Dynamic Patterns of a Driven Hanging Chain: Swinging, Jumping, Knots.

Andrew Belmonte^1, Michael J. Shelley^2, Shaden T. Eldakar^1, and Chris H. Wiggins^2

When shaken periodically at one end, a hanging chain or string displays a startling variety of distinct dynamic
behaviors, depending on its length and the amplitude and frequency of the shaking. We find experimentally that
instabilities occur in tongue-like regions of parameter space. The unstable states observed include swinging and rotating
pendular motion, and also more complex, chaotic states. Mathematically, the dynamics are described by a nonlinear wave
equation. Linear stability analysis predicts instabilities within the well-known resonance tongues; their boundaries
agree well with our experiment. Full numerical simulations of the 3D dynamics reproduce and elucidate many aspects of
the experiment, indicating for instance that the kinetic energy of the entire chain is periodically zero in the pendular
state, and that sharp gradients in tension occur in the complex states. Experimentally the chain is also observed to tie
knots in itself, some quite complex, which modify its subsequent dynamics; however the trefoil knot does not remain tied,
but slips off the free end. These occurrences are beyond the reach of the current analysis and simulations.

^1 W. G. Pritchard Laboratories, Department of Mathematics, Pennsylvania State University, University Park, PA 16802

^2 The Courant Institute of Mathematical Sciences, New York University, New York, NY 10012