Closing a catenary loop: the lariat chain, the string shooter, and the heavy elastica

TL;DR

This work analyzes the closed-loop equilibria of axially moving strings under gravity and drag, clarifying the role of drag and bending elasticity in forming loop-shaped configurations. It synthesizes historical results (e.g., Gregory) and recent analyses (ChakrabartiHanna16), identifies two critical bifurcations at $D=1$ and $D=2$, and shows that loop solutions exist only for moderate to high drag, with low-drag regimes needing bending regularization. By introducing bending stiffness (heavy elastica) and deriving global momentum/pseudomomentum balances, the authors demonstrate how small amounts of elasticity drastically alter loop shapes and enable physically admissible patching. The study also corrects misinterpretations in the literature, provides detailed numerical and analytical approaches, and situates the problem within broader contexts such as lariat chains and chain fountains, highlighting the practical importance of regularization for real systems.

Abstract

We review, critique, and extend results related to the problem of closed loop shape equilibria of a string shooter, a type of catenary consisting of steady, axially moving configurations of an inertial, inextensible, perfectly flexible string in the presence of gravity and drag forces. We highlight recurring misconceptions, and relate to similar problems, including the lariat (no gravity), chain fountain (not closed), and heavy \emph{elastica} (bending stiffness). We focus on the difficulty inherent to continuing a catenary through a vertical orientation, necessary to close a loop, which difficulty changes in nature as the system undergoes bifurcations with increasing drag. We construct solutions by implementing available analytical results, and numerically generate additional solutions with added bending stiffness. We briefly discuss global balances of linear, angular, and pseudo-momentum for this system.

Figures (5)

• Figure 1: A string shooter in operation, and its inventor StringLauncher. The string moves axially clockwise with speed $v$. The launch mechanism provides a force ${\bf{R}}$. (Annotations added to a still from YeanyYouTube, flipped horizontally for comparison with other figures in the present paper.)
• Figure 2: Catenary loops with low, moderate, and high drag ($D = 0.90, 1.2958, 2.50$) and launch angle $\theta(0) = \frac{\pi}{4}$. (a) Shapes. (b) Arc length $s$, (c) shifted tension $\sigma - v^2$, and (d) curvature $d_s\theta$, as functions of angle $\theta$. The moderate drag curvature cusp in (d) reaches the axis at $\theta=-\tfrac{\pi}{2}$ by orthogonal approach.
• Figure 3: Catenary loops with moderate drag $D = 1.2958$ and launch angles $\theta(0) = -\frac{\pi}{3}, 0, \frac{\pi}{4}, \frac{\pi}{3}$. (a) Shapes. (b) Arc length $s$, (c) shifted tension $\sigma - v^2$, and (d) curvature $d_s\theta$, as functions of angle $\theta$. Curves in (d) intersect on the axis at $\theta=-\tfrac{\pi}{2}$.
• Figure 4: Stiff catenary loops with bending stiffness $E = 0.00254$ and zero, low, and moderate drag ($D = 0, 0.90, 1.10$) and a kink-free vertical orientation at the support (launch angle $\theta(0)=\frac{\pi}{2}$, return angle $\theta(1)=-3\frac{\pi}{2}$), alongside a classical static non-stiff ($D=0, E=0$) catenary with an angle cutoff of $\pm0.05$. (a) Shapes. (b) Arc length $s$, (c) shifted tension $\sigma - v^2 + E\left(d_s\theta\right)^2$, and (d) curvature $d_s\theta$, as functions of angle $\theta$. The curvature of the catenary is approximately $-79.933$ at $-\pi$ and $0$, and approximately $-0.200$ at the patching point.
• Figure 5: Stiff catenary loops with moderate drag $D = 1.2958$, launch angle $\theta(0) = \frac{\pi}{4}$, and different different return angles or stiffnesses ($E=0.01$, $\theta(1) = -1.308\pi$), ($E=0.001$, $\theta(1) = -1.308\pi$), ($E=0.01$, $\theta(1) = -\pi$), alongside a non-stiff ($E=0$) catenary with the same drag and launch angle, and the same return angle ($\theta(1) = -1.308\pi$) as two of the stiff catenaries. (a) Shapes. (b) Arc length $s$, (c) shifted tension $\sigma - v^2 + E\left(d_s\theta\right)^2$, and (d) curvature $d_s\theta$, as functions of angle $\theta$.