Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 196

From this, the differential equation is

d 2 y

dx 2 1 q ¼ S y E J ;

ð 3 : 17 Þ

where q stands for the radius of curvature. With the abbreviation

r

S
E J

x ¼

in ( 3.17 ), it is

y 00 ¼ y x 2 : ð 3 : 18 Þ

The trial solution for this differential equation is

y ¼ c 1 e x x þ c 2 e x x

ð 3 : 19 Þ

with the deviations

y 0 ¼ x ð c 1 e x x c 2 e x x Þ
y 00 ¼ x 2 ð c 1

ð 3 : 20 Þ

e x x þ c 2 e x x Þ:

The boundary conditions are

for x ¼ 0 y ¼ 0

y 00 ¼ 2 = D for x ¼ x 0

because at the point x 0 , the radius of curvature of the tape is equal to the radius
R = D/2 of the sheave in the middle of the tape. From the boundary conditions we
get the constants

2
D x 2 ð e x x 0 e x x 0 Þ :

c 1 ¼ c 2 ¼

With these constants and ( 3.19 ) and ( 3.20 ), the equations of the bending line
and its deviations are

2
D x 2

sinh xx
sinh xx 0

y ¼

D x cosh xx

2

y 0 ¼

ð 3 : 21 Þ

sinh xx 0

y 00 ¼ 2

D sinh xx sinh xx 0 :