Page 123 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

120 Chapter 1 Fundamentals oF Vibration

k 4

k 5 k 6
R u

k 3

k 2

k 1

FiGure 1.70 A combination of torsional and linear springs.

1.11 A machine of mass m = 500 kg is mounted on a simply supported steel beam of length
l = 2 m having a rectangular cross section 1depth = 0.1 m, width = 1.2 m2 and Young’s
2
11
modulus E = 2.06 * 10 N>m . To reduce the vertical deflection of the beam, a spring of
stiffness k is attached at mid-span, as shown in Fig. 1.71. Determine the value of k needed to
reduce the deflection of the beam by
a. 25 percent of its original value.
b. 50 percent of its original value.
c. 75 percent of its original value.
Assume that the mass of the beam is negligible.

m

k

l

FiGure 1.71 A machine mounted on a spring-supported beam.
1.12 A bar of length L and Young’s modulus E is subjected to an axial force. Compare the spring
constants of bars with cross sections in the form of a solid circle (of diameter d), square (of
side d), and hollow circle (of mean diameter d and wall thickness t = 0.1d). Determine
which of these cross sections leads to an economical design for a specified value of axial
stiffness of the bar.