Page 61 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

58 Chapter 1 Fundamentals oF Vibration

l
E, A, I
F W k 3EI
l 3
x(t)
d
W mg
x(t)
W

(a) Cantilever with end force (b) Equivalent spring
FiGure 1.25 Spring constant of a cantilever beam.

Solution: We assume, for simplicity, that the self weight (or mass) of the beam is negligible and
the concentrated load F is due to the weight of a point mass 1W = mg2. From strength of materials
[1.26], we know that the end deflection of the beam due to a concentrated load F = W is given by
Wl 3
d = (E.1)
3EI
where E is the Young’s modulus and I is the moment of inertia of the cross section of the beam about
the bending or z-axis (i.e., axis perpendicular to the page). Hence the spring constant of the beam is
(Fig. 1.25(b)):
W 3EI
k = = (E.2)
d l 3

Notes:
1. It is possible for a cantilever beam to be subjected to concentrated loads in two directions at its
end—one in the y direction 1F y 2 and the other in the z direction 1F z 2—as shown in Fig. 1.26(a).
When the load is applied along the y direction, the beam bends about the z-axis (Fig. 1.26(b))
and hence the equivalent spring constant will be equal to

3EI zz
k = (E.3)
l 3
When the load is applied along the z direction, the beam bends about the y-axis (Fig. 1.26(c))
and hence the equivalent spring constant will be equal to
3EI yy
k = (E.4)
l 3
2. The spring constants of beams with different end conditions can be found in a similar manner
using results from strength of materials. The representative formulas given in Appendix B can
be used to find the spring constants of the indicated beams and plates. For example, to find the
spring constant of a fixed-fixed beam subjected to a concentrated force P at x = a (Case 3
in Appendix B), first we express the deflection of the beam at the load point 1x = a2, using
b = l - a, as
2
2
2 2
2
P1l - a2 a Pa 1l - a2 1al - a 2
2
y = [3al - 3a - a1l - a2] = (E.5)
6EIl 3 3EIl 3