Kurtosis

Kurtosis is a fourth-order statistical moment that measures the "tailedness" or impulsiveness of a signal's amplitude distribution. In machinery diagnostics, elevated kurtosis is one of the earliest indicators of developing faults such as bearing spalling, gear tooth cracks, and impact-related defects.

Definition

The kurtosis of a signal x(t) is the normalized fourth central moment:

K = E[(x − μ)⁴] / (E[(x − μ)²])²

where μ is the mean and E[·] denotes the expectation operator. This is often called the "kurtosis" directly. An alternative convention subtracts 3 to produce the "excess kurtosis," which equals zero for a Gaussian distribution.

Reference Values

Kurtosis (K)	Excess Kurtosis	Distribution Type	Mechanical Interpretation
3.0	0.0	Gaussian (mesokurtic)	Normal operating condition; random vibration without impacts
3.0 – 4.0	0.0 – 1.0	Slightly leptokurtic	Early-stage fault; occasional low-energy impacts beginning
4.0 – 6.0	1.0 – 3.0	Moderately leptokurtic	Developing fault; clear impulsive events present
6.0 – 10.0	3.0 – 7.0	Highly leptokurtic	Advanced fault; strong periodic impacts dominate the signal
> 10.0	> 7.0	Extremely leptokurtic	Severe damage; isolated high-energy impacts or bursts

A perfectly healthy machine with purely Gaussian random vibration has K = 3.0. Values significantly above 3.0 indicate the presence of transient impulsive events. In practice, trending relative changes is more reliable than using absolute thresholds.

Bearing Diagnostics

Kurtosis is particularly effective for bearing condition monitoring because bearing defects produce short-duration impact pulses that are superimposed on the background vibration:

Healthy bearing: K ≈ 3.0 — vibration is random and Gaussian.
Incipient spall: K rises to 4–5 as micro-impacts from the small defect create occasional amplitude spikes.
Growing defect: K increases further (5–8) as impacts become more energetic and regular.
Severe damage: K may paradoxically decrease as the defect grows so large that impacts overlap and the signal becomes more uniformly noisy.

In late-stage bearing failure, the kurtosis may return toward 3.0 despite severe damage. Always use kurtosis in combination with other indicators (RMS level, envelope spectrum, crest factor) for a complete diagnostic picture.

Spectral Kurtosis

Spectral kurtosis extends the concept to the frequency domain, computing the kurtosis of each frequency bin across successive FFT blocks:

SK(f) = E[|X(f)|⁴] / (E[|X(f)|²])² − 2

Frequency bins with high spectral kurtosis indicate bands where impulsive energy is concentrated. This is used to automatically select the optimal bandpass filter for envelope analysis (the "kurtogram" approach).

Parameters

Parameter	Options	Description
Input Channel	Any signal channel	Vibration signal for kurtosis computation.
Block Size	1024 – 65536 samples	Number of samples per kurtosis calculation block.
Convention	Standard (K), Excess (K−3)	Whether the Gaussian reference of 3 is subtracted.
Bandpass Filter	Optional: f_low–f_high	Pre-filter to focus on a specific frequency range.
Overlap	0% – 90%	Block overlap for kurtosis vs time computation.

Kurtosis vs RPM

Tracking kurtosis as a function of rotational speed reveals speed-dependent impulsiveness changes. Bearing defect frequencies are proportional to shaft speed, and resonance bands excited by impacts shift accordingly. The kurtosis vs RPM plot helps identify:

Critical speeds where structural resonances amplify impact responses
Speed ranges where the defect repetition rate coincides with a structural mode
Minimum and maximum speed thresholds for reliable fault detection

Comparison with Related Metrics

Metric	Definition	Sensitivity
Kurtosis	Normalized 4th moment	Highly sensitive to isolated peaks; best for early detection
Crest Factor	Peak / RMS	Sensitive to single extreme peaks; less robust statistically
RMS Level	Root mean square	Responds to overall energy increase; less sensitive to early faults
Skewness	Normalized 3rd moment	Detects asymmetric impacts (e.g., one-sided rubs)

Practical Tips

Establish a baseline kurtosis value for each measurement point during known-good operating conditions. Trend deviations from this baseline rather than relying on absolute thresholds.
Use a bandpass filter to isolate the frequency range of interest before computing kurtosis. Broadband kurtosis may dilute the impulsive signature with irrelevant noise energy.
For continuous monitoring, compute kurtosis on short overlapping blocks (e.g., 1-second blocks with 50% overlap) to track rapid changes in impulsiveness.
Combine kurtosis with envelope spectrum analysis: use kurtosis to detect and trend the onset of a fault, and the envelope spectrum to identify its type and location.