Steady states of elastically-coupled extensible double-beam systems
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'''' -...
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| Format: | Article |
| Language: | English |
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AIMS Press
2016-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | http://www.aimspress.com/article/10.3934/Math.2017.1.28/fulltext.html |
| _version_ | 1819268075229282304 |
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| author | Filippo Dell’Oro Claudio Giorgi Vittorino Pata |
| author_facet | Filippo Dell’Oro Claudio Giorgi Vittorino Pata |
| author_sort | Filippo Dell’Oro |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-23T21:27:17Z |
| format | Article |
| id | doaj.art-fcf8b2b57f2f4cd69147a04a1b38471c |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-23T21:27:17Z |
| publishDate | 2016-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-fcf8b2b57f2f4cd69147a04a1b38471c2022-12-21T17:30:35ZengAIMS PressAIMS Mathematics2473-69882016-12-0121286910.3934/Math.2017.1.28Steady states of elastically-coupled extensible double-beam systemsFilippo Dell’Oro0Claudio Giorgi1Vittorino Pata21 Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy2 DICATAM, Universit`a degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy1 Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, ItalyGiven $\beta\in\mathbb{R}$ 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.http://www.aimspress.com/article/10.3934/Math.2017.1.28/fulltext.htmlCoupled-beams structures| steady states| bifurcations| buckling |
| spellingShingle | Filippo Dell’Oro Claudio Giorgi Vittorino Pata Steady states of elastically-coupled extensible double-beam systems AIMS Mathematics Coupled-beams structures| steady states| bifurcations| buckling |
| title | Steady states of elastically-coupled extensible double-beam systems |
| title_full | Steady states of elastically-coupled extensible double-beam systems |
| title_fullStr | Steady states of elastically-coupled extensible double-beam systems |
| title_full_unstemmed | Steady states of elastically-coupled extensible double-beam systems |
| title_short | Steady states of elastically-coupled extensible double-beam systems |
| title_sort | steady states of elastically coupled extensible double beam systems |
| topic | Coupled-beams structures| steady states| bifurcations| buckling |
| url | http://www.aimspress.com/article/10.3934/Math.2017.1.28/fulltext.html |
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