Snapping of elastic strips with controlled ends

Abstract

Snapping mechanisms are investigated for an elastic strip with ends imposed to move and rotate in time. Attacking the problem analytically via Euler's elastica and the second variation of the total potential energy, the number of stable equilibrium configurations is disclosed by varying the kinematics of the strip ends. This result leads to the definition of a 'universal snap surface', collecting the sets of critical boundary conditions for which the system snaps. The elastic energy release at snapping is also investigated, providing useful insights for the optimization of impulsive motion. The theoretical predictions are finally validated through comparisons with experimental results and finite element simulations, both fully confirming the reliability of the introduced universal surface. The presented analysis may find applications in a wide range of technological fields, as for instance energy harvesting and jumping robots.

Elastica catastrophe machine: theory, design and experiments

“…Therefore, the generic planar section of S K at fixed values of d has shape and physical meaning definitely similar to those of the catastrophe locus of the classical Zeeman machine (see Fig. 1 left) having two canonical and two dual cusps, see [8] and [25]. A critical configuration with a certain sign of curvature at the two ends is characterized by symmetric angle θ S with the same sign.…”

“…7 It is worth mentioning that the present nomenclature differs from that used by some authors [7,17] defining the projection C C of the catastrophe set in the control (force) plane as bifurcation set and the snap-back set S K as catastrophe set. 8 It is noted that the 'catastrophe set' C P is a curve within the physical plane X l − Y l , obtained as the projection of the 'catastrophe set' C 3D P collecting the critical rotation angle Θ C l as third physical coordinate…”

“…Published in Journal of the Mechanics and Physics of Solids (2020) doi: 10.1016/j.jmps.2019.103735 The points B 1 (magenta in Fig. 4, left) common to both C (+) P and C (−) P subsets are present only for effective sets and always correspond to supercritical pitchfork bifurcations as the limit case displaying a non-snapping elastica [8]. Contrarily, the sharp corner points B 2 (yellow in Fig.…”

“…In particular, the triads {d, θ A , θ S } can be related to a unique or two different stable configurations through a function S K (d, θ A , θ S ) as [8] S K (d, θ A , θ S ) > 0 ⇔ monostable domain: one stable configuration,…”

“…The transition between the bistable and monostable domains (13) occurs for the set of critical conditions of snap-back for one of the two stable configurations, differing by the sign of curvature at the two ends. Such a condition can be represented through the concept of universal snap surface (restricted here to type 1 only [8]), which can be expressed in the following implicit form…”