细长杆中Greenhill不稳定性的磁稳定化与滞回切换
Magnetic Stabilization of the Greenhill Instability and Hysteretic Switching in Slender Rods
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中文总结 AI 辅助
本文通过三维变分理论与实验，研究均匀磁场对细长铁磁杆Greenhill不稳定性的稳定作用，并揭示永磁体旋转引起的滞回切换机制，为场稳定细丝和双稳态致动器提供设计规则。
中文摘要 AI 辅助
在块体铁磁体中，磁能比弹性能弱六个数量级。细长性消除了这一差异。在细铁磁杆中，两种能量随长径比以不同方式缩放，即使在外加中等磁场下，普通镍丝也会发生非线性变形。我们建立了一个在重力和磁载荷作用下铁磁Kirchhoff杆的三维变分理论，并通过分岔分析、计算和实验进行研究。均匀磁场可以稳定垂直杆在自重下的屈曲，即经典的Greenhill不稳定性。我们给出了夹持和简支支撑的尖锐判据，以Airy型主特征值表示，并精细描述了分岔景观的特征，其与非磁性对应物具有定性不同的特征。当受到永磁体产生的磁场作用时，我们识别出丰富的分岔景观，因为磁体沿半径略大于杆的静止长度的圆拟静态旋转。我们证明不稳定性仅通过平面模式发生，并且每次跳跃都是非简并的鞍结。我们还定位了产生滞回的两个折叠。在$50\\,\mu$m镍丝上的实验确认了Greenhill阈值，在约$30$ mT下对夹持和简支支撑均实现再稳定，以及滞回环的显著特征，如切换角度、宽度和折叠几何形状，均准确。我们的分析和实验为场稳定细丝和远程切换的双稳态致动器提供了原则性设计规则。我们的分析还提供了一种良态数值方法，在切换事件附近有效，而直接数值最小化在此处严重病态。
英文摘要
In a bulk ferromagnet, magnetic energy is six orders of magnitude weaker than elastic energy. Slenderness removes this disparity. In a thin ferromagnetic rod, the two energies scale differently with aspect ratio, and an ordinary nickel wire deforms nonlinearly even under modest remote fields. We develop a three-dimensional variational theory of a ferromagnetic Kirchhoff rod under gravity and magnetic loading and study it through bifurcation analysis, computation, and experiment. A uniform field can stabilize a vertical rod against buckling under its own weight, the classical Greenhill instability. We give sharp criteria for clamped and pinned supports in terms of an Airy-type principal eigenvalue, and give a fine description of the character of the bifurcation landscape, which has qualitatively different features than its non-magnetic counterpart. When subject to the field generated by a permanent magnet, we identify a rich bifurcation landscape as the magnet is rotated quasistatically along a circle of radius slightly larger than the rest length of the rod. We prove that instability occurs only through a planar mode and that each snap is a nondegenerate saddle-node. We also locate the two folds that produce hysteresis. Experiments on $50\,μ$m nickel wires confirm the Greenhill threshold, restabilization at ~$30$ mT for both clamped and pinned supports, as well as the salient features of the hysteresis loop such as switching angles, width, and geometry of folds accurately. Our analysis and experiments offer principled design rules for field-stabilized filaments and remotely switched bistable actuators. Our analysis also offers a well-conditioned numerical method is valid near the switching events, where the direct numerical minimization is strongly ill-conditioned.
发表机构
- Indian Institute of Science（印度科学研究所）
- University of Utah（犹他大学）
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