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Networks of geometrically exact beams: Well-posedness and stabilization

  • * Corresponding author: Charlotte Rodriguez

    * Corresponding author: Charlotte Rodriguez

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex

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  • 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 & Coron in [2] that the zero steady state of this network is exponentially stable for the $ H^1 $ and $ H^2 $ 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.

    Mathematics Subject Classification: 35L50, 35R02, 93D15.

    Citation:

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  • Figure 1.  Left: tree-shaped network with $ N = 8 $ edges, $ \mathcal{N}_S = \{0, 3, 5, 6, 7, 8\} $ and $ \mathcal{N}_M = \{ 1, 2, 4 \} $. Right: star-shaped network with $ N = 4 $ edges, $ \mathcal{N}_S = \{0, 2, 3, 4\} $ and $ \mathcal{N}_M = \{ 1 \} $

    Figure 2.  A multiple $ n $ and the incident edges and nodes

    Figure 3.  The $ i $-th beam in its different configurations $ \Omega_s^i $, $ \Omega_c^i $ and $ \Omega_t^i $

    Figure 4.  Characteristic curves $ (\mathbf{x}(t), t) $ with $ \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}(t) = \lambda(\mathbf{x}(t)) $, where either $ \lambda(s)>0 $ or $ \lambda(s)<0 $ for all $ s \in [0, \ell] $

    Figure 5.  Example of choice of $ \rho $ and the weight functions

    Figure 6.  Recovering the single beam case: example of choice of $ \rho $ and weight functions

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