“Ehrich [1] described the interaction of two signals close together in frequency through a non-linear system. Although he did not use the terms modulation and beating, he appears to have been the first to document and describe the use of sidebands and an oscillating envelope in the diagnosis of a mechanical system. Eshleman [2] expanded on Ehrich and discussed the possibility of misalignment, looseness, rubs, and other phenomena caused by system response non-linearity. He also described the generation of sidebands in the vibration spectra. He raised the subject of using this characteristic as a quantitative diagnostic tool as an area of fruitful future research. Both Ehrich and Eshleman used the Fourier transform to determine the frequency content of truncated signals. Eisenmann [3] and Taylor [5] present explanations of the characteristics of simple modulation and beating. Several case histories are given that add depth to the explanation. Taylor [5] also very briefly touches on the non-linear aspects of misalignment. Other authors, Mitchel [4] and Crawford [6], have discussed the appearance of sidebands in spectra and oscillating waveforms and their use in diagnosing machinery faults. Okubo [13] gives an excellent paper on the effect of non-linearity on transfer functions and, consequently, Nyquist and Bode’ plots. This paper will expand on these results.

How mechanical systems respond to forces: The frequency response plot that we are all familiar with, Figure 1, comes with the understood assumption that it only applies to linear systems. Figure 2 is the frequency response map of a system whose response characteristic changes with increasing force.

Here we will look at the effect of the force response characteristic of a system on the vibrations that we observe. Figure 3 shows the force response characteristic that might be associated with misalignment, a rub, or looseness. The soft and progressive change in restraint of misalignment with some types of couplings like gear, Bendix disk, or elastomer donut is more subtle. A non-linear model for misalignment can do a creditable job of explaining its apparently inconsistent characteristics [7].

Modeling a non-linear system : A power series can be used to describe the force response characteristic of the progressively more non-linear systems, as illustrated in Figure 3. It has the form:

A + BX + CX2 + DX3 + …… MXN EQ(1)

Where A,B,C,D,M, and N are arbitrary coefficients; X is the independent variable, in our case force. It may be a single force or the sum of independent forces.”