The most common method used to analyze the drafting process is the so-called “geometrical” method. Many theories on the drafting have been developed by this method or by following its geometrical concept.
It is common knowledge that the irregularity of slivers thickness has statistic characteristics. It is very difficult to clarify by the geometrical method the relation between this statistic irregularity and the drafting process. This articles deal with an “analytical” method which we have built up (instead of the geometrical method) to solve these complex problems and to calculate some characteristics of the irregularity found in drafted slivers.
The analytical method has made it clear, among other things, that the drafting process has an oscillatory behavior; that its resonance wave length is about 2_??_3 times the mean fiber length; and that the oscillatory behavior differs according to the staple diagram of drafted slivers and is most conspicuous when square-cut staple fiber are drafted.

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Draft-cut length and draft-cut wave are theoretically and experimentally described in this article. Draft-cut length is affected by the draft ratio, breaking elongation and the draft-cut position in which a fiber is broken. Major causes for draft-cut wave are the change and distribution of the breaking position of a fiber in the draftcut zone.

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To discuss the properties of sliver irregularity produced by roller drafting, the author has completed the following procedures:
As the first step, stochastic treatment was made to analyze the behaviors of the fiber in the draft zone. By using this formula, the distribution of the velocity change point, ρ(x), was deduced as a special case of the general representation.
As the second step, on the basis of the above-mentioned interpretation, a gene-ralized draft equation which is the differential equation for the fiber density of the front end, n(x t), was deduced. The two-velocity model method is considered an approximate equation constituting a simplified case of the generalized draft equation.
By using those equations, the sliver irregularity of the weight per unit length was studied from two points of view. One is the causal interpretation. The other is the statistical interpretation. The causal interpretation means the “one to one correspondence” of the sliver irregualrity.
As the last step, physical quantity representing the behavior of the fiber in the draft zone, was related to the spectral density used in the theory of the sliver structure.

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A new instrument that measures drafting force continuously and eliminates short term variations in sliver weight has been developed by the authors (Figs. 1, 2).
The electronic instrument consists of magnetic power amplifiers (Fig. 3) with a loop transfer function H (S) as given by eq. (6).
The instrument has been found to successfully reduce short term sliver weight irregularities. The effect of loop gain and time constant on sliver irregularity has also been investigated. Also by changing the corner frequency ω_a of the instrument, the effect of ω_a on the wave length of sliver irregularities has been obtained (see Fig. 18).

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The authors studied theoretically the behavior of fiber bundles as they are delivered from the front rolls of a ring spinning frame and twisted. The causes of the long-term twist irregularity found in some yarns and their effects on yarn properties have been analyzed.

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