TABLE OF CONTENTS In this chapter, so far we have seen that many problems in random vibration can be solved by the use of the power spectral density function alone, together with an understanding of probability density and distribution functions. There are some problems, however, for which this will not be sufficient. Before looking at these, we must first introduce a few more topics, such as correlation functions, and how they are related to spectral density functions, including cross- power spectral density functions.
Before considering autocorrelation and cross-correlation functions, we should briefly consider correlation, a basic concept in statistics. The correlation coefficient is often applied to pairs of properties, such as the height and weight of men in a given population. The idea can be extended to include sampled values of two time histories, x( t) and y( t), say. The correlation coefficient for simultaneous pairs of samples of x and y, taken from x( t) and y( t), where the mean values are zero, is defined as:
| (10.64) |  |
where ? x and ? y are the standard deviations of x and y, respectively and ? xy = ? xy ? is known as the covariance. It can be seen from Eq. (10.64), incidentally, why ? xy is sometimes called the normalized covariance.
The correlation coefficient, ? xy, is...
