Page 447 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

434 Random Vibrations Chap. 13

Figure 13.5-7. Time delay for auto-
correlation.
shows a block diagram for the determination of the autocorrelation. The signal
x(t) is delayed by r and multiplied, after which it is integrated and averaged. The
delay time r is fixed during each run and is changed in steps or is continuously
changed by a slow sweeping technique. If the record is on magnetic tape, the time
delay r can be accomplished by passing the tape between two identical pickup
units, as shown in Fig. 13.5-7.
Cross correlation. Consider two random quantities x{t) and y{t). The
correlation between these two quantities is defined by the equation
RxyiT) = E[x{t)y{t + t )] =( x (t )y ( t + t ))
(13.5-3)
= lim rj. f x{t)y{t + t ) dt
T-*oo i
/
- T 2
which can also be called the cross correlation between the quantities x and y.
Such quantities often arise in dynamical problems. For example, let x{t) be
the deflection at the end of a beam due to a load F^{t) at some specified point.
yU) is the deflection at the same point, due to a second load F2U) at a different
point than the first, as illustrated in Fig. 13.5-8. The deflection due to both loads is
then z{t) = (t) + y(C, and the autocorrelation of z{t) as a result of the two
x
loads is
+ t ))
= ( [ x ( / ) -I-y(i)][j^(i + t ) + y { t + t )])
= x { t ) x { t -f- t)> + { x { t ) y { t t)) (13.5-4)
{
+ { y { t ) x { t + t)) + { y { t ) y { t + t)>
= +^xy('^) +^yy('^) +^y(' ^)

Figure 13.5-8.