Page 136 - Intro Predictive Maintenance

Vibration Monitoring and Analysis 127

ω = 2πf
2π

π

x
π 2π
X 0
t

T

Figure 7–8 Illustration of vibration cycles.

w
VPM =
p

By definition, velocity is the first derivative of displacement with respect to time. For
a harmonic motion, the displacement equation is:

(
X = X 0 sin w t)
The first derivative of this equation gives us the equation for velocity:

dX
˙
v = = X = w X cos w t)
(
0
dt
This relationship tells us that the velocity is also harmonic if the displacement is har-
monic and has a maximum value or amplitude of -wX 0.
By definition, acceleration is the second derivative of displacement (i.e., the first deriv-
ative of velocity) with respect to time:

2
dX
˙˙
()
a = 2 = X = w 2 X 0 sin w t
dt
2
This function is also harmonic with amplitude of w X 0.
Consider two frequencies given by the expression X 1 = asin(wt) and X 2 = bsin(wt +
f), which are shown in Figure 7–9 plotted against wt as the X-axis. The quantity, f,
in the equation for X 2 is known as the phase angle or phase difference between the