Rotating strings

Abstract. Analytical expressions are provided for the configurations of an inextensible, flexible, twistable inertial string rotating rigidly about a fixed axis. Solutions with trivial radial dependence are helices of arbitrary radius and pitch. Non-helical solutions are governed by a cubic equation whose roots delimit permissible values of the squared radial coordinate. Only curves coplanar with the axis of rotation make contact with it. The classical problems of thin strings have attracted theoretical interest since at least the seventeenth century. They include the catenary, velaria, lintearia, and motion in a radial potential [1]. Closely related to the last are the rigid rotations of a string about a fixed axis. This problem and its generalizations, including additions of a uniform gravitational force, air drag, end masses, and tangential motion of material elements along the string, have been considered in diverse contexts by engineers, physicists, and mathematicians. There have been abstract studies [2–8], as well as models of textile manufacture [9–17], payload manipulation by space-, air-, and marine craft [18–25], jump (skipping) ropes and wind turbines [26–30]. Whether thought of as tethers whirling drogues, yarn balloons, or troposkeins, these flexible bodies adopt their configurations through a simple and ubiquitous process. Thus, it is rather surprising that a fully three-dimensional analytical treatment of the bare-bones problem remains unpublished. This situation is likely due to the imposition of certain boundary conditions that severely restricted the solution space explored by previous researchers. This note details solutions of the unaugmented problem, with general boundary conditions. Given a time-dependent curve X(s, t) parametrized by arc length s, the balance of inertia and line tension in an inextensible, perfectly flexible and twistable string of uniform mass density µ is described by the vector wave equation and metrical constraint µ∂2tX = ∂s (σ∂sX) , (1) ∂sX · ∂sX = 1, (2) where the stress σ is a multiplier field enforcing the constraint [7,31–37]. I will use two frames to describe the curve. The first is the moving triad of unit vectors (̂t, n̂, b̂), a