Cepstrum Analysis in Vibration Diagnostics

Definition: What is a Cepstrum?

Cepstrum analysis is an advanced signal processing technique that provides information about periodic structures within a frequency spectrum. The name “cepstrum” is an anagram of “spectrum,” which hints at its nature: it is effectively the “spectrum of a spectrum.” It is calculated by taking the logarithm of a frequency spectrum and then performing an inverse Fourier transform on the result. This process highlights periodic patterns in the spectrum, such as families of harmonics or sidebands, which might be difficult to discern by looking at the spectrum directly.

In the cepstrum plot, the x-axis is called quefrency (an anagram of frequency) and has units of time. Peaks on this axis, called rahmonics, indicate the period (in seconds) of the repeating patterns in the original spectrum.

Why Use Cepstrum Analysis?

While a standard FFT spectrum is excellent for identifying individual frequency components, it can become cluttered and difficult to interpret when a machine fault generates a large number of harmonics and sidebands. Cepstrum analysis simplifies this by consolidating an entire family of repeating frequencies into a single, clear peak. Its primary uses are:

• Detecting Harmonic Families: It can easily identify a fundamental frequency and its harmonics, even if the fundamental itself is weak or absent in the spectrum.
• Identifying Sideband Families: It excels at finding sidebands, which are often low in amplitude and buried in noise. Cepstrum can clearly show the presence of sidebands and measure their spacing.
• Separating Source and Path Effects: In some applications, it can help separate the vibration source signal from the structural response of the machine.
• Echo Detection: It can identify echoes or reflections in a signal.

Key Applications in Machine Diagnostics

1. Gearbox Diagnostics

This is the most common and powerful application of cepstrum analysis. A damaged gear tooth will modulate the gear mesh frequency (GMF), creating sidebands around the GMF peak spaced at the rotational speed of that gear. In a complex gearbox with multiple shafts and gears, the spectrum can be a confusing mix of different GMFs and sidebands. The cepstrum cuts through this complexity:

• A peak in the cepstrum at the quefrency corresponding to the gear’s rotational period (1 / RPM) is a clear indicator of a fault on that specific gear.
• The amplitude of the cepstrum peak can be trended to monitor the progression of the gear wear.

2. Rolling Element Bearing Analysis

Similar to gearboxes, bearing defects also generate sidebands. For example, a defect on the inner race will create sidebands spaced at the shaft speed around the inner race defect frequency (BPFI) and its harmonics. The cepstrum can help confirm the presence of these patterns, especially when they are not obvious in the spectrum.

3. Turbomachinery Analysis

In turbines and compressors, cepstrum can be used to identify blade pass frequency harmonics and diagnose issues related to blade damage or aerodynamic problems.

How to Interpret a Cepstrum Plot

1. Calculate Rotational Periods: Before looking at the cepstrum, calculate the time periods of the main rotating components (e.g., for a shaft at 1800 RPM or 30 Hz, the period is 1/30 = 0.033 seconds).
2. Look for Peaks at Known Periods: Examine the cepstrum for significant peaks (rahmonics) that line up with these calculated time periods.
3. Identify Harmonics: Look for peaks at integer multiples of the fundamental quefrency. These indicate the presence of strong harmonic families in the original spectrum.
4. Trend Amplitudes: Monitor the amplitude of the cepstrum peaks over time. An increasing amplitude indicates a worsening condition.

Cepstrum analysis is a powerful tool that requires experience to use effectively, but for complex machinery like gearboxes, it can provide diagnostic clarity that is unmatched by spectrum analysis alone.

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