Steady states of elastically-coupled extensible double-beam systems

TL;DR

This paper analyzes the steady states of a nonlinear, elastically-coupled double-beam system under axial loads, providing explicit conditions for solutions and revealing complex symmetric and nonsymmetric equilibrium behaviors.

Contribution

It derives necessary and sufficient conditions for equilibrium solutions and explicitly characterizes their Fourier modes, including the emergence of nonsymmetric solutions.

Findings

Solutions have at most three nonvanishing Fourier modes.

Nonsymmetric solutions can occur despite system symmetry.

Elastic energy distribution can be uneven among solutions.

Abstract

Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in dimensionless form $$\begin{cases} u"" - \Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases} $$ describing the equilibria of an elastically-coupled extensible double-beam system subject to evenly compressive axial loads. Necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes. In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutions for which the elastic energy fails to be evenly distributed. Such a feature turns out to be of some relevance in the analysis of the longterm dynamics, for it may lead up to nonsymmetric energy exchanges between the two beams, mimicking the transition from vertical to torsional oscillations.