Page 454 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
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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
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