Engineering Mechanics

Chapter 26 : Helical Springs and Pendulums „„„„„ 533

Notes. 1. We have considered only two springs for the sake of simplicity. But this principle
may be extended for any no. of springs.

2.All the other relations regarding periodic time, frequency etc. are also applicable in this
case.

Example 26.6. A block of mass 50 kg supported by two springs connected in series hangs
from the ceiling. It can move between smooth vertical guides. The spring constants are 4 kN /m and
6 kN /m as shown in Fig. 26.3.

Fig. 26.3.
The block is pulled 40 mm down from its position of equilibrium and then released. Determine
(a) period of vibrations, maximum velocity and acceleration of the block.
(b) quantities in (a) above, when the block is supported by the springs connected in parallel.
Solution. Given : Mass (m) = 50 kg = 0.005 t, Stiffness of first spring (s 1 ) = 4 kN/m ;
Stiffness of second spring (s 2 ) = 6 kN /m and displacement (r) = 40 mm = 0·04 m.

(a) When the springs are connected in series

We know that spring constant of an equivalent spring,

12

11 1 1110 1

ss s46242·4

= + =+= =

or s = 2·4 kN/m

and deflection of the spring 0.05 9.8 0·204 m
2·4

mg

s

×

δ= = =

∴ Period of vibrations,

0·204

22 0·91s

9·8

t

g

δ

=π =π = Ans.

We know that the angular velocity of the block,

22

6·9 rad/s

t 0·91

ππ

ω= = =

∴ Maximum velocity,

vmax = ωr = 6·9 × 0·04 = 0·276 m/s Ans.

and maximum acceleration, amax = ω^2 r = (6·9)^2 × 0·04 = 1·9 m/s^2 Ans.