Page 24 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 1.2 Periodic Motion 11

The Fourier series can also be represented in terms of the exponential
function. Substituting
cos + e
sin
in Eq. (1.2-1), we obtain

x{t) = y + E [i(«„ - ifcje'“”' + {{a„ + ib„)e
n = 1
(1.2-4)
= T + ¿
n = l
= E

where
^0 ~ 2^0
(1.2-5)
= i(«« - iK)
Substituting for and from Eq. (1.2-3), we find to be
1 t/2
c„ = ~ r x(t)(coso)„t - i sin (0„t)dt
^ • '- t/2
(1.2-6)
^ • '- t/2
Some computational effort can be minimized when the function x(t) is
recognizable in terms of the even and odd functions:
x{t) =E{t) + 0 ( 0 (1.2-7)
An even function E(t) is symmetric about the origin, so that E(t) = E ( - t \ i.e.,
cosiot = cos(-ii>0. An odd function satisfies the relationship 0(t) = —0( —t),

i.e., sin coi = —sin( —coi). The following integrals are then helpful:
E{t) sin (i)Jdt = 0
J - t/2
( 1.2-8)
0 ( 0 cos (ojdt = 0

When the coefficients of the Fourier series are plotted against frequency
the result is a series of discrete lines called the Fourier spectrum. Generally plotted
are the absolute values I2c„| = ^a l and the phase (/>„ = tan~^ an
example of which is shown in Fig. 1.2-2.