AI Insight
This study investigates the flutter instability of a flexible rod (modeled as a Cosserat rod) immersed in a viscous fluid and subjected to a follower force, focusing on the weakly nonlinear regime near the onset of instability. Using a multiple-scale perturbation expansion, the authors derive a Stuart-Landau amplitude equation that analytically characterizes the emergence of stable limit-cycle oscillations from a Hopf bifurcation. The analysis confirms a supercritical Hopf bifurcation in which the oscillation amplitude grows as the square root of the distance from the critical force threshold, consistent with results from fully nonlinear simulations.
Why it matters
This analytical framework provides a rigorous reduced-order model for predicting self-sustained oscillations in soft elastic structures, with direct relevance to the design and control of pressure-driven soft robotic arms and bio-inspired propulsion systems operating at low Reynolds numbers.
arXiv:2605.14702v1 Announce Type: cross
Abstract: We study the weakly nonlinear saturation of the flutter instability of a planar Cosserat rod in a viscous fluid driven by a terminal follower force. This instability, established in our preceding work as a Hopf bifurcation of a non-self-adjoint linear operator, produces stable limit-cycle oscillations in the fully nonlinear overdamped dynamics. Here we derive an analytical description of the emergence of this limit cycle near threshold. Working close to the critical follower force, we perform a multiple-scale expansion about the compressed straight base state and systematically remove secular growth at higher orders. Solvability at cubic order, enforced using the adjoint eigenmode of the non-Hermitian operator, yields a Stuart-Landau amplitude equation for the critical oscillatory mode. The Landau coefficients are expressed as explicit inner products involving the critical eigenmode, its adjoint, and quadratic corrections. The resulting reduced theory predicts a supercritical Hopf bifurcation with a steady-state tip oscillation amplitude scaling as the square root of the distance from threshold. These predictions rationalize the near-threshold scaling observed in nonlinear simulations and provide an analytical normal form for the onset of self-sustained beating in pressure-driven soft robotic arms at low Reynolds number.
Source: Weakly nonlinear analysis of Hopf bifurcations in the elastohydrodynamics of Cosserat rods