Nonlinear Dynamics of a Geometrically Exact Slender Rod Subjected to a Moving Mass

In this paper, a geometrically exact nonlinear formulation of a planar Cosserat rod is presented under the assumptions of inextensibility and unshearability. The Cosserat rod model is discretised using the generalised-alpha method. This numerical method is shown to exhibit good convergence and stability for all models presented in this paper. The discretised nonlinear model is tested and verified for a simple cantilever problem. The discretised Cosserat model is extended by deriving equations that couple a moving mass to the rod by using interpolation and an approximate delta function. The numerical method remains stable under mass-rod coupling and shows no signs of shear locking commonly encountered in FEM formulations. The moving mass-rod model is applied to an arch problem, with interesting nonlinear phenomena observed.  Instabilities of the arched rod during in-plane collapse are investigated for various moving-mass weights and initial velocities. Adjusting these parameters produces a stabilising (delaying) effect on arch failure. The critical initial velocity of the moving mass required to traverse the whole arch by reaching the apex and subsequently the right-hand support is found. The moving mass-rod model is also applied to the classical Brachistochrone problem to determine the minimum descent time of a mass falling down a nonlinear, flexible rod. For all moving mass-rod tests, energy is shown to be conserved up to discretisation error. We provide comparisons between moving mass, moving load, nonlinear and linear models, highlighting contrasting behaviour to illustrate the effects overlooked when nonlinearity and moving-mass inertia are ignored.