Abstract Summary
In this talk we discuss the aspects of non-material modelling of a flexible structure moving between two qualitatively different domains with a configurational force, acting at the boundary. In particular, we consider the dynamics of a flexible rod, partially inserted into an inclined rigid channel with little or no friction. Initially, the rod will slide downwards because of the gravitational force. The free segment vibrates with growing frequency and decreasing amplitude as the rod is injected further. The falling will be decelerated by the longitudinal contact force, which is acting at the interface between the free segment and the part inside the channel. This configurational force is expressed in terms of the work needed to straighten the curved rod to pull it into the channel. Eventually it will outweigh gravity such, that injection will change to ejection. Under circumstances the rod will fully eject out of the opening. The formulation of this dancing rod problem was first suggested by the authors of [1], who also performed physical experiments and provided a fascinating video of the dynamic process as supplementary material. Being restricted to the case of vanishing inertia of the rod with a concentrated mass at its tip, their mathematical model is based on the equations of Newtonian mechanics and reproduces the experimental observations. The numerical approach of the present contribution features a finite element discretization of the deformation of the free segment with respect to a normalized coordinate. The material length of the free segment is considered as an additional degree of freedom. This results into a mixed Eulerian-Lagrangian kinematic description of a sliding beam in the spirit of [2,3]: while the material particles move across the boundaries of the elements, we still manage to obtain expressions for the potential and the kinetic energy of the entire rod. Numerical experiments using both, classical Kirchhoff rod model and shear deformable Simo-Reissner theory demonstrate the existence of the critical initial length of the free segment, which, in dependence on other parameters, determines, whether the motion is quasi-periodic or the rod will completely eject out of the channel. Further investigations on the configurational force feature analytical solution of the contact problem for a flexible beam, confined in a channel with small width, which allows to consistently account for frictional interaction between the rod and the channel in the general setting. References [1] Armanini, C., Dal Corso, F., Misseroni, D., & Bigoni, D. (2019). Configurational forces and nonlinear structural dynamics. Journal of the Mechanics and Physics of Solids, 130, 82-100. [2] Humer, A., Steinbrecher, I., & Vu-Quoc, L. (2020). General sliding-beam formulation: A non-material description for analysis of sliding structures and axially moving beams. Journal of Sound and Vibration, 480, 115341. [3] Vetyukov, Y. (2018). Non-material finite element modelling of large vibrations of axially moving strings and beams. Journal of Sound and Vibration, 414, 299-317.
