Page 148 - Intro Predictive Maintenance

Vibration Monitoring and Analysis 139

2
dX dX 2
=- 2m -w X
dt 2 dt
or

2
dX dX 2
+ 2m +w X = 0
dt 2 dt
or
2
2
D + 2mD + w = 0
which has a solution of:

1
d t
X = Ae + B e d t 2
where:

d 1 =- + m 2 -w 2
m
m
d 2 =- - m 2 -w 2
There are different conditions of damping: critical, overdamping, and underdamping.
Critical damping occurs when m equals w. Overdamping occurs when m is greater than
w. Underdamping occurs when m is less than w.

The only condition that results in oscillatory motion and, therefore, represents a
mechanical vibration is underdamping. The other two conditions result in periodic
motions. When damping is less than critical (m < w), then the following equation
applies:

X 0 - t m
X = e (a cos a 1 t + msin a 1 t)
1
a 1
where:

2
a 1 = w - m 2
Forced Vibration—Undamped. The simple systems described in the preceding two
sections on free vibration are alike in that they are not forced to vibrate by any excit-
ing force or motion. Their major contribution to the discussion of vibration funda-
mentals is that they illustrate how a system’s natural or resonant frequency depends
on the mass, stiffness, and damping characteristics.

The mass-stiffness-damping system also can be disturbed by a periodic variation of
external forces applied to the mass at any frequency. The system shown in Figure 7–12
is increased in complexity by adding an external force, F 0, acting downward on the
mass.