Unevenness in the Appearance of Spun Yarns

Abstract

Object The relation between the apparent unevenness of a spun yarn and the apparent unevenness on the black board is discussed theoretically. The apparent unevenness on the black board is expressed by the total length of its various widths. Results1. The probability of the presence of uneven areas exceeding m in number in a band made of infinitely long, parallel yarns to the number of M is given as: P(_??_m, M)=1-∑<m-1><i=0> (M i)(α/β)i/(1+α/β)M, where α is the input density and β the output density of an uneven area in a yarn.2. The total length T of apparent uneven areas shown on a black board which has widths to the number of N and which is L in length is obtained as follows: T=L(N-M+1)[P(_??_m, M)-∑<m-1><j=1>P(_??_m, M-j)-{P(_??_m', M+1)-∑<M><j=1>p(_??_m', M+1-j)}]where m is the discrimination threshold for an uneven area having widths to the number of M, and m' is the discrimination threshold for widths to the number of M+1.3. The total length T of apparent uneven areas in ideal yarns and the total length T of service yarns Nos. 1 and 3 are calculated by using the foregoing results.4. The length distribution of apparent uneven areas is the transition probability of a high order in Markov's chain.