Page 99 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

96 Chapter 1 Fundamentals oF Vibration
Thus Eq. (1.70) can be written as

a e invt + e -invt e invt - e -invt
x1t2 = 0 + ba ¢ ≤ + b ¢ ≤ r
a
2 n = 1 n 2 n 2i
a ib
= e i102vt ¢ 0 - 0 ≤
2 2

a n ib a n ib
+ a be invt ¢ - n ≤ + e -invt ¢ + n ≤ r (1.82)
n = 1 2 2 2 2

where b = 0. By defining the complex Fourier coefficients c and c as
0
-n
n
a - ib
c = n n (1.83)
n
2
and
a + ib
c -n = n 2 n (1.84)

Eq. (1.82) can be expressed as

x1t2 = a c e invt (1.85)
n
n = -
The Fourier coefficients c can be determined, using Eqs. (1.71)–(1.73), as
n
t
a n - ib n 1
c = = x1t23cos nvt - i sin nvt4dt
n
2 t L
0
t
1
= x1t2e -invt dt (1.86)
t L
0

1.11.3 The harmonic functions a cos nvt or b sin nvt in Eq. (1.70) are called the harmonics of
n
n
Frequency order n of the periodic function x(t). The harmonic of order n has a period t>n. These har-
n
n
n
n
spectrum monics can be plotted as vertical lines on a diagram of amplitude (a and b or d and f )
versus frequency 1nv2, called the frequency spectrum or spectral diagram. Figure 1.56
shows a typical frequency spectrum.