B R AT I O N B ASICS
V I B R AT I O N B ASICS
Vibration is defined as the oscillatory movement of a mass about a reference position characterized by displacement, velocity, and acceleration. Displace- ment (s) is the distance that the mass moves from its reference position in meters (see Figure 13-34), velocity (v) is the speed at which the mass moves in meters per second (m/sec), and acceleration (a) is the rate of change of the
mass’s velocity per second (m/sec 2 ). Table 13-9 displays the equations for these vibration motion parameters. Vibration also involves other parameters, including frequency, amplitude, and wave form. Vibration can be mathemati- cally defined in terms of periodic motion of a mass from a reference position by:
s t = s max sin ω t where:
s t = the position and distance of movement in meters (displacement) s max = the maximum displacement in meters (peak displacement)
ω = the angular frequency in radians per second
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S ECTION PLC Process Data Measurements C HAPTER 4 Applications
and Transducers 13
Peak Distance
Period T 1 + s max
Time ( t)
Displacement – s max
Figure 13-34. Displacement.
dt 2 S e c o n d d e r i v a it v e o f d i s p l a c e m e n t Table 13-9. Motion parameters associated with vibration.
dt
The angular frequency can also be expressed as angular velocity where ω =
2 π f . In vibration, velocity is the first derivative of displacement, while acceleration is the first derivative of velocity (or the second derivative of displacement):
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S ECTION PLC Process Data Measurements C HAPTER 4 Applications
and Transducers 13
All three vibration terms—displacement, velocity, and acceleration—have the same periodic frequency. Another important term in vibration monitoring
is the peak acceleration, which is frequency squared ( ω 2 ) times the peak
displacement (s 0 ):
peak = s 0 ω
This peak acceleration equation indicates that acceleration can become large even with very small displacement, since the displacement term is multiplied by the square of the frequency. Thus, acceleration can easily reach
a level of several g values (1g = 9.8 m/sec 2 ), creating a potentially destructive vibration. Table 13-10 lists the characteristics of several types of vibration.
E X AM PLE 1 3 -1 2
A steam pipe in a heat batching system (see Figure 13-35) vibrates at a frequency of 8 cycles per second (8 Hz) with a peak displacement of 10 mm (1 cm or 0.01 m). (a) Find and plot the displacement equation indicating the period, and (b) calculate the peak accelera-
tion in m/sec 2 and its equivalent in g units.
Steam Pipe vibrating at 8 Hz
with peak displacement of 10 mm
Figure 13-35. Heat batching system.
S OLU T I ON
(a) Figure 13-36 presents the graph of displacement versus time of vibration, which is given mathematically by the equation:
s t = s max sin ω t = s max sin 2 π ft = ( . )sin 0 01 28 π () t
= ( . )sin 0 01 . 50 265 t
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0 H (Courtesy of PMC/BETA Corp., Natick, MA) 1
2 Table 13-10. T Vibration identification guide. E
S ECTION PLC Process Data Measurements C HAPTER 4 Applications
and Transducers 13
20 mm displacement 10 mm peak
+10 mm
Time ( t)
–10 mm 125 msec period
8 Hz frequency
Figure 13-36. Displacement versus time of vibration.
This system has a period ( T ) equal to:
T ==
= . 0 125 sec
f 8 Hz (b) The peak acceleration is:
peak = ω s max
= 2 ( 2 π fs )
max
= 2 ( π 2 8 0 01 )(.) = 2 ( . 50 265 0 01 )(.) = 2 . 25 66 m/sec
This value in g units is:
a peak = ( . 25 266 m/m )
2 98 . m/m
= . 2 578 g
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