Page 66 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.7 sprinG elements 63

A B T

0.3 m 0.2 m 0.25 m
1 2 0.15 m 3

A B
2 m 3 m

FiGure 1.30 Propeller shaft.

two segments 12 and 23 are to be considered as series springs. The spring constants of segments
) are given by
12 and 23 of the shaft (k t 12 and k t 23
4 4 9 4 4
GJ 12 Gp1D 12 - d 12 2 180 * 10 2p10.3 - 0.2 2
= = =
k t 12
l 12 32l 12 32122
6
= 25.5255 * 10 N@m>rad
4 4 9 4 4
GJ 23 Gp1D 23 - d 23 2 180 * 10 2p10.25 - 0.15 2
= = =
k t 23
l 23 32l 23 32132
6
= 8.9012 * 10 N@m>rad
Since the springs are in series, Eq. (1.16) gives
k 6 6
6
k t 12 t 23 125.5255 * 10 218.9012 * 10 2
= = = 6.5997 * 10 N@m>rad
k t eq 6 6
125.5255 * 10 + 8.9012 * 10 2
k t 12 + k t 23
■
equivalent k of hoisting drum
example 1.7
A hoisting drum, carrying a steel wire rope, is mounted at the end of a cantilever beam as shown in
Fig. 1.31(a). Determine the equivalent spring constant of the system when the suspended length of
the wire rope is l. Assume that the net cross-sectional diameter of the wire rope is d and the Young’s
modulus of the beam and the wire rope is E.

Solution: The spring constant of the cantilever beam is given by
3EI 3E 1 Eat 3
3
k b = = ¢ at ≤ = (E.1)
b 3 b 3 12 4b 3