Page 434 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.2 Time Averaging and Expected Value 421

It is the average or mean value of a quantity sampled over a long time. In the case
of discrete variables x^, the expected value is given by the equation

E[x] = lim ^ ¿ X , (13.2-3)

Mean square value. These average operations can be applied to any
variable such as x^{t) or x(t) • y(t). The mean square value, designated by the
notation or E[x^(t)], is found by integrating x^(t) over a time interval T and
taking its average value according to the equation

£’[x ^(/)] = x^ = lim -i f X^dt (13.2-4)
7’ —>00 ' J

Variance and standard deviation. It is often desirable to consider the
time series in terms of the mean and its fluctuation from the mean. A property of
importance describing the fluctuation is the variance which is the mean square
value about the mean, given by the equation

a lim i f (x —x)^ dt (13.2-5)
I J(\
By expanding the above equation, it is easily seen that
.2
( x y (13.2-6)
so that the variance is equal to the mean square value minus the square of the
mean. The positive square root of the variance is the standard deviation, a.
Fourier series. Generally, random time functions contain oscillations of
many frequencies, which approach a continuous spectrum. Although random time
functions are generally not periodic, their representations by Fourier series, in
which the periods are extended to a large value approaching infinity, offers a
logical approach.
In Chapter 1, the exponential form of the Fourier series was shown to be

x(t) = -I- £ + c*e- ") (13.2-7)
n= 1
This series, which is a real function, involves a summation over negative and
positive frequencies, and it also contains a constant term The constant term
is the average value of x{t) and because it can be dealt with separately, we exclude
it in future considerations. Moreover, actual measurements are made in terms of
positive frequencies, and it would be more desirable to work with the equation

c(0 = Re E ( 13.2-8)