In this paper, we systematically study the dynamic snap-through behavior of a pre-deformed elastic ribbon by combining theoretical analysis, discrete numerical simulations, and experiments. By rotating one of its clamped ends with controlled angular speed, we observe two snap-through transition paths among the multiple stable configurations of a ribbon in three-dimensional (3D) space; this is different from the classical snap-through of a two-dimensional (2D) bistable beam. Our theoretical model for the static bifurcation analysis is based on the Kirchhoff rod equations, and dynamical simulations are conducted using the Discrete Elastic Rods (DER) algorithm. The planar beam model is also employed for the asymptotic analysis of dynamic snap-through behaviors. The results show that, since the snap-through processes of both planar beams and 3D ribbons are governed by a saddle–node bifurcation, the same scaling law for the delay applies. We further demonstrate that by controlling the velocity of end-rotation, distinct snap-through pathways can be realized. In this way, we may selectively skip specific modes and, moreover, particular final modes can be strategically achieved. Through a parametric study using numerical simulations, we construct general phase diagrams for both mode skipping and selection of snapping ribbons. The work serves as a benchmark for future investigations on dynamic snap-through of thin elastic structures and provides guidelines for the novel design of intelligent mechanical systems.

中文翻译：

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利用弹性带中的动态分叉进行模式跳跃和选择


在本文中，我们通过理论分析、离散数值模拟和实验相结合，系统地研究了预变形弹性带的动态弹跳行为。通过以受控角速度旋转其夹紧端之一，我们观察到三维 (3D) 空间中带的多个稳定配置之间的两个快速过渡路径；这与二维 (2D) 双稳态光束的经典突跳不同。我们的静态分岔分析理论模型基于基尔霍夫杆方程，并使用离散弹性杆 (DER) 算法进行动态模拟。平面梁模型也用于动态弹跳行为的渐近分析。结果表明，由于平面光束和 3D 带状体的突跳过程均受鞍节点分叉控制，因此适用相同的延迟缩放定律。我们进一步证明，通过控制末端旋转的速度，可以实现不同的突跳路径。这样，我们可以选择性地跳过特定模式，并且可以策略性地实现特定的最终模式。通过使用数值模拟的参数研究，我们构建了模式跳跃和折断带选择的通用相图。这项工作为未来薄弹性结构动态突弹的研究奠定了基准，并为智能机械系统的新颖设计提供了指导。