## Waterfall Analysis: Frequency Spectrum or Order Spectrum?

This article addresses two basic approaches to analyzing rotating machinery during transient (sweeping rpm) conditions.  The first is the traditional method which uses Frequency (FFT) Spectrum analysis at target rpm values throughout a run up or a coast down condition.  As discussed in a previous article on frequency resolution and smearing in waterfall analysis, there is a trade off between the rate of the rpm change, the highest order of interest, and the analysis frequency resolution.  These same considerations must also be addressed when using a different approach using Order Spectrum analysis. However, better estimates of orders are obtainable when using this second method.

## Background

The signal used for this example analysis was an artificially created square wave digitized at 2048 samples/second which was swept in 10 seconds from 10 Hz to 100 Hz (corresponding to the fundamental rotation frequency of 600 to 6000 rpm).  This simulated a signal that has frequency components of 1X, 3X, 5X, 7X, and so on… times the fundamental frequency.  A second signal was also created to serve as a tachometer signal.  This signal was a sinusoid digitized at 2048 samples/second which was also swept from 10 Hz to 100 Hz in the same period of 10 seconds.

## Waterfall Analysis – Frequency Spectrum

For the waterfall shown in Figure 1 a frequency spectrum analysis was performed.  This type of analysis takes as input data that has been digitized at a constant sampling rate.  In this case it was 2048 samples/second. The resulting waterfall can be displayed either as a speed-frequency waterfall or as a speed-order waterfall; the latter form with orders along the X-axis was chosen to better see the order components of this particular signal.  This waterfall was calculated with 1 Hz frequency line spacing.

Figure 1: Spectrum waterfall from time (non-synchronously) sampled data

In order to illustrate what a typical spectrum would look like a single spectrum from this waterfall data at a simulated speed equivalent to 3000 rpm is shown below in Figure 2.  The graph displays the individual analysis lines from the FFT and clearly shows a fundamental frequency of 50 Hz (3000 rev/minute ÷ 60 seconds/min) together with odd multiples of the fundamental frequency which is indicative of a square wave function.

Figure 2: Frequency spectrum from (non-synchronously) sampled time data

The effects of leakage are evident on the 1st, 3rd, and 5th orders.  The higher orders show additional broadening of the peaks due to the smearing effects caused by the constantly changing rpm during the acquisition of the time block used to calculate the FFT.  This phenomenon was discussed in a previous waterfall article.

## Waterfall Analysis – Order Spectrum

The calculation of an order spectrum is accomplished by converting the time based signal to a revolution based signal.  This is referred to as synchronously sampled data, meaning that there is a fixed number of data samples in each revolution of the measured specimen.  For analysis example used previously, the rotational sampling frequency was specified to be 64 samples/revolution.  In other words, no matter what the rpm of the measured device is, there will always be 64 data samples per revolution.

The conversion was accomplished in the software by taking the non-synchronously sampled time data and changing it to synchronously sampled data by interpolating the data at time instants (relative to the tachometer signal) corresponding to the equivalent angle positions of a rotational signal sampled at 64 times/revolution.  Using the Nyquist frequency analogy in the time domain, the highest order that can be analyzed in the order domain is the 32nd order (which is ½ the number of samples in each revolution – the equivalent of a Nyquist order point).  An order resolution of ¼ order was chosen which meant that each block of synchronously sampled data to be analyzed covered 4 revolutions (256 points) of data.  The angle domain order resolution is the reciprocal of the length of the FFT block expressed in revolutions and is analogous to the time domain frequency resolution which is the reciprocal of the length of the FFT time block expressed in seconds. The resulting order waterfall is shown in Figure 3.

Figure 3: Order Waterfall from synchronously sampled data

A single order spectrum extracted at 3000 rpm is shown in Figure 4.

Figure 4: Order Spectrum from (synchronously) sampled angle data

As in Figure 2 above, the order spectrum data is displayed showing the lines calculated using FFT methods.  Because the number of samples is fixed for each revolution, any signals which are related to the fundamental rotation frequency are periodic within the captured block of data used to calculate the FFT.  The calculated lines correspond to rotational orders.  In this case, the calculated lines fall directly on the FFT lines so there is no leakage, and because the sampling frequency is continuously changing with the specimen’s rpm, there is no smearing of the data.  Any frequency components that are not periodic within a sampled record used to calculate the FFT will show significant leakage.

## Summary

The classical method of calculating waterfall and order cut information from continuously variable rpm data can be significantly improved by either synchronously sampling the data during the acquisition process or synchronously re-sampling the data to provide rotation angle data that is periodic within the sampled record used to calculate the FFT spectrum at each rpm in the waterfall.  The analysis software used to synchronously resample data and calculate synchronously sampled waterfalls and orders are included in the DATS Rotating Machinery Analysis Suite.

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#### John Mathey

Senior Technical Specialist at Prosig
John Mathey graduated with a MS degree from the University of Toledo in 1972. John has over 35 years of experience with instrumentation, measurement, and analysis. Twenty-five of those years were spent at Ford Motor Company solving and providing training for vehicle noise, vibration, and harshness (NVH) issues. He is now a technical specialist at Prosig USA, Inc. where he provides technical support to Prosig customers in the U.S.A.

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