Nodal profile control for networks of geometrically exact beams

TL;DR

This paper establishes the existence, uniqueness, and controllability of solutions for networks of geometrically exact beams, introducing an intrinsic model that simplifies analysis and control design for large motions.

Contribution

It introduces an intrinsic first-order hyperbolic model for beam networks, proves semi-global existence and uniqueness, and demonstrates local controllability of nodal profiles.

Findings

Existence and uniqueness of solutions for the IGEB model.

Local exact controllability of nodal profiles in networks with cycles.

Transfer of results from IGEB to GEB model via nonlinear transformation.

Abstract

In this work, we consider networks of so-called geometrically exact beams, namely, shearable beams that may undergo large motions. The corresponding mathematical model, commonly written in terms of displacements and rotations expressed in a fixed basis (Geometrically Exact Beam model, or GEB), has a quasilinear governing system. However, the model may also be written in terms of intrinsic variables expressed in a moving basis attached to the beam (Intrinsic GEB model, or IGEB) and while the number of equations is then doubled, the latter model has the advantage of being of first-order, hyperbolic and only semilinear. First, for any network, we show the existence and uniqueness of semi-global in time classical solutions to the IGEB model (i.e., for arbitrarily large time intervals, provided that the data are small enough). Then, for a specific network containing a cycle, we address the problem of local exact controllability of nodal profiles for the IGEB model -- we steer the solution to satisfy given profiles at one of the multiple nodes by means of controls applied at the simple nodes -- by using the constructive method of Zhuang, Leugering and Li [Exact boundary controllability of nodal profile for Saint-Venant system on a network with loops, in J. Math. Pures Appl., 2018]. Afterwards, for any network, we show that the existence of a unique classical solution to the IGEB network implies the same for the corresponding GEB network, by using that these two models are related by a nonlinear transformation. In particular, this allows us to give corresponding existence, uniqueness and controllability results for the GEB network.