Page 64 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.7 sprinG elements 61
W k d
1 1
k 1
k 1 k 1
W
1 d
2 2
k 2 W k d
k 2
k 2
1 d
st d W
2 d
W W

(a) (b) (c)
FiGure 1.28 Springs in series.

If k denotes the equivalent spring constant, then for the same static deflection,
eq
W = k d (1.14)
eq st
Equations (1.13) and (1.14) give

k d = k d = k d
2 2
eq st
1 1
or
k d k d
eq st
eq st
d = and d = (1.15)
1
k 1 2 k 2
Substituting these values of d and d into Eq. (1.12), we obtain
2
1
k d + k d
eq st
eq st
k 1 k 2 = d st
That is,
1 1 1
= + (1.16)
k eq k 1 k 2
Equation (1.16) can be generalized to the case of n springs in series:

1 1 1 1
= + + g + (1.17)
k eq k 1 k 2 k n
In certain applications, springs are connected to rigid components such as pulleys, levers,
and gears. In such cases, an equivalent spring constant can be found using energy equiva-
lence, as illustrated in Examples 1.8 and 1.9.