Page 60 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.7 sprinG elements 57
1.7.3 As stated earlier, any elastic or deformable member (or element) can be considered as a
spring spring. The equivalent spring constants of simple elastic members such as rods, beams,
Constants of and hollow shafts are given on the inside front cover of the book. The procedure of finding
the equivalent spring constant of elastic members is illustrated through the following
elastic elements examples.

spring Constant of a rod
example 1.3
Find the equivalent spring constant of a uniform rod of length l, cross-sectional area A, and Young’s
modulus E subjected to an axial tensile (or compressive) force F as shown in Fig. 1.24(a).

Solution: The elongation (or shortening) d of the rod under the axial tensile (or compressive) force
F can be expressed as
d s Fl
d = l = el = l = (E.1)
l E AE
change in length d force F
where e = = is the strain and s = = is the stress induced in the rod.
original length l area A
Using the definition of the spring constant k, we obtain from Eq. (E.1):
force applied F AE
k = = = (E.2)
resulting deflection d l
The significance of the equivalent spring constant of the rod is shown in Fig. 1.24(b).

F

l d
(a)

F
AE d
k
l
(b)

FiGure 1.24 Spring constant of a rod.
■

spring Constant of a Cantilever beam
example 1.4
Find the equivalent spring constant of a cantilever beam subjected to a concentrated load F at its end
as shown in Fig. 1.25(a).