Theoretical and Numerical Investigation of a Special Case of a Rotating Chain: The Helicoseir Problem

A physically simple but mathematically cumbrous problem of rotating heavy chain with one fixed top point is studied. Nonlinear partial differential equations of motion in 3D case are derived as well as ones for 2D shapes of relative equilibrium. The equations are found very complicated and solved numerically. A linear case of small displacements is analyzed in terms of Bessel functions. The qualitative and quantitative behaviour of the problem is discussed with the help of bifurcation diagram. Dynamics of the two-dimensional model near the equilibrium positions is studied with the help of simulation using the absolute nodal coordinate formulation (ANCF); the equilibriums are found unstable. The reason of instability is explained using a variational principle. Analysis of 3D motion in the rotating frame shows that Coriolis inertia forces can stabilize the motion. 3D numerical simulation using ANCF shows that the spatial motion of the helicoseir is stable and looks like self-excited oscillations near the flat instable configurations that were obtained before.