Page 106 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.11 harmoniC analysis 103
x(t) can be expressed as
t
x1t2 = A ; 0 … t … t (E.3)
t
Equation (E.3) is shown in Fig. 1.54(a). To compute the Fourier coefficients a n and b n , we use
Eqs. (1.71)–(1.73):

2p>v 2p>v 2 2p>v
v v t v A t
a 0 = x1t2 dt = A dt = ¢ ≤ = A (E.4)
p L 0 p L 0 t p t 2 0
2p>v 2p>v
v # v t #
a n = x1t2 cos nvt dt = A cos nvt dt
p L 0 p L 0 t
2p>v 2p>v
Av # A cos nvt vt sin nvt
= t cos nvt dt = J + R
pt L 0 2p 2 n 2 n 0
= 0, n = 1, 2, c (E.5)

2p>v 2p>v
v # v t #
b n = x1t2 sin nvt dt = A sin nvt dt
p L 0 p L 0 t
2p>v 2p>v
Av # A sin nvt vt cos nvt
= t sin nvt dt = J - R
pt L 0 2p 2 n 2 n 0
A
= - , n = 1, 2, c (E.6)
np
Therefore the Fourier series expansion of x(t) is
A A A
x1t2 = - sin vt - sin 2 vt - c
2 p 2p

A p 1 1
= J - bsin vt + sin 2vt + sin 3vt + c r R (E.7)
p 2 2 3
The first three terms of the series are shown plotted in Fig. 1.54(b). It can be seen that the approxima-
tion reaches the sawtooth shape even with a small number of terms.

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numerical Fourier analysis
example 1.20
The pressure fluctuations of water in a pipe, measured at 0.01-second intervals, are given in Table 1.1.
These fluctuations are repetitive in nature. Make a harmonic analysis of the pressure fluctuations and
determine the first three harmonics of the Fourier series expansion.