Yarn Geometry - Idealized yarn structure - Textile point

Yarn Geometry - Idealized yarn structure

Figure: (a) Idealized helical yarn geometry, (b) Opened out diagram of cylinder at radius r and (c) Opened out diagram of yarn surface

Assumptions –

1.      The yarn is assumed to be circular and uniform in cross-section.

2.      The yarn is composed of consecutive layers of different radial.

3.      A fiber in the center will follow a straight line.

4.      The density of packing (no of fiber in cross-section) fiber will remain constant.

Let,

R = Yarn radius (cm)

r = Radius of cylinder containing helical path of a particular fiber (cm)

T = Yarn twist per unit length

h = Length of yarn having one turn of twist

α = Helical angle of twist at yarn surface in degree

θ = Corresponding helical angle at radius r in degree

l = Length of fiber in one turn of twist at radius r (cm)

L = Length of fiber in one turn of twist at radius R (cm)

So, clearly

By cutting the concentric cylinders along a line parallel to the yarn axis and then opening the cylinder out flat, it is possible to obtain,

It is sometimes useful to establish a cylindrical polar co-ordinate system. The length along the yarn axis is denoted by z, the angular rotation about this axis by ϕ and the radial distance from the axis by r. The equations of fiber following a uniform helix in the idealized yarn are then

The length ‘q’ along the fiber is given by