Dynamics of snapping beams and jumping poppers

• Published in: Europhysics letters Vol. 105; no. 2; pp. 24001 - p1-24001-p6
• Main Authors: Pandey, A., Moulton, D. E., Vella, D., Holmes, D. P.
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences, IOP Publishing and Società Italiana di Fisica 01.01.2014; IOP Publishing
• Subjects: 46.32.+x; 46.70.De; 62.20.D; Arches; Beams (structural); Concentrated loads; Deformation; Dynamics; Elastic beams; Elastic buckling; Elastic deformation; Frequency stability; Instability; Mathematical analysis; Postbuckling; Propagation modes; Scaling laws; Shells; Shells (structural forms); Sound propagation; Sound waves; Stability; Stretchability; Wave propagation
• ISSN: 0295-5075; 1286-4854
• DOI: 10.1209/0295-5075/105/24001

We consider the dynamic snapping instability of elastic beams and shells. Using the Kirchhoff rod and Föppl-von Kármán plate equations, we study the stability, deformation modes, and snap-through dynamics of an elastic arch with clamped boundaries and subject to a concentrated load. For parameters typical of everyday and technological applications of snapping, we show that the stretchability of the arch plays a critical role in determining not only the post-buckling mode of deformation but also the timescale of snapping and the frequency of the arch's vibrations about its final equilibrium state. We show that the growth rate of the snap-through instability and its subsequent ringing frequency can both be interpreted physically as the result of a sound wave in the material propagating over a distance comparable to the length of the arch. Finally, we extend our analysis of the ringing frequency of indented arches to understand the "pop" heard when everted shell structures snap-through to their stable state. Remarkably, we find that not only are the scaling laws for the ringing frequencies in these two scenarios identical but also the respective prefactors are numerically close; this allows us to develop a master curve for the frequency of ringing in snapping beams and shells.