Page 455 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 455
442 Random Vibrations Chap. 13
Fourier transform of x{t), which can be evaluated from the equation
X ( f ) r dt (13.7-2)
— oc
Like the Fourier coefficient C„, X( f ) is a complex quantity which is a continuous
function of / from —oo to Foo, Equation (13.7-2) resolves the function x(t) into
harmonic components X(f), whereas Eq. (13.7-1) synthesizes these harmonic
components to the original time function x(t). The two previous equations above
are referred to as the Fourier transform pair.
Fourier transform (FT) of basic functions. To demonstrate the spectral
character of the FT, we consider the FT of some basic functions.
Example 13.7-1
x(t) (a)
From Eq. (13.7-1), we have
/ oo X{f )e‘"^l‘ df
Recognizing the properties of a delta function, this equation is satisfied if
X{f) =AS ( f - f „ ) (b)
Substituting into Eq. (13.7-2), we obtain
dt (c)
The FT of x(t) is displayed in Fig. 13.7-1, which demonstrates its spectral character.
Xif)
Adif-fn)
-f-
^ Figure 13.7-1. FT of
Example 13.7-2
x(t) = a„cos(2vf„t) (a)
Because
cos277/„r =
the result of Example 13.7-1 immediately gives
X( f ) = Y i ^ i f - f „ ) + S ( f + fn)] (b)
Figure 13.7-2 show s that X{f) is a tw o-sided function o f / .