# Rieter

### Twist formulas

#### Index

To elucidate several relationships involved in twisting, two yarns are considered below in a theoretical model. One yarn is assumed to be double the thickness of the other  [21]. Consider for each case a single fiber f and f', respectively (Fig. 61). Prior to twisting, these fibers lie at the periphery on the lines AC, A'C', respectively.

Assume that the yarns are clamped at the lines AG (A'G') and CD (C'D') and are each turned once through 360°. Then the fibers take up new positions indicated by the lines AEC and A'E'C', respectively. Each fiber can adopt this helical disposition only if its length is increased. However, owing to the greater diameter of yarn II, the extension of fiber f' must be significantly higher than that of fiber f.

The difference becomes clear if the yarns are rolled on a plane, whereupon two triangles (ABC and AB'C') are derived, each with the same height H. Fiber f has extended from H to l , while fiber f' has extended from H to L. The greater extension in yarn II also implies greater tension and thus more pressure towards the interior. The strength of yarn II is considerably greater than that of yarn I.

Fiber extensions in the yarn can be measured only with difficulty, so that they cannot be used as a scale of assessment of the strength to be expected. Such a scale could, however, probably be provided by an angle, for example, the angle γ of inclination to the axis. From the above considerations, it follows that yarn II has a higher strength than yarn I. Yarn II also has a greater inclination angle γ than yarn I.

The strengths (F) are proportional to the inclination angles:

$\frac {F_I}{F_{II}} = \frac { \gamma_1}{ \gamma_2}$

In other words, the greater the angle of inclination, the higher the strength. If the two yarns are to have the same strength, then the inclination angles must be the same, so that γ1 = γ2 (all other influencing factors being ignored here). This is only possible if the height of each turn in yarn I is reduced from H to h.

In the given example, yarn I must therefore have twice as much twist as yarn II (Fig. 62).

Fig. 61 – Winding of two fibers (f and f’ ) in yarns of different thickness

Fig. 62 – Number of turns of twist in thin yarns