Extended Hamilton’s principle applied to geometrically exact Kirchhoff sliding rods

Abstract

This article addresses the dynamic modeling of geometrically exact sliding Cosserat rods. Such systems need to consider non-material time-varying domains to which the Lagrangian view point of solid mechanics is inappropriate. In the article here presented, we use the geometrically exact model of inextensible Kirchhoff rods along a non-material domain whose time variations are not necessarily imposed but are governed by the dynamics, i.e. depend on the configuration of the rod. To progress through derivation, we use the variational calculus on Lie group introduced by Poincaré, and apply it to an extension of Hamilton’s principle holding for open rod systems, which is derived in the article. This extended variational principle uses a moving non-material tube across which the material rod slides. The resulting closed formulation of sliding rods dynamics takes the form of a set of non-material Cosserat–Poincaré’s partial differential equations governing the time-evolution of the cross-section pause of the non-material tube, coupled with an ordinary Lagrange’s differential equation for the sliding motion of the rod across the tube. While emphasize is on the dynamic formulations, the modeling approach is numerically illustrated on a few examples related to the so called sliding spaghetti problem.