Paper: Falling Slinkys in JSV
Our manuscript entitled “Falling vertical chain of oscillators, including collisions, damping, and pretensioning” has been published in the Journal of Sound and Vibration. This work was authored by Raymond Plaut, and co-authored by Andy Borum, Douglas Holmes, and David Dillard.
A recent topic of interest has been the “levitating Slinky”. The Slinky spring is held at its top and hangs in a vertical equilibrium configuration. The top is then released and the Slinky falls downward. During an initial period of time, the bottom of the Slinky does not move. A similar phenomenon occurs if an elastic bar is held at its top and then released. Such a falling Slinky can be modeled by a discrete set of rigid masses, each one representing a coil (turn). The masses in the model are connected by massless springs, and also by dashpots if internal damping is not negligible. As for adjacent coils of a Slinky, adjacent masses of the model cannot penetrate each other, which introduces geometrical constraints on the system.
A chain of point masses connected by linear springs and sometimes dashpots is considered. The chain hangs in a vertical equilibrium configuration, held by its top mass. Then the top mass is released, and the chain falls. Internal damping, modeled by the dashpots, causes the bottom mass to move faster. As the system falls, upper masses sometimes accelerate faster than gravitational acceleration, and collisions may occur between adjacent masses. The types of collisions treated here include elastic, inelastic, and perfectly inelastic (in which colliding masses often stick together thereafter). The unstretched lengths of the springs, and a compressive force caused by pretensioning, may significantly affect the characteristics of the motion. Analytical and numerical results are presented for cases involving a few masses, and some generalizations are made for systems with an arbitrary number of masses. Also, the vertical chain may be used to model the motion of a falling Slinky after release at its top end. The bottom of the continuous Slinky does not move until the coils above it have collapsed onto it, and the collapse time is estimated here using the discrete chain model. For a metal Slinky with 86 masses, the estimated time is close to that previously obtained by a continuous elastic analysis.
