Page 444 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.5 Correlation 431

and its probability density, by differentiation, is
1
p(x) Ul <A

= 0 |x| > A
For the wide-band record, the amplitude, phase, and frequency all vary
randomly and an analytical expression is not possible for its instantaneous value.
Such functions are encountered in radio noise, jet engine pressure fluctuation,
atmospheric turbulence, and so on, and a most likely probability distribution for
such records is the Gaussian distribution.
When a wide-band record is put through a narrow-band filter, or a resonance
system in which the filter bandwidth is small compared to its central frequency / q,
we obtain the third type of wave, which is essentially a constant-frequency
oscillation with slowly varying amplitude and phase. The probability distribution
for its instantaneous values is the same as that for the wide-band random function.
However, the absolute values of its peaks, corresponding to the envelope, will have
a Rayleigh distribution.
Another quantity of great interest is the distribution of the peak values. Rice^
shows that the distribution of the peak values depends on a quantity N^)/2M,
where is the number of zero crossings, and 2M is the number of positive and
negative peaks. For a sine wave or a narrow band, is equal to 2M, so that the
ratio N^)/2M = 1. For a wide-band random record, the number of peaks will
greatly exceed the number of zero crossings, so that N^^/2M tends to approach
zero. When N^^/2M = 0, the probability density distribution of peak values turns
out to be Gaussian, whereas when N^^/2M = 1, as in the narrow-band case, the
probability density distribution of the peak values tends to a Rayleigh distribution.

13.5 CORRELATION
Correlation is a measure of the similarity between two quantities. As it applies to
vibration waveforms, correlation is a time-domain analysis useful for detecting
hidden periodic signals buried in measurement noise, propagation time through
the structure, and for determining other information related to the structure’s
spectral characteristics, which are better discussed under Fourier transforms.
Suppose we have two records, x^U) and X2U), as shown in Fig. 13.5-1. The
correlation between them is computed by multiplying the ordinates of the two
records at each time t and determining the average value ( Xj(/)x2( 0 ) ^>y dividing
the sum of the products by the number of products. It is evident that the
correlation so found will be largest when the two records are similar or identical.

^See Ref. [8].