  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 \cite{BC2016} 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.
