Zhu Xintong, Pomerenk Olivia, Ristroph Leif
Applied Mathematics Laboratory, Courant Institute of Mathematical Sciences, Department of Mathematics, New York University, New York, NY 10012.
Proc Natl Acad Sci U S A. 2025 Jan 7;122(1):e2411588121. doi: 10.1073/pnas.2411588121. Epub 2024 Dec 30.
Mechanical systems with moving points of contact-including rolling, sliding, and impacts-are common in engineering applications and everyday experiences. The challenges in analyzing such systems are compounded when an object dynamically explores the complex surface shape of a moving structure, as arises in familiar but poorly understood contexts such as hula hooping. We study this activity as a unique form of mechanical levitation against gravity and identify the conditions required for the stable suspension of an object rolling around a gyrating body. We combine robotic experiments involving hoops twirling on surfaces of various geometries and a model that links the motions and shape to the contact forces generated. The in-plane motions of the hoop involve synchronization to the body gyration that is shown to require damping and sufficiently high launching speed. Further, vertical equilibrium is achieved only for bodies with "hips" or a critical slope of the surface, while stability requires an hourglass shape with a "waist" and whose curvature exceeds a critical value. Analysis of the model reveals dimensionless factors that successfully organize and unify observations across a wide range of geometries and kinematics. By revealing and explaining the mechanics of hula hoop levitation, these results motivate strategies for motion control via geometry-dependent contact forces and for accurately predicting the resulting equilibria and their stability.
具有移动接触点的机械系统——包括滚动、滑动和碰撞——在工程应用和日常体验中很常见。当一个物体动态探索移动结构的复杂表面形状时，分析此类系统的挑战会变得更加复杂，比如在呼啦圈运动这种常见但却不太为人理解的情境中就会出现这种情况。我们将这项活动作为一种独特的对抗重力的机械悬浮形式进行研究，并确定物体绕旋转体滚动时稳定悬浮所需的条件。我们结合了机器人实验，其中包括在各种几何形状表面上旋转呼啦圈，以及一个将运动和形状与所产生的接触力联系起来的模型。呼啦圈的平面内运动涉及与身体旋转的同步，这表明需要有阻尼和足够高的启动速度。此外，只有对于具有“臀部”或特定表面斜率的物体才能实现垂直平衡，而稳定性则要求物体呈带有“腰部”且曲率超过临界值的沙漏形状。对该模型的分析揭示了无量纲因子，这些因子成功地组织并统一了在广泛的几何形状和运动学范围内的观察结果。通过揭示和解释呼啦圈悬浮的力学原理，这些结果激发了通过与几何形状相关的接触力进行运动控制以及准确预测由此产生的平衡及其稳定性的策略。
