Linear and Non-Linear Systems
To assist in understanding the transmission of vibration through a machine, it is instructive to investigate the concept of linearity and what is meant by linear and non-linear systems. Thus far, we have discussed linear and logarithmic amplitude and frequency scales, but the term “linear” also refers to the characteristics of a system which can have input and output signals. A “system” is any device or structure that can accept an input or stimulus
in some form and produce a corresponding output or response. Examples of systems are tape recorders and amplifiers, which operate on electrical signals, and mechanical structures, whose inputs are vibration forces, and whose outputs are vibration displacements, velocities, or accelerations.
Definition of Linearity
A system is said to be linear if it meets the following two criteria:
1. If input x to the system results in output X, then an input of 2x will produce output of 2X. In other words, the magnitude of the system output is proportional to the magnitude of the system input.
2. If input x produces output X, and input y produces output Y, then an input of x + y will produce an output of X + Y. In other words, the system handles two simultaneous inputs independently, and they do not interact within the system. Implicit in these criteria is the fact that a linear system will not produce any frequencies in the output that are not present in the input.
Note that there is nothing in these criteria that says the system output is the same as the system input, or even that it resembles the system input. For instance, the input could be an electric current, and the output could be a temperature. In the case of mechanical structures such as machines, we will consider the input to be a vibratory force and the output to be the measured vibration itself.
Non-Linearities in Systems
Absolutely perfect linearity does not exist in any real system. There are many different types of non-linearity, and they exist in varying degrees in all mechanical systems, although many actual systems approach linear behavior, especially with small input levels. If a system is not perfectly linear, it will produce frequencies in its output that do not exist in its input. An example of this is a stereo amplifier or tape recorder that produces harmonics of its input signal. This is called “harmonic distortion“, and it degrades the quality of the music being reproduced. Harmonic distortion almost always gets much worse at high signal levels. An example of this is a small radio that sounds relatively “clean” at low volume levels, but sounds harsh and distorted at high volume levels.
Many systems are very nearly linear in response to small inputs, but become non-linear at higher levels of excitation. Sometimes a definite threshold exists in which input levels only a little above the threshold result in gross non-linearity. An example of this is the “clipping” of an amplifier when its input signal level exceeds the voltage or current swing capacity of its power supply. This is analogous to a mechanical system where a part is free to move until it hits a stop, such as a loose bearing housing that can move a little before being stopped by the mounting bolts.
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.