The Relationship Between The Time Domain And The FFT:
Frequency Modulation
So is it safe to assume that each of the previous signals were generated by a machine that is generating vibration at 1x rpm and 2x rpm (i.e. a reciprocating compressor) ? Or could there be another explanation for the signal shape seen on those pages (which is really what is being analyzed - the signal shape) ? Let's return to our discussion of the actual, real-life vibration signal we looked at a few pages back. 
  • We discussed how there can be some variation in the free rotation of the shaft - a momentary "binding" action that occurs as the shaft rotates through a particular portion of it's rotation. 
  • That phenomenon could occur for a number of reasons. In that situation, we considered the possibility of the gears being improperly set. That would create more resistance to rotation when the teeth were bottomed out than opposite that point. It would momentarily slow down the rotation.
Let's examine the 'frequency modulated' signal shown here:

Figure 1: A "Frequency Modulated" Signal
Figure 2:
A Bearing Undergoing Frequency 
Modulation During Each Rotation
  • Let's first examine the positive-going portion of the signal. The bottom of the cycle (the '-' peak) first occurs at about 19 msec. The '+' peak occurs at about 32 msec so it takes a total of about 13 msec to move from the "-" peak to the "+" peak. The reciprocal of the period will give us the frequency during that portion of the signal: 
    • 1/(0.013 x 2) x 60 = 2308 cpm (the 13 msecs is multiplied by 2 to calculate a full cycle).
  • Now let's examine the negative going peak. From the "+" peak at 32 msec, the signal descends to a "-" peak by about 53 msec - a total of 21 msec. For that portion of the signal, the shaft rotates:
    • 1/(0.021 x 2) x 60 = 1429 cpm.
  • Yet if we simply calculate the total time for one cycle (peak to peak), we measure from 19 msec to 53 msec - about 34 msec. 
    • 1/(0.034) x 60 = 1765 cpm.
This is called frequency modulation. What is happening here may or may not be evident if we were to analyze the time domain signal - it will depend on the resolution (yes, time domain is just as dependent on resolution as the FFT is), the time sample, number of bytes, etc. But remember, the question we are discussing here is how will the FFT treat this phenomenon ?
The FFT only deals in pure sinusoids. So how will it account for the frequency modulation we see here ? We will unquestionably have a peak around 1765 cpm but the signal is not a pure sinusoid - it is distorted by the frequency modulation we see. How does the FFT mathematically explain this phenomenon ?
In other words, what combination of simple sinusoids, when combined, will generate the signal we see above ?