Page 457 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

444 Random Vibrations Chap. 13

xit)

I"
L 0 L
2 2

Figure 13.7-5. Rectangular pulse and its spectra.

Note that the FT is now a continuous function instead of a discontinuous function.
The product X X which is a real number, is also plotted here. Later it will be shown
to be equal to the spectral density function.

FTs of derivatives. When the FT is expressed in terms of co instead of /,
a factor 1 /2 tt is introduced in the equation for xit):

(13.7-3)

• dt (13.7-4)

This form is sometimes preferred in developing mathematical relationships. For
example, if we differentiate Eq. (13.7-3) with respect to i, we obtain the FT pair:

x(l) = X j "

‘
icoX{co) = / x { t)e -‘“> dt
Thus, the FT of a derivative is simply the FT of the function multiplied by ico\
F T [ i( 0 ] = /iu F T [x (0 ] (13.7-5)
Differentiating again, we obtain

F T [ i( 0 ] = - cu2FT[x( 0 ] (13.7-6)
These equations enable one to conveniently take the FT of differential equations.
For example, if we take the FT of the differential equation
my X cy + ky = x( t)