Non-Linearities in Rotating Machines
As has been discussed, the vibration of a machine is actually its response to forces caused by moving parts in the machine. We measure the vibration at various locations on the machine, and deduce from these vibrations the magnitude of the forces. In measuring the frequency of the vibration, we assume the forces occur at the same frequency as the response, and that the measured levels are proportional to the magnitudes of the forces. This rationale assumes that the machine is linear in its response to forcing functions, and this is a reasonable assumption for most machines.
However, as a machine wears and clearances increase, or if it develops cracks or loose parts, its response will no longer be linear, and the result is that the measured vibration can be quite different in character from the forcing functions. For instance, an unbalanced rotor imparts a sinusoidal force at a frequency of 1X to the bearing, and this force does not contain any other frequency. If the mechanical structure of the machine is non-linear, this sinusoidal force will be distorted, and the resulting vibration will occur at harmonics of 1X as well as 1X. The extent and magnitude of the harmonic content of the vibration is a measure of the degree of non-linearity of the machine. For instance, the vibration of a journal bearing contains greater and greater numbers and magnitudes of harmonics as the bearing clearance increases.
Flexible couplings are non-linear when misaligned, and this is the reason their vibration signature contains a strong second harmonic of 1X. Worn couplings that are misaligned often produce a strong third harmonic of 1X. When forces acting at different frequencies interact in a non-linear way in a machine, the result is the generation of sum and difference frequencies -- new frequencies that are not present in the forcing functions themselves. These sum and difference frequencies are the sidebands found in spectra of defective gearboxes, rolling element bearings, etc. In the case of a gearbox, one forcing frequency is the gear mesh and another is the rpm of the gear. If the gear is eccentric or otherwise misshapen, the rpm will modulate the gear mesh resulting in sidebands. Modulation is always a non-linear process, creating new frequencies that do not exist in the forcing functions.
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