What are crest factor and kurtosis, and how do you read them for bearings?

Definitions: crest factor and kurtosis

Crest factor is the peak value divided by the RMS value: crest = peak / RMS. For a sinusoid, the peak is √2 times the RMS, so the crest factor is about 1.414. Short, sharp impacts raise the peak without changing the RMS much, which increases the crest factor.

Kurtosis is based on the fourth moment of the amplitude distribution: kurtosis = E[(x − μ)^4] / σ^4. It is about 3 for Gaussian noise and 1.5 for a continuous sinusoid. Impulsive signals give larger values. Some software reports excess kurtosis (kurtosis − 3), which is 0 for Gaussian noise. Check which definition your tool uses.

A small signal worked by hand

For arithmetic only, take a hypothetical record of 10 samples with mean 0. Two samples are +3 and −3 m/s², and the other eight are 0.

• Peak = 3. Mean square = (3² + 3²) / 10 = 18 / 10 = 1.8. RMS = √1.8 ≈ 1.342.
• Crest factor = 3 / 1.342 ≈ 2.236, which is exactly √5.
• Fourth moment = (3^4 + 3^4) / 10 = 162 / 10 = 16.2. σ^4 = 1.8² = 3.24.
• Kurtosis = 16.2 / 3.24 = 5.0.

A continuous sinusoid gives about 1.5. This record gives 5.0, which looks impulsive. Ten samples are too few for real conclusions; the numbers only illustrate the arithmetic.

Interpretation and common mistakes

In early bearing damage, impacts increase, and both crest factor and kurtosis can rise. As damage spreads, the impacts blend together, RMS rises, and the two indicators may fall back. Watching one number alone can therefore mislead.

• Using one value as a fixed threshold. There is no universal kurtosis limit; compare with your own baseline.
• Comparing directly with sinusoid or Gaussian values. Measurement type, load and sensor position change the result.
• Computing on raw unfiltered data. Low-frequency offset and unrelated components can distort the result, so a band-pass filter is usually applied first.

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