Vibration Time Wave form : Multiple Frequencies-Non Linear System (Sum and Difference Frequencies ) .

Another type of amplitude modulation occurs when one component is eccentric. One example of sum and difference frequencies is gear eccentricity. When one gear is eccentric or out of round, the amplitude of gear mesh frequency increases when the high place or places go into mesh. If the gear has only one high place, the signal amplitude will be higher once each revolution. In either case, amplitude modulation is caused by the eccentric gear. The associated spectra contain a spectral line at gear mesh frequency with side bands of gear speed. If the gear has more than one high place, then the difference frequency between the gear mesh frequency and the side bands is equal to the number of high places times the speed of the problem gear. If two high places are present, the difference frequency is two times gear speed. Three high places would generate a difference frequency of three times gear speed, four high places would generate four times gear speed, etc.

If the eccentric gear has not caused looseness, side bands will occur at gear mesh frequency
plus gear speed or multiples of gear speed. In other words, the side bands will be on the high side of gear mesh frequency. The frequencies add, in this case, because the phase relationship between the carrier and the modulator is constant. As stated earlier, the machine is behaving in a linear manner.
One Revolution of Gear with Four Eccentricities.

Sum and Difference Frequency with No Phase Shift.

Sum and Difference Frequency with Phase Shift.

When a gear or geared shaft system is loose, the looseness causes the modulator to subtract from the carrier because the two frequencies are out of phase. When the two frequencies are in phase, they add. Looseness causes an out-of-phase condition. Eccentricity is an in-phase condition.
If gear eccentricity has caused looseness (non linearity) associated with the problem gear, side band scan occur on both sides of gear mesh frequency. If looseness is the more severe problem, the amplitudes of the side bands will be higher on the low side of gear mesh frequency. If eccentricity is the more severe problem, the amplitudes of the side bands will be higher on the high side of gear mesh frequency. If looseness is the only problem, then the side bands occur only on the low side of gear mesh frequency.
The frequencies subtract when looseness is present because the phase relationship between the signals is not constant, which means the machine is acting in a nonlinear manner.
The principles described for gear mesh frequency apply to other generated frequencies such as blade or vane pass frequencies, bearing frequencies, frequencies from multiple defects, and frequencies from bars or corrugations on press rolls.