Steady states of elastically-coupled extensible double-beam systems

Filippo Dell'Oro; Claudio Giorgi; Vittorino Pata; Filippo Dell'Oro; Claudio Giorgi; Vittorino Pata

doi:10.3934/Math.2017.1.28

AIMS Mathematics

2017, Volume 2, Issue 1: 28-69. doi: 10.3934/Math.2017.1.28

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Research article

Steady states of elastically-coupled extensible double-beam systems

1.
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
2.
DICATAM, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy

Received: 05 December 2016 Accepted: 12 December 2016 Published: 21 December 2016

Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in 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-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.
- Coupled-beams structures,
- steady states,
- bifurcations,
- buckling
Citation: Filippo Dell'Oro, Claudio Giorgi, Vittorino Pata. Steady states of elastically-coupled extensible double-beam systems[J]. AIMS Mathematics, 2017, 2(1): 28-69. doi: 10.3934/Math.2017.1.28

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Abstract

Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in 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-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.

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