Page 462 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.8 FTs and Response 449

Another useful relationship can be found by multiplying H{a)) by its conju­
gate The result is
H{co)H%co) =
X{co)X^{co)
or
Y{(o)Y%co) =\H{co)\ X{co)X*{co) (13.8-4)
Thus, the output power spectrum is equal to the square of the system transfer
function multiplied by the input power spectrum. Obviously, each side of the
previous equation is real and the phase does not enter in.
We wish now to examine' the mean square value of the response. From Eq.
(13.7-8), the mean square value of the input x{t) is
-------- -0 0 -o o 1
X ^ = f S A f ^ ) d f = l \im X ( f ) X * { f ) d f
j
The mean square value of the output y{t) is
= f S ^ ( f ^ ) d f = f lim j Y { f ) Y * ( f ) d f
J —rr T —>O0
Substituting yy* = \Hif)\^XX*, we obtain

= f \ n i n r lim j X { f ) X * { f ) df
T ^ oo ^

(13.8-5)
-00
= / \ H{ f ) \ ^ S Af ^ ) d f
which is the mean square value of the response in terms of the system response
function and the spectral density of the input.
In these expressions, S(f are the two-sided spectral density functions over
both the positive and negative frequencies. Also, S(f^) are even functions. In
actual practice, it is desirable to work with spectral densities over only the positive
frequencies. Equation (13.8-5) can then be written as

= f \ H { f ) f s A f A d f (13.8-6)
•'n
and because the two expressions must result in the same value for the mean square
value, the relationship between the two must be
S i f A = S i f ) - 2 S { f A (13.8-7)
Some authors also use the expression
r\H{<o)\^SA<o)da^ (13.8-8)
•'n
Again, the equations must result in the same mean square value so that
2TrS((o) = S( f ) (13.8-9)