Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 119

the monograph of M. F. Glushko 1966. They reported from measurements of rope
lay length showing the twist and the stresses of hoist ropes in deep mine shafts.
In the following the rotary angle will be derived on the base of ( 2.77 ). That has
the advantage that the constants for the different wire ropes from Table 2.6 can be
used. Transforming ( 2.77 ) the twist angle is

x ¼ du

M c 1 d S
c 2 d 2 dx ¼ S þ c 3 G d 4 :

ð 2 : 84 Þ

The rope tensile force increases from the lower end with the rope length x and
the angle b F between the horizontal and the secant of the small rope bow
(S C m g L cos b F as normal in practice) approximately

S S 0 þ m g x sin b F

with the tensile force S 0 on the lower end and the length-related rope mass
m (exactly for b F = 90). Equation ( 2.84 ) is with that

M c 1 d ð S 0 þ m g x sin b F Þ

du¼ dx : ð 2 : 85 Þ

c 2 d 2 ð S 0 þ m g x sin b F Þþ c 3 G d 4

By integrating the rotatry angle u is

u ¼ c 1 x

M
c 2 d 2 m g sin b F

c 1 c 3 d G
c 2

c 2 d

þ

m g sin b F

ln½c 2 d 2 ð S 0 þ m g x sin b F Þþ c 3 G d 4 þ B : ð 2 : 86 Þ

As preproposed the rotary angle u is u = 0 for x = 0 and u = 0 for x = L.
From this and ( 2.86 ) the torque M and the constant B can be derived. The torque

is

M ¼ c 1 c 3 G d 3
c 2

c 1 d m g L sin b F

: ð 2 : 87 Þ

c 2 S 0 þ c 3 G d 2
c 2 S 0 þ c 2 m g L sin b F þ c 3 G d 2

ln

Then with ( 2.86 ) the rotary angle is

c 2 m g x sin b F
c 2 S 0 þ c 3 G d 2 þ 1

ln

u ¼ c 1 x
c 2 d c 1 L d

: ð 2 : 88 Þ

c 2 c 2 m g L sin b F

c 2 S 0 þ c 3 G d 2 þ 1

ln

The maximum rotary angle occurs for the rope length