Civil Engineering Reference
In-Depth Information
1.0E+000
1.0E-001
a=1 s-1
a=50 s-1
1.0E-002
1.0E-003
1.0E-004
1
10
100
1000
Frequency (Hz)
Figure 1.8
Fourier transform of time functions shown in
Figure 1.7
1.4.3
Stochastic (random) motion. Fourier transform for a finite time
T
As pointed out above, a stochastic process gives data, which in a strict sense never
exactly repeat themselves.
Figure 1.9
may be thought of as time samples of such
processes. It could for example be the sound pressure measured at a given distance from
a sound source of stochastic nature such as a jet engine, a nozzle for compressed air or a
waterfall to mention a few examples. It is important to realize that such time histories or
time functions, in practice, certainly being of finite length, are just samples of an infinite
number of functions which could be attributed to the actual physical process. Collections
of identical types of source will each give a different time function. All these possible
time functions taken together make up what one in a strict sense calls a stochastic
process. A collection of such time functions is what is called an
ensemble.
In practice, however, just
one
such time history may be sufficient in our data
analysis. The reason for this is that processes that represent physical phenomena often
are
ergodic
. This means that we may extract all the necessary information from one
single time function. This does not imply, however, that we will not experience non-
linear phenomena when dealing with sound and vibration data analysis, analysis that
demands ensemble averaging.
Lets make
x
(
t
) represent one such stationary time function, existing in a theoretical
sense for all values of time
t
. Then we get
