The Phenomenon of Steady-State String Motion

Abstract

The paper examines the phenomenon of steady-state motion for a string traveling with constant velocity along an invariant curve under gravity in a viscous medium. This technically important phenomenon has been known in the literature for about 120 years and may be applied in high-speed turbines, the textile industry, etc. The conditions for the phenomenon’s existence are found. Concepts of two critical string velocities as well as sub, super, and hypercritical domains are introduced. The analytical solutions for the nonlinear differential equations and arbitrary constants for the general boundary conditions are found. The theoretical results are very close to the experimental ones.

On the dynamics of axially moving strings

“…Since we only study steady state solutions, we use the equations of motion derived in [3] which are in dimensionless variables…”

“…1 and Fig. 2, κ denotes the curvature, Q the shear force perpendicular to the string axis and H * is a horizontal force component including velocity effects ( [3]). With L, denoting the (constant) length of the cable between the highest points of the rolls, the following scalings have been performed…”

“…Since this problem has also many practical applications such as the motion of cable cars and lifts or of the paper in paper milling machines there are also many investigations available. We mention as relevant for our investigation [1,3,4].…”