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If a macroscopic body of mass $m$ moves according to a certain law of motion like, for example, $$x(t)=A\cos(2\pi ft)$$ then what physical interpretation can be attributed to the Fourier transform of $x(t)$?

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  • $\begingroup$ Just a check re your comments. Do you mean the Fourier transform or the Fourier series? They require different methods to tackle and I interpret them in different ways. A Fourier Transform is like $\mathcal{F}[1]=\int_{-\infty}^\infty e^{-i \omega x} dx=2\pi\delta(\omega)$. A Fourier series is like $f(x)=\sum_{-\infty}^\infty A_n e^{i \omega n x}$. $\endgroup$
    – user12029
    Nov 6 '15 at 6:47
  • $\begingroup$ "Do you mean the Fourier transform or the Fourier series?" I mean the Fourier transform because $x(t)$ can be any function, not necessarily a periodical one. $\endgroup$ Nov 6 '15 at 7:43
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It tells you what is the spectral content of the motion.

Some examples of when this might be interesting:

  • The object being measured is a point on a guitar string---then it would tell you what note is being played

  • The object being measured is a planet. Then the peak frequency of the motion is the inverse of the planet's year.

  • The object being measured is part of a mechanical actuator---then the spectral response due to an applied force impulse (or driving signal) could tell you about the requirements for a control system for that actuator to remain stable. Generally in a linear system if you can measure the spectrum of the response to a know input with known frequency content then you can predict the response to other inputs.

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In this case, the Fourier transform will be a dirac pointing the unique frequency of vibration $f$ (or indeed 2, at $-f$ and $f$), with amplitude $A$ and phase $0$.

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  • $\begingroup$ I am interested in finding a physical interpretation of $F[x(t)]$. I know how the Fourier transform of a cosine function looks like. $\endgroup$ Nov 6 '15 at 1:18
  • $\begingroup$ what do you expect so different to the notion of spectrum ? one space have the vibrating motions. The other is the set of "vibrations numbers". You directly see modes in that space, isn't it something big enough ? $\endgroup$ Nov 6 '15 at 8:13
  • $\begingroup$ BTW your example is a bit borring (too simple). the membrane of a loudspeaker, or vibrating exchangers pipes would be more rich. $\endgroup$ Nov 6 '15 at 8:15

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