyses, for the latter providing an original method to treat the edges for the three possible boundary conditions in Timoshenko beam lattices. The influence of long-range interactions is analysed, and the way shear effect affects the shape vibration modes of the discrete model is interestingly illustrated, finally concluding that some of the new developed continuum models accurately capture the behaviour of the lattice.