Analysis of a reduced-order nonlinear model of a multi-physics beam

Abstract

This article studies nonlinear behaviors of a two-dimensional beam with piezoelectric patches on it. Governing electromechanical equations of the beam taking into account several coupled piezoelectric patches, at different positions, are derived. Then, spatiotemporal variables of the system are separated. A methodology is proposed, for the detection of different mode functions and corresponding frequencies of the multi-physics beam, via using space equations of the system. As a representative example, a homogeneous beam with a single piezoelectric patch is considered. The paper is followed by consideration of two particular cases: (i) a single mode of the system is in resonance with the direct lateral-base excitation and (ii) two modes of the system present an internal resonance, while the first one is in resonance with the lateral-base excitation. For both cases, the electromechanical system equations are projected on its targeted mode(s). The temporal equations are treated via a multiple scale method leading to detections of its fixed points. The effects of one of the nonlinear coefficients of the piezoelectric patch, on the overall responses of the multi-physics beam, in terms of changing its behavior from hardening to softening (or vice versa), are discussed and commented upon. Moreover, it is shown that the piezoelectric patch is able to control the targeted mode(s) of the system.