Page 454 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 454
Sec. 13.7 Fourier Transforms 441
Figure 13.6-10. Fourier coefficients
versus n.
The mean square value is determined from the equation
— 1 fT
= lim 7 ^ / x^{t) dt
T^oo — T
= l i m ^ i dt
T—00-^^ J Y ^
^
oo ^
n = \
and because x^= dw, the spectral density function can be repre
sented by a series of delta functions:
00 ^
^ /( ^ ) = H —y ^ d ( iv - no)^^)
n^l
13.7 FOURIER TRANSFORMS
The discrete frequency spectrum of periodic functions becomes a continuous one
when the period T is extended to infinity. Random vibrations are generally not
periodic and the determination of its continuous frequency spectrum requires the
use of the Fourier integral, which can be regarded as a limiting case of the Fourier
series as the period approaches infinity.
The Fourier transform has become the underlying operation for the modern
time series analysis. In many of the modern instruments for spectral analysis, the
calculation performed is that of determining the amplitude and phase of a given
record.
The Fourier integral is defined by the equation
x(t) = J X(f)e'^^^'df (13.7-1)
In contrast to the summation of the discrete spectrum of sinusoids in the Fourier
series, the Fourier integral can be regarded as a summation of the continuous
spectrum of sinusoids. The quantity X( f ) in the previous equation is called the