A method for detecting and characterizing faults in a rotating machine involves analysis of selected aspects of the evolution of the spectral distribution function (SDF) of its vibrations. The method is based on the proposition that long-term accumulation of damage (e.g., bearing wear or spalling of gear teeth) is accompanied by long-term changes in the SDF, and that these changes are revealed as differences among SDFs computed from short-observation-time vibration data. Data displays produced according to older vibration-analysis methods, including methods based on the conventional power spectral density (PSD), tend to display large components of random noise and other artifacts that often make interpretation difficult. The present method based on the SDF gives more robust indications of changes, yet entails little more computation than do methods based on the conventional PSD.

The SDF is derived from the PSD. It is a monotonically increasing function that bears the same relation to the PSD as a cumulative probability distribution to the probability density function.

Consider a vibration signal X*(t)*, where *t* is time. Let X*(t) *be sampled at intervals of Δ*t *to obtain the time series X* _{j}* = X

*(j*Δ

*t)*, where

*j*is an integer. Let the time series comprise a total of

*N*samples. Let the time series be transformed as

for any integer *k* from 1 to *N*. The periodogram representation at *M* = (*N/2*) + 1 points is given by

for *k* = 1 or *M* and

for any integer *k* from 2 to *M − *1, where the superscript asterisk denotes the complex conjugate. This representation is an *M*-line discrete estimate of the power spectral density, for frequencies *f _{k}* = (

*k*− 1) / (

*N*Δ

*t*).

The sum of all terms in the periodogram is the mean-square amplitude of the time series — a measure of the total signal energy. The spectral distribution function (SDF) in this situation is defined as the partial sum

for any integer *k* from 1 to *M*. Thus,* F _{M }*is the mean-square amplitude, and

*F*is an estimate of the energy at and below frequency

_{k}*f*= (

_{k}*k −*1) / (

*N*Δ

*t*). The SDF can be normalized by

*F*, to obtain a normalized spectral distribution function (NSD) that is an estimate of the fraction of energy at and below a given frequency.

_{M}The method was applied in a test case involving vibration data from a helicopter tail-rotor gearbox. Data were taken under three conditions that might be encountered in a long-term progression of damage; (1) a baseline condition (no faults), (2) spalls on two gear teeth, and (3) half of one gear tooth removed. For each test condition, the vibration measurements were digitized and phase-averaged with respect to shaft rotation, then *N* = 46,880 samples were acquired at a rate of 46,880 Hz (thus making the observation time 1 second).

Figure 1 shows the PSDs and NSDs for the three test conditions. The PSDs are characterized by a few tones rising above background noise. Although there are differences among the PSDs, they are difficult to identify by casual visual inspection. The differences among the NSDs are more obvious and are easily quantifiable. The discrete jumps in the NSD indicate the tones, while the finite-sloped portions indicate the buildup of frequency-dependent continuous random noise.

The quantifiable indicators in the NSD are the frequencies that bound specified relative energy levels; for example, the first and ninth deciles are marked by the dashed lines in Figure 1. A more meaningful way to display these indicators is to juxtapose and join plots of the frequencies, parameterized by the relative energy levels, for sequential observation times. The resulting combination of plots (see Figure 2) amounts to a contour plot of constant relative energy levels in frequency-time space. Such contour plots are simple and easy to read, and can be stored for use in subsequent diagnoses.

*This work was done by Sanford Davis of *Ames Research Center*. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com under the Information Sciences category. This invention is owned by NASA, and a patent application has been filed. Inquiries concerning nonexclusive or exclusive license for its commercial development should be addressed to*

###### the Patent Counsel, Ames Research Center (650) 604-5104.

*Refer to ARC-14240.*

##### NASA Tech Briefs Magazine

This article first appeared in the April, 1999 issue of *NASA Tech Briefs* Magazine.

Read more articles from the archives here.