44—A Mathematical Model for the Ideal Sliver and its Applications to the Theory of Roller Drafting

The first part of the paper gives a precise mathematical formulation to Martindale’s ideas on sliver structure. Thus, an ideal sliver has been defined as an infinitely long structure composed of straight fibres which are parallel to the sliver axis and are so distributed along the axis that the probability of finding one fibre-end (right or left) in a small length δx at x (x being measured from arbitrary origin on the sliver axis) is vδx + o(δx). It has been shown that the various well-known properties of an ideal sliver, like the correlogram, V(L), B(L) and the spectral density can be deduced from the above definition in a consistent manner.The second part is devoted to a study of the effect of roller drafting on the structure of an ideal sliver. In order to do this, the following assumptions about the movement of fibres in the drafting zone have been made : the input sliver is composed of fibres of the same length L ;each fibre moves with the surface speed (v1 of the back rollers until it abruptly attains the surface speed (v2) of the front rollers and continues to move thereafter with the latter speed ;the motion of a floating fibre is influenced by that of the other fibres in its neighbourhood only by virtue of its overlaps with them ;The joint distribution of the transition times of any two fibres depends on their marginal distributions and their times of leaving the back rollers in a prescribed manner.It has been shown on the basis of the above assumptions that drafting waves are, in general, a consequence of the correlations between the transition times of fibres in the drafting zone. Further, taking the correlation coefficient of the transition times of any two fibres which leave the back rollers at specified times to be a linear and decreasing function of the lag between these times, a theoretical justification for the observed increase in irregularity has been given. In fact, it has been shown that where n1(t) is the number of fibres held by the front rollers, n0(t)) is the number of fibres held by the back rollers at time t, v = average number of right-hand ends of fibres per unit length of the ingoing sliver.