(^544) „„„„„ A Textbook of Engineering Mechanics
Example 26.14. A block whose weight = W is suspended by means of a spring whose spring
constant = k 2 from the end of a rigid weightless beam, which is of length l and hinged to a wall by a
frictionless connection at its left. It is held in a horizontal position by a spring whose spring constant
= k 1. attached to it at a distance b from the hinge end to the ceiling as shown in Fig. 26.9.
Fig. 26.9.
Prove that (f), the natural frequency of vibration of the system in Hz is given by :
12
2
12
1
(^2) [(/)]
gk k
n
Wk l b k

π +
Solution. Fig. 26.9 shows OA the cantilever beam in the horizontal position with springs of
constant k 1 and k 2 and weight W attached at A. A little consideration will show, that if the weight W is
removed, the cantilever will spring upwards and A will occupy a new position A' as shown in the
figure.
Let T = Tension in the spring k 1.
Taking moments about O and equating the same,
T × b = W × l
or T Wl
b
= ...(i)
∴ Deflection of spring k 1 (at B) due to tension T
11
TWl
kkb
== ...(ii)
From the geometry of the figure, we find that
Deflection atBADeflection at
bl

∴ Deflection at A
2
1
1
×Deflectionat
lWl
lBWlkb
bbkb
===⎛⎞
⎜⎟
⎝⎠
and deflection of spring k 2 (at A) due to weight W
2
W
k