Necessary modification of the Euler-Eytelwein formula for knitting machines

The Euler–Eytelwein formula, developed by Leonhard Euler (1707–1783) and Johann Alber Eytelwein (1764–1848), describes the friction of a rope or flat belt surrounding a cylindrical drum. It has been used for 200 years in order to describe friction of belts in machines, of ropes around bollards and other situations. However, this rope friction equation does not fit the situation in knitting machines where the diameters of yarn and needles are of the same order. As our experimental results illustrate, the Euler–Eytelwein formula has to be modified for this application, leading particularly to altered yarn tensions due to friction. In our article, we show the results of yarn tension measurements on different knitting machines and demonstrate the validity of the modified formula. Furthermore, we compare our calculation with an earlier attempt to adapt the Euler–Eytelwein formula to knitting machines.