Capture into slow-invariant-manifold in the fluid-structure dynamics of a sprung cylinder with a nonlinear rotator

We investigate the dynamics of a two-dimensional circular cylinder mounted on a linear spring, restricted to move in the cross-flow direction and undergoing vortex-induced vibration, incorporating a strongly nonlinear (i.e., non-linearizable) internal element consisting of a mass that is free to rotate about the cylinder axis and whose angular motion is restrained by a linear viscous damper. The conjunction of the essentially nonlinear inertial coupling with the dissipative element makes the internal attachment behave as a nonlinear energy sink that is able to extract and dissipate energy from the motion of the cylinder and (indirectly) the surrounding fluid. At the intermediate Reynolds number Re=100, we find that the cylinder with rotator undergoes repetitive cycles of slowly decaying oscillations interrupted by chaotic bursts; during the slowly decaying portion of each cycle, the dynamics of the cylinder is regular and can lead to significant vortex street elongation with partial stabilization of the wake. We construct a reduced-order model of the fluid-structure interaction dynamics based on the data obtained by direct numerical simulation, and employ analytical techniques such as complexification/averaging and the multiple-scales method to show that the strongly modulated response is the manifestation of a resonance capture into a slow invariant manifold (SIM) that leads to targeted energy transfer from the cylinder to the rotator. Capture into the SIM corresponds to transient cylinder stabilization, whereas escape from the SIM leads to chaotic bursts. Hence, the action of the nonlinear rotator on the resonance dynamics of the fluid-structure interaction is clarified.