  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.
