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Steady states of elastically-coupled extensible double-beam systems

1 Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
2 DICATAM, Universit`a degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy

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Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary modelin dimensionless form\begin{align*}\begin{cases}u'''' - \Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,d s\Big)u'' +k(u-v) = 0\\v'''' - \Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,d s\Big)v'' -k(u-v) = 0\end{cases}\end{align*}describing the equilibria of an elastically-coupled extensible double-beamsystem subject to evenly compressive axial loads.Necessary and sufficient conditionsin 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 solutionsfor 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 nonsymmetricenergy exchanges between the two beams, mimickingthe transition from vertical to torsional oscillations.
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