Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 95

2.2.4.2 Longitudinal Oscillation of a Hanging Mass

A mass hanging on a wire rope can be made to oscillate along the axis of the rope.
Without taking the damping into consideration, the angular frequency is

r

c S
M

x 0 ¼

and the frequency

r :

c S
M

f 0 ¼ x 0

1
2 p 2 p ¼

ð 2 : 57 Þ

Here it is presupposed that the rope mass is much smaller than the hanging mass
M and can be neglected. The wire rope as a spring has the spring constant

c S ¼ E S ð r lower ; r upper Þ A
L

ð 2 : 58 Þ

with the rope elasticity module Es (r lower , r upper ), the metallic rope cross-section A
and the rope length L. When the stress amplitude changes, the rope elasticity
module will be nearly constant if the middle stress remains the same. The rope
elasticity module

E S ð r lower ; r upper Þ ¼E S ð r m r a Þ ð 2 : 58a Þ

with the amplitude r a and with the middle rope tensile stress

r m ¼ M g
A

ð 2 : 58b Þ

can be evaluated using ( 2.50 ) and ( 2.52 ) or Tables 2.3 and 2.4 .
The frequency of the hanging mass—neglecting the rope damping and other
dampings—can be calculated with the help of the Excel-program SEILELA2.XLS.
In addition to the frequency, the damping of the longitudinal vibrations is of
interest. Wehking et al. ( 1999 ) have made some decay tests. Figure 2.16 shows the
test situation. A main mass M and a dropping mass M A hang on a wire rope with
the diameter d = 10 mm and the length l = 12 m. After cutting the thin rope
between the main mass M and the dropping mass M A the main mass swings with
decreasing amplitude.
Figure 2.17 shows the typical behaviour of a decay test. Only the tests with
wire ropes which were ten times loaded before will be considered here. With the