Page 103 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

100 Chapter 1 Fundamentals oF Vibration
Since cos [2p12n - 12>4] = 0 for n = 1, 2, 3, c , and sin [2p12n - 12>4] =
n + 1
1-12 for n = 1, 2, 3, c, Eq. (1.95) reduces to
t 4A 1-12 n + 1 2p12n - 12t
x ¢t + ≤ = a cos (1.96)
1
4 p n = 1 12n - 12 t
which can be identified to be the same as Eq. (1.92).

1.11.6 In some practical applications, the function x(t) is defined only in the interval 0 to t as
half-range shown in Fig. 1.59(a). In such a case, there is no condition of periodicity of the function,
expansions since the function itself is not defined outside the interval 0 to t. However, we can extend
the function arbitrarily to include the interval -t to 0 as shown in either Fig. 1.59(b) or
Fig. 1.59(c). The extension of the function indicated in Fig. 1.59(b) results in an odd func-
tion, x 1t2, while the extension of the function shown in Fig. 1.59(c) results in an even
1
function, x 1t2. Thus the Fourier series expansion of x 1t2 yields only sine terms and that
2
1
of x 1t2 involves only cosine terms. These Fourier series expansions of x 1t2 and x 1t2
1
2
2
x(t)

t
0 t

(a) Original function

(t)
x 1

t
t 0 t

(b) Extension as an odd function

x (t)
2

t
t 0 t

(c) Extension as an even function

FiGure 1.59 Extension of a
function for half-range expansions.