Page 100 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.11 harmoniC analysis 97
d n f n

0 0
v 2v 3v 4v 5v 6v 7v 8v 9v Frequency (nv) v 2 v 3v 4v 5 v 6v 7v 8v 9v Frequency (nv)

FiGure 1.56 Frequency spectrum of a typical periodic function of time.

1.11.4 The Fourier series expansion permits the description of any periodic function using either
time- and a time-domain or a frequency-domain representation. For example, a harmonic function
Frequency- given by x1t2 = A sin vt in time domain (see Fig. 1.57(a)) can be represented by the
amplitude and the frequency v in the frequency domain (see Fig. 1.57(b)). Similarly, a peri-
domain odic function, such as a triangular wave, can be represented in time domain, as shown in
representations Fig. 1.57(c), or in frequency domain, as indicated in Fig. 1.57(d). Note that the amplitudes
d and the phase angles f corresponding to the frequencies v can be used in place of the
n
n
n
amplitudes a and b for representation in the frequency domain. Using a Fourier integral
n
n
(considered in Section 14.9) permits the representation of even nonperiodic functions in
x(t) x(v)
A sin (vt f ) A
0
A
x 0
0 t 0 v
v
A
(a) (b)

a (coef cients of cosine terms in Eq. (1.70))
n
a
a 0 1 a
x(t) 2 a 3 a 4
0 v n
v vv 2vv 3vv 4v
1
3
2
4
A
b n (coef cients of sine terms in Eq. (1.70))
x 0
0 t b 1
b 2
(c) b 3 b 4
0 v n
v vv 2vv 3vv 4v
1
4
3
2
(d)
FiGure 1.57 Representation of a function in time and frequency domains.