Page 455 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
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442                                       Random Vibrations  Chap. 13

                              Fourier transform  of  x{t), which can be  evaluated from  the  equation

                                                    X ( f )  r          dt              (13.7-2)
                                                            — oc
                              Like  the  Fourier coefficient  C„, X( f )  is  a complex quantity which  is  a  continuous

                              function  of  /   from  —oo  to  Foo,  Equation  (13.7-2)  resolves  the  function  x(t)  into
                              harmonic  components  X(f),  whereas  Eq.  (13.7-1)  synthesizes  these  harmonic
                              components  to  the  original time  function  x(t).  The  two previous  equations  above
                              are  referred to as the  Fourier transform pair.

                                   Fourier transform  (FT) of basic functions.  To  demonstrate  the  spectral
                              character of the  FT, we consider the  FT of some  basic functions.
                              Example  13.7-1
                                                           x(t)                              (a)
                                   From  Eq. (13.7-1), we  have
                                                               /  oo  X{f )e‘"^l‘ df

                                   Recognizing the  properties of a delta function,  this equation  is satisfied if
                                                         X{f)  =AS ( f - f „ )               (b)
                                   Substituting into Eq. (13.7-2), we obtain

                                                                           dt                (c)

                                   The  FT of x(t) is displayed  in  Fig.  13.7-1, which  demonstrates its spectral character.
                                              Xif)

                                                           Adif-fn)


                                    -f-
                                                                  ^   Figure 13.7-1.  FT of
                              Example  13.7-2
                                                         x(t)  = a„cos(2vf„t)                (a)
                                   Because
                                                     cos277/„r  =
                                   the  result of Example  13.7-1  immediately gives
                                                    X( f )  =  Y i ^ i f - f „ ) + S ( f  +  fn)]  (b)
                                   Figure  13.7-2  show s  that  X{f)   is  a  tw o-sided  function  o f  / .
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