Networks of geometrically exact beams: Well-posedness and stabilization

Abstract

<p style='text-indent:20px;'>In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) – in the sense that large motions (deflections, rotations) are accounted for in addition to shearing – and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin &amp; Coron in [<xref ref-type="bibr" rid="b2">2</xref>] that the zero steady state of this network is exponentially stable for the <inline-formula><tex-math id="M1">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.</p>