13—the Friction of a Yarn Lapping a Cylinder

An approximate form of the equation relating the on-going tension, T 1, to the off-going tension, T 2 as a yarn laps a cylindrical object for an angle θ can be written in the form T 2/T 1 = eαθ While this bears a superficial resemblance to the equation T 2/T 1 = eμθ, which is applicable to a material which obeys Amontons' laws of friction (i.e. μ is constant), it must be emphasized that α is not a constant but depends on the radius of curvature, ρ, of the cylinder and on the on-going tension. The conditions under which this equation can be used without introducing errors in T 2/T 1 greater than 5% are investigated. It is shown that the approximation is sufficiently close when αθ ≤ 0·5 and that, for a given value of θ, this condition is most likely to be fulfilled when the cylinder and/or yarns are rough and the yarns are lubricated. For a given yarn—cylinder combination, the value of α decreases as the value of (ρ/T 1) decreases, as the temperature increases, and as the velocity decreases.