Initially straight slender elastic filaments or rods with constrained ends buckle and form stable two-dimensional shapes when prestressed by bringing the ends together. Beyond a critical value of this prestress, rods can also deform off plane and form twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such deformed filaments subject to nonconservative active follower forces and fluid drag. We find that softly constrained filaments that are clamped at one end and pinned at the other exhibit stable two-dimensional planar flapping oscillations when active forces are directed toward the clamped end. Reversing the directionality of the forces quenches the instability. For strongly constrained filaments with both ends clamped, computations reveal an instability arising from the twist-bend-activity coupling. Planar oscillations are destabilized by off-planar perturbations resulting in twisted three-dimensional swirling patterns interspersed with periodic flipping or reversal of the swirling direction. These striking swirl-flip transitions are characterized by two distinct timescales: the time period for a swirl (rotation) and the time between flipping events. We interpret these reversals as relaxation oscillation events driven by accumulation of torsional energy. Each cycle is initiated by a fast jump in torsional deformation with a subsequent slow decrease in net torsion until the next cycle. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by nonconservative active forces. Taken together, our results suggest avenues by which prestress, elasticity, and activity may be used to design synthetic macroscale pumps or mixers.

中文翻译：

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预应力有源细丝的三维非线性动力学：拍打，涡旋和翻转

最初带有约束端的细长弹性细丝或棒弯曲并在将端头放在一起预紧时形成稳定的二维形状。除此预应力的临界值外，杆还可以在平面外变形并形成扭曲的三维平衡形状。在这里，我们分析了这种变形长丝在非保守主动跟随力和流体阻力作用下的三维不稳定性和动力学。我们发现，当作用力指向被夹紧的一端时，被一端夹紧并固定在另一端的受约束的细丝会表现出稳定的二维平面振摆振荡。反转力的方向性可以消除不稳定。对于两端受夹紧的强约束灯丝，计算揭示了由扭曲-弯曲-活动耦合引起的不稳定性。平面振动由于平面外扰动而不稳定，从而导致扭曲的三维旋流图案散布着旋涡方向的周期性翻转或反转。这些引人注目的漩涡-翻转过渡具有两个不同的时间尺度：漩涡（旋转）的时间段和翻转事件之间的时间。我们将这些逆转解释为由扭转能量积累驱动的弛豫振荡事件。每个循环都是通过扭转变形的快速跳跃而开始的，随后净扭力的缓慢减小直到下一个循环。我们的工作揭示了当弱惯性强阻尼杆由于非保守作用力而变形时的丰富时空图案。在一起