Relation between Taper Angle and Amount of Yarn on a Beam

Relation between Taper Angle and Amount of Yarn on a Beam: 

Let,
s = Traverse length.
L = Axial
d = Empty beam dia.
D = Full beam dia.

dm = (D+d)/2 = mean dia. 

Where,

X = Tape distance 

α = Taper angle 

v = Volume of yarn stored on beam. 

Let, s > x so as to maintain stability 

       ∏D2L       ∏d2L

V = ………. -  ………….

         4               4

    ∏L

= ……..  (D2 – d2)

     4

          D+d       D-d

= ∏L  (…….) (……..)

             2           2

From figure, it is clear that 

         D+d     D-d 

dm = ……. & ………. = x tan α 

           2         2 

So, v = π L dm (x tanα) 

V > π L dm S tan α if, x > s 

V < π L dm S tan α if, x < s 

So, V ∞ S tan α if α = 90° then V = α 

So unlimited amount of yarns can be wound if flange stays perpendicular to beam barrel. Practically this is impossible. But this type of package permit’s to wind high amount of yarn.