Chain paradoxes

Abstract

For nearly two centuries, the dynamics of chains have offered examples of paradoxical theoretical predictions. Here, we propose a theory for the dissipative dynamics of one-dimensional continua with singularities which provides a unified treatment for chain problems that have suffered from paradoxical solutions. These problems are duly solved within the present theory and their paradoxes removed-we hope.

Impact-induced acceleration by obstacles

“…In the present work, we confirm the existence of the effect in a ball chain and obtain quantitative data on trajectories and constitutive behavior, revealing multiple, at times competing, mechanisms and a complicated dependence on the angle of the impacted surface. We resolve some outstanding questions, and lend support to conjectured explanations and theoretical modeling from the work of both aforementioned groups as well as a dissipative shock model proposed by Virga [37].…”

“…We observed no clear connection between the onset of deviation and velocity or distance from the free end at impact, and no characteristic delay time. There may be a weak correlation between drop height (impact velocity) and total separation, but if so, this does not arise from an earlier onset of deviation, but rather from a dependence of incremental rate of deviation on velocity, something predicted by existing theories [34,35,37]. However, we find a characteristic delay length, such that a length of approximately 0.15 m of chain (45 beads) experiences impact before the onset of deviation.…”

“…The functional form (2) and the use of a single constitutive parameter to describe direct ball chain coiling impact, and by extension pickup [18,19,37], seem well supported by the data. We note that equation (2) is actually integrable [37,41]. Substituting F 2 h h = ¢ ( ) ( ) in (2), we obtain a first order linear differential equation,…”

“…Existing theory corresponds to impact on flat surfaces. The arguments presented by Hamm and Géminard [34] for a coiling chain, Ruina and co-workers [35] for a link, and Virga [37] for a continuum containing a dissipative shock all give rise to an equation of motion for the free length y(t) of the chain of the form y g y ÿ 1 0 , 1…”

Acquiring momentum: simple strategies by simple objects

“…Perhaps the most lucid explanations of the physics underlying the anomalous behavior are given in [4] and [5]. Agreement with theory, particularly [6], appears to be very good.…”