Page 74 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.8 mass or inertia elements 71
Pinion, mass moment of inertia J 0

u
R

Rack, mass m x

FiGure 1.37 Translational and rotational masses in a
rack-and-pinion arrangement.

Case 2: Translational and Rotational Masses Coupled Together. Let a mass m having
#
a translational velocity x be coupled to another mass (of mass moment of inertia J ) having
#
0
a rotational velocity u, as in the rack-and-pinion arrangement shown in Fig. 1.37.
These two masses can be combined to obtain either (1) a single equivalent translational
mass m or (2) a single equivalent rotational mass J , as shown below.
eq
eq
1. Equivalent translational mass. The kinetic energy of the two masses is given by
1 # 1 #
2
T = mx + J u 2 (1.22)
2 2 0
and the kinetic energy of the equivalent mass can be expressed as
1 #
T = m x 2 (1.23)
eq
2 eq eq
# # # #
Since x eq = x and u = x>R, the equivalence of T and T gives
eq
#
#
#
1 m x = 1 mx + 1 x 2
2
2
0
2 eq 2 2 J ¢ ≤
R
That is,
J
m eq = m + 0 (1.24)
R 2
# # # #
2. Equivalent rotational mass. Here u eq = u and x = uR, and the equivalence of T and
T leads to
eq
#
#
1 J u = 1 m1uR2 + 1 # 2
2
2
0
2 eq 2 2 J u
or
J eq = J + mR 2 (1.25)
0