Analysis of a reduced order nonlinear model of a multi-physics beam
Résumé
This article studies nonlinear behaviors of a two-dimensional beam with
piezoelectric patches on it. Governing electro-mechanical equations of the beam taking
into account several coupled piezoelectric patches, at different positions are derived.
Then, spatio-temporal 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 electro-mechanical 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.