Page 82 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.6 Vibration Isolation 69

M
Figure 3.6-2.

Comparison of the preceding equation with Eq. (3.5-8) shows that
Ft X
TR =
F, Y
When the damping is negligible, the transmissibility equation reduces to
1
TR = (3.6-3)
1
where it is understood that the value of to be used is always greater than
]/2. On further replacing by A /g, where g is the acceleration of gravity and A
is the statical deflection, Equation (3.6-3) can be expressed as
1
TR =
(277/)“ A / g - 1
To reduce the amplitude X of the isolated mass m without changing TR, m
is often mounted on a large mass M, as shown in Fig. 3.6-2. The stiffness k must
then be increased to keep the ratio k/{m 4- M) constant. The amplitude X is,
however, reduced because k appears in the denominator of Eq. (3.6-la).
Because in the general problem the mass to be isolated may have 6 DOF
(three translation and three rotation), the designer of the isolation system must use
his or her intuition and ingenuity. The results of the single-DOF analysis should,
however, serve as a useful guide. Shock isolation for pulse excitation is discussed in
Sec. 4.5 in Chapter 4.
Example 3.6-1
A machine of 100 kg mass is supported on springs of total stiffness 700 kN /m and has
an unbalanced rotating clement, which results in a disturbing force of 350 N at a
speed of 3000 rpm. Assuming a damping factor of C = 0.20, determine (a) its
amplitude of motion due to the unbalance, (b) the transmissibility, and (c) the
transmitted force.
Solution: The statical deflection of the system is
100 X 9.81
= 1.401 X 10 ' m = 1.401 mm
700 X 10'
and its natural frequency is
f 9.81 = 13.32 Hz
" 277 1.401 X 10