Nonlinear dynamics and onset of steady precession of a ring on a vertical rod.

We report on the unexpected precessional motion that occurs when a small, rigid ring is rotated on a vertical smooth rod. Using high-speed imaging, two distinct regimes of motion are observed experimentally. (i) Oscillatory motion when the ring has a single contact with the rod (transient motion). (ii) Steady state (terminal velocity) when the ring has two instantaneously stationary contacts with the rod. These two regimes and the transition between them are analyzed through qualitative, analytical, and numerical means. One key feature of the steady-state motion is energy dissipation through rolling resistance and air drag. Our model predicts that steady-state motion occurs only when certain geometric conditions (the ratio of the radii of a ring and a rod, and the tilt angle) are fulfilled. As the steady-state regime is often found to be the preferred final state, the dynamic stability of the single contact configuration is investigated through bifurcation, phase-space, and linear stability analyses. Our numerical simulation accounts for the stick-slip transition at the contact point, and is in good agreement with experimental data across all experiments. Our findings have potential applications in various engineering fields, such as studying the vibrational detachment of threaded fasteners and developing gravity-driven centrifuges.