Dynamic analysis and non-standard continualization of a Timoshenko beam lattice

Abstract

In this paper, a Timoshenko beam lattice, made up of a chain of masses and straight segments, is proposed, considering bending and shear deformation by means of linear rotational and transverse springs, respectively. Different standard and non-standard continualization methods are applied to it, highlighting here for the first time the suitability of taking the coupled discrete governing equations as a starting point for deriving new continuum models. Several novel low order non-classical continuum models are obtained, with the aim of reliably capturing size-effects and reflecting the dispersive behaviour of the discrete system. Low order governing equations prevents the need for extra boundary conditions when finite (bounded) solids are treated. An extensive analysis of the transition frequency, which initiates the shear propagation spectrum, has been carried out, examining its influence for the discrete and non-standard continuum models. The natural frequencies of a finite solid with two different boundary conditions are obtained through an edge treatment applied here for the first time to this kind of lattices, thus making it possible to solve the clamped-free edges configuration. The reliability of these approaches is evaluated by comparing their dynamic behaviours with that of the discrete system (taken as a reference), through both dispersion and vibration analyses, some of the new proposed continuum models successfully capturing the behaviour of the discrete one, even for high wavenumbers. Moreover, the appearance of physical inconsistencies is examined.