Modes of vibration, stability and internal resonances on non-linear piezoelectric small-scale beams

Abstract

This paper presents a non-linear piezoelectric size-dependent Rayleigh's beam model, based on modified couple stress theory, a non-classical continuum theory that is sensitive to the size effect. Hamilton principle is employed to determine the partial differential governing equations and the boundary conditions. Free periodic vibrations are studied, with the goal of investigating the modes of vibration of piezoelectric small-scale beams in the non-linear regime, a study that appears to be lacking in the literature. The method of multiple scales and a shooting method are applied to solve the equations of motion; the stability of the solutions computed by the shooting method is investigated. A hinged-hinged beam is chosen as an example to delineate the non-linear size dependent free vibration behaviour. For the first time in this specific problem, secondary branches due to bifurcations are found. An important novelty of the work is that these secondary branches are determined using the shooting method, which does not suffer from the drawbacks of popular alternatives, as perturbation methods and the method of harmonic balance.