# Cepstrum: analysis of harmonic families

The Cepstrum is a tool used to investigate periodic structures in frequency spectra. The method is the result of calculating the Inverse Fourier Transform (IFT) of the logarithm of the estimated signal spectrum. Find out more about this analysis!

## Cepstrum: How does it work?

To understand the concept, it’s helpful to think of a machine in operation. Naturally, this asset will produce vibrations derived from movement or contact between its components.

If the machine is functioning properly and shows no signs of failure (even at an initial stage), it will exhibit certain vibration patterns. Now, if one of the gears, for example, has a defect such as wear or backlash, the vibration pattern will change.

New vibration patterns can indicate where the fault lies. That is, if it appears at low frequency, it may be associated with certain components, while it will be associated with others if the signals are at high frequency. This situation is reflected in the Cepstrum analysis.

In the Dynamox platform, it is represented as follows:

### What is Cepstrum Analysis?

Cepstrum is a technique used to identify harmonic families and sidebands in complex and consistent signals. Instead of analyzing just one component of the spectrum, it analyzes families of harmonics and signal modulations.

It is useful, therefore, for analyzing the entire frequency range as a whole.

Thereby, through Cepstrum analysis, it is possible to analyze periodicities as if they were a waveform. It is a very useful tool for analyzing gears, especially gear defects.

For those familiar with vibration analysis, the terminology used in Cepstrum analysis may seem a bit different. Check it out:

• Spectral Analysis is equivalent to Cepstral Analysis
• Spectrum is equivalent to Cepstrum
• Frequency (Hz) is equivalent to Quefrency (s)
• Harmonic is equivalent to Rahmonic

Find out more about Cepstrum in the video (available in portuguese):

## How to use this type of analysis?

Gear faults sometimes may not be evident in the acceleration spectrum. For example, an analyst who does not have access to Cepstrum but suspects issues in a gearbox, for instance, tends to look at the acceleration signal to identify a resonance region.

That is, mapping the carrier frequency and from there, analyzing the sidebands (the modulating frequency). Then, it is necessary to subtract the carrier frequency from the modulating frequency and thus find the specific frequency that will identify the presence of a backlash frequency, gear frequency, rotation frequency of a certain shaft and so on. But with Cepstrum, this is not necessary.

To detect a fault with Cepstrum, the first step is to have the value of the Gear Mesh Frequency (GMF). Normally, the GMF presents sidebands of the operating velocity. And around it, all peaks are of low amplitude when there is no failure since there is no excitation of these frequencies. The GMF is given by:

#### GMF = Number of Gear Teeth x Gear Velocity (RPM)

By defining this value, you look at the Cepstrum analysis, leaving aside the carrier frequency and focusing on the behavior of the sidebands. It is these sidebands, therefore, that will define the severity of the defect.

That is, the more sidebands and the higher their energy, the more severe the failure. Thus, by disregarding the normal and expected frequency of meshing, the analysis is focused only on what should not be there.

So, the analyst can then apply the tool only to the frequencies identified in the sidebands and generate a report from that. In fact, this behavior in the acceleration spectrum may seem quite stable and not raise awareness of the need for inspection.

Therefore, vibration monitoring of reducers tends to require the use of more than one tool or type of analysis. The combination allows the detection of failures in the early stages, since gear defects are already advanced when they appear at velocity data.

### Case Analysis: the use of Cepstrum

One of the most common applications for Cepstrum is gear analysis, which aids in detecting faults, as demonstrated in a case involving a cooling tower gearbox. The identification was made possible due to the continuous online monitoring of the asset by Dynamox sensors. These sensors provide data accessible via WebPlatform, enabling the execution of Cepstrum analysis.

The analysis was applied to verify the increase in vibration after the component returned from repair. Thus, along with other spectral analyses, Cepstrum allowed for the diagnosis of irregular hunting tooth wear.

The video highlights the importance of using analysis tools together. Additionally, it demonstrates an application of Cepstrum to highlight the “undesirable” frequency of what is expected in the asset’s vibration pattern.

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