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Fig. 3. Distribution Examples and Kurtosis Values
with an extreme case corresponding to the harmonic signal and a kurtosis value of 1.5. A leptokurtic signal is char-acterized as having spaced pulses which are indicative of speech signals. The signal used as an example was created by speaking “A-B-C-D-E” into a microphone. However, in the case of gear damage, it could be produced with multiple gear tooth faults.
As discussed previously, the FM4 CI is the kurtosis of a modi.ed signal called the difference signal. This is the orig-inal TSA signal with its gear mesh frequencies, harmonics, and .rst-order sidebands removed. Other condition indica-tors, namely M6A, and M8A, also operate on the difference signal. (Ref. 23) The next section gives additional insight into this modi.ed signal, how it is determined, and the ex-pectations when using synthesized signals as those obtained from vibration separation.
The Difference Signal
Many of the CIs investigated look at the statistical prop-erties of the difference signal, d(t). If the signal to be tested represent a full shaft cycle, then the period, Tcycle = NΔt, where N is the number of points in a cycle, and Δt is the sampling rate. Therefore, when taking a standard FFT, the frequency resolution, Δ f = fsamp/Nˉ= 1/Tcycle, is equal to the shaft frequency, fsha ft . This says that all Fourier coef-.cients are computed at integer multiples of the shaft fre-quency. This means, for example, if the .rst gear mesh frequency and its .rst-order sidebands are removed, this would involve modifying only three points. In general, the gear mesh frequencies are at index values corresponding to GMF =(1: n .1)·Nteeth + 1 where n = fNyquist / fsha f t , and the left side band (LSB) and right side band (RSB) would correspond to LSB = GMF .1 and RSB = GMF +1 respec-tively. Again, this represents the case for which a full cycle is used, which is usually the case when testing a complete carrier cycle. However, when analyzing other signals, more general approaches are needed.
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