17—a Theory of Roller Drafting

A mathematical theory of the drafting of a sliver of fibres whose length is negligible has been given by Cox. A corresponding theory for fibres of finite length requires a knowledge of the time taken by a fibre to pass from the back rollers to the front ones, and this in turn depends on the mechanical action between contiguous fibres. In this paper, it is assumed that the probability that a floating fibre has the speed of the back or front rollers is proportional to the number of contacts it has with other fibres moving with these speeds; and an integral equation is deduced for the distribution of the fibres in the sliver between the rollers. Some typical nurrierical results are given. The first part of the paper gives the theory for a sliver composed of fibres that are all of the same length; the second part deals with a sliver in which there is a fibre-length distribution.