Page 26 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 1.3 Vibration Terminology 13

Figure 1.3-1. Average value of a
rectified sine wave.
The square of the displacement generally is associated with the energy of the
vibration for which the mean square value is a measure. The mean square value of
a time function x{t) is found from the average of the squared values, integrated
over some time interval T:

lim \ t ) d t (1.3-2)
t /:
For example, if x{t) = A sin (ot, its mean square value is
A^ rT\
j I , 2
.2 _ lim -jT j ^ (1 - coslcot) at = 2 A
^

The root mean square (rms) value is the square root of the mean square
value. From the previous example, the rms of the sine wave of amplitude A is
A / = 0.707^1. Vibrations are commonly measured by rms meters.
The decibel is a unit of measurement that is frequently used in vibration
measurements. It is defined in terms of a power ratio.

dB = 101og,o(^
(1.3-3)
= 101og,„(|)

The second equation results from the fact that power is proportional to the square
of the amplitude or voltage. The decibel is often expressed in terms of the first
power of amplitude or voltage as

dB = 20 log j (1.3-4)
Thus an amplifier with a voltage gain of 5 has a decibel gain of
201ogio(5) = +14

Because the decibel is a logarithmic unit, it compresses or expands the scale.
When the upper limit of a frequency range is twice its lower limit, the
frequency span is said to be an octave. For example, each of the frequency bands
in Figure 1.3-2 represents an octave band.