Page 445 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

432 Random Vibrations Chap. 13

Figure 13.5-1. Correlation between
x^{t) and X2U).

Figure 13.5-2. Function x(t)
shifted by r.
For dissimilar records, some of the products will be positive and others will be
negative, so their sum will be smaller.
Next, consider the case in which X2U) is identical to x^it) but shifted to the
left by a time r, as shown in Fig. 13.5-2. Then, at time t, when x^ is x(t), the value
of X2 is x(t + r), and the correlation is given by x{t)x{t + r)). Here, if r = 0,
(
we have complete correlation. As r increases, the correlation decreases.
It is evident that this result can be computed from a single record by
multiplying the ordinates at time t and t + r and determining the average. We
then call this result the autocorrelation and designate it by R{t). It is also the
expected value of the product x{t)x{t -t- r), or

R{ t) = E[x{t)x{t + t)] ={x{t)x{t + t))
(13.5-1)
= lim i i ^ x{t)x{t -h t) dt
T-^oc i •'-r/2
When T = 0, this definition reduces to the mean square value:

/?(0) = P = (13.5-2)
Because the second record of Fig. 13.5-2 can be considered to be delayed with
respect to the first record, or the first record advanced with respect to the second
record, it is evident that R{r) = R { - t) is symmetric about the origin r = 0 and is
always less than i?(0).
Highly random functions, such as the wide-band noise shown in Fig. 13.5-3,
soon lose their similarity within a short time shift. Its autocorrelation, therefore, is
a sharp spike at r = 0 that drops off rapidly with ± r as shown. It implies that
wide-band random records have little or no correlation except near r = 0.