Exploiting dynamic bifurcation in elastic ribbons for mode skipping and selection

TL;DR

The paper addresses dynamic snap-through in pre-deformed elastic ribbons actuated by end rotation, revealing two distinct transition paths and the possibility to skip or select modes through dynamic loading. It combines an anisotropic Kirchhoff rod model, a discrete Elastic Rods (DER) numerical framework, AUTO-based static bifurcation analysis, and tabletop experiments to map static and dynamic behavior in 3D. Key findings include that all snap-through events are governed by saddle-node (fold) bifurcations, with robust delayed-bifurcation scaling ($T_s\sim \dot{\mu}^{-1/5}$, $\Delta\alpha\sim \dot{\alpha}^{4/5}$ in underdamped and $\Delta\alpha\sim \dot{\alpha}^{2/3}\nu^{2/3}$ in overdamped regimes), and that dynamic loading enables controlled mode skipping and mode selection via inertia and damping. The results yield general phase diagrams for mode skipping and selection, providing design principles for intelligent mechanical systems based on thin elastic structures.

Abstract

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, which is different from the classical snap-through of a two-dimensional (2D) bistable beam. Our theoretical model for the static bifurcation analysis is derived based on the Kirchhoff equations, and dynamical numerical 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 the saddle-node bifurcation, the same scaling law for the delay applies. We further demonstrate that, in elastic ribbons, by controlling the rotating velocity at the end, distinct snap-through pathways can be realized by selectively skipping specific modes, 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.

Figures (16)

• Figure 1: Problem setup. (A) A naturally straight elastic ribbon with length $L$ (in $x$ direction), width $W$ (in $y$ direction), and thickness $h$ (in $z$ direction). (B) The ribbon undergoes out-of-plane buckling to form mode $U_+$ (or $U_-$) when subjected to compression $\Delta L$ in the $x$ direction. (C) The buckled ribbon undergoes supercritical pitchfork bifurcation to form mode $U_+S_-$ or $U_+S_+$ by further applying shear $\Delta W$ to the right end in $y$ direction. (D) The snap-through buckling of the bifurcated ribbon can be triggered by rotating one end with angle $\alpha$ to achieve the mode transition through two possible paths: (E) Path $1$: $U_+S_- \rightarrow U_-S_+$ and, (F) Path $2$: $U_+S_+ \rightarrow U_+S_- \rightarrow U_-S_+$. (The nomenclatures are based on the midpoint orientation of the ribbon.)
• Figure 2: Snapshots of the elastic ribbon (top view) during the snap-through transition process for: (A) Path $1$: $U_{+}S_{-} \rightarrow U_{-}S_{+}$ and, (B) Path $2$: $U_{+}S_{+} \rightarrow U_{+}S_{-} \rightarrow U_{-}S_{+}$, which include (i)-(iii) Experimental configurations and, (iv)-(vi) the corresponding simulated configurations.
• Figure 3: Static bifurcation diagrams of an elastic beam. (A) A schematic diagram of the rotation-induced snap-through of beam. (B) The normalized midpoint height, $z(1/2) / L$, as a function of the rotational angle, $\alpha$, for a planner beam with different pre-compression, $\Delta L / L$. (C) The rescaled midpoint height, $Z(1/2) = z(1/2)/ L \cdot (\Delta L / L)^{-1/2}$, as a function of the rescaled rotational angle, $\mu = \alpha \cdot (\Delta L / L)^{-1/2}$. The symbols are from the discrete simulations and the lines from the theoretical solution. The fold points are marked as black dots.
• Figure 4: Static bifurcation diagrams of an elastic ribbon with pre-compression $\Delta L / L = 0.40$ and pre-shear $\Delta W / L = 0.37$. Snap-through process for the (A) Path 1, $U_+S_+ \rightarrow U_-S_-$, and, (B) Path 2, $U_+S_- \rightarrow U_+S_+ \rightarrow U_-S_-$. The black solid lines (stable equilibrium) and grey dashed line (unstable equilibrium) are from the theoretical solutions, and the blue symbols are from the discrete numerical simulations. The fold points are marked as black dots.
• Figure 5: Parametric analysis for the static bifurcation diagrams of an elastic ribbon. (A) The normalized midpoint height, $z(1/2) / L$, as a function of the rotational angle, $\alpha$, for the ribbon with fixed pre-shear, $\Delta W / L = 0.37$, and different pre-compression, $\Delta L / L \in [0.35, 0.39]$. (B) The critical rotational angle, $\alpha_{c}$, as a function of the pre-compression, $\Delta L / L$. (C) The normalized midpoint height, $z(1/2) / L$, as a function of the rotational angle, $\alpha$, for the ribbon with fixed pre-compression, $\Delta L / L = 0.40$, and different pre-shear, $\Delta W / L \in [0.35, 0.39]$. (D) The critical rotational angle, $\alpha_{c}$, as a function of the pre-shear, $\Delta W / L$. In (B) and (D), the solid lines are from the theoretical model, and the symbols are obtained from the discrete simulation.