RMS, crest factor and kurtosis: vibration features explained clearly

A vibration signal contains thousands of data points per second. Monitoring needs a few numbers that summarise what matters: the vibration features. The best known are RMS, crest factor and kurtosis. They are easy to calculate, but not easy to interpret. A feature can rise markedly with incipient bearing damage and fall again in the final stage. This article explains each feature with its formula, shows what it reveals about bearing damage and where it fails.

What a feature is and what it is not

A feature condenses a measurement window into a single number. Information is lost in the process, and deliberately so. RMS tells you something about the energy, but nothing about the shape of the signal. Kurtosis tells you something about the shape, but nothing about the magnitude. No single feature fully describes the condition of a spindle.

All features in this article refer to a measurement window with N samples x₁ to x_N. RMS and peak value have the unit of the input signal, i.e. m/s² or g for an accelerometer. Crest factor and kurtosis are dimensionless ratios. This is a practical advantage: they do not depend on the sensor gain.

RMS: the energy of the vibration

RMS, short for root mean square, is the square root of the mean of the squared samples:

It measures the mean energy of the vibration in the window. That is why standards such as ISO 10816 and ISO 20816 assess the vibration condition of entire classes of machines using an RMS value, in that case of vibration velocity. In our calculation, RMS includes any DC component of the signal.

Strength: RMS is stable and responds reliably to anything that increases the vibration overall: unbalance, loose parts, advanced bearing wear.

Limit: incipient bearing damage produces short, weak impacts. Their energy is small compared with the overall vibration of a running spindle. RMS therefore usually only changes once the damage is advanced. It is a good watchdog for gross changes, but not an early warning system for bearings. More on standard limits in the article Vibration analysis on machine tools and ISO limits.

Peak value and crest factor: how far the impacts stand out

The peak value is the largest absolute value in the window. On its own it says little, because a single knock during clamping or a tool change drives it up. It becomes interesting in relation to RMS. This ratio is called the crest factor:

For a pure sine wave, the crest factor is exactly √2, i.e. about 1.41. The more short, high peaks a signal contains, the larger it becomes. That is exactly what happens with early bearing damage: each time a rolling element passes over the defect, an impact is produced that stands out clearly from the rest of the signal, while RMS barely moves.

Limit: the crest factor falls again in the final stage of bearing damage. As the defect spreads, the impacts become more numerous and smear. The overall vibration, and with it RMS, rises, and the peaks no longer stand out as clearly. A falling crest factor is therefore not an all-clear. It may mean that the damage is advanced. If you look only at the crest factor, you can mistake severe damage for an improvement.

Kurtosis: the measure of impulsiveness

Kurtosis describes how strongly a distribution is shaped by rare, extreme values. We use Pearson's kurtosis, i.e. without subtracting 3:

Because the deviations are raised to the fourth power, rare large excursions count very heavily. Three reference values help with interpretation:

The vibration of a healthy spindle under load is a mixture of speed-related components and broadband noise. Its kurtosis is usually close to the reference values. When impacts from bearing damage are added, it rises markedly.

Limit: kurtosis has the same weakness as the crest factor. In the final stage, when the signal is dominated by many impacts and broadband noise, it approaches the value for noise again. There is also a second weakness: it responds to any kind of impact, including process impacts during interrupted cuts, as the tool enters and leaves the cut, or during a tool change. So an elevated kurtosis tells you that impacts are present, but not where they come from.

Band features: energy where the damage is

Time-domain features see the whole signal. Band features look specifically at a frequency range or a single frequency. This lets them answer the question that RMS and kurtosis leave open: where does the change come from?

In our system we use two kinds of band features:

• Orders: the amplitude at once and twice per revolution in the order spectrum. The first order typically rises with unbalance, the second often with alignment or mounting problems. See Order analysis and unbalance.
• Envelope amplitudes at the bearing frequencies: the amplitude in the envelope spectrum at the defect frequency of the outer race (BPFO), inner race (BPFI), cage (FTF) and at twice the rolling element frequency (BSF). These features show not only that impacts are present, but also at what repetition rate. This means they can be assigned to a bearing component. The basics are covered in the article Detecting rolling bearing damage early.

Limit: band features are only as good as the data they are based on. The bearing frequencies need the correct geometry from the datasheet and the current speed. The envelope needs a suitably chosen frequency band around a structural resonance. And the spectrum needs sufficient frequency resolution for neighbouring lines to be separated. More on this in the article FFT, spectrum and resolution.

Why we combine several features

Every feature has a range in which it is strong and one in which it fails. Together they cover the progression of bearing damage much better:

This leads to a simple rule: one feature alone is not enough. A falling crest factor with rising RMS is a different picture from a falling crest factor with constant RMS. Only the combination makes the progression readable. That is why, in our system, the edge box calculates RMS, peak value, crest factor, kurtosis, the first and second orders and the four envelope amplitudes from every measurement window. How the edge box does this at the machine is described in the article The edge box at the machine.

Why we assess against the learned baseline rather than fixed limits

There is no fixed limit for kurtosis or crest factor that applies to all spindles. Installation, sensor position, speed and machining shape the base level of each machine. A spindle roughing with interrupted cuts has a different kurtosis from one doing finishing.

Our system therefore learns, for each feature, the learned baseline, i.e. the normal state of this spindle, separately for each measurement point and speed range, from at least 20 measurements in healthy condition. The median and a robust spread are stored. Each new measurement is expressed as a deviation in multiples of this spread; only increases count. This gives the levels Normal, Noticeable, Warning and Alarm. An alert is only raised once a level persists over several consecutive measurements. That way, a single impact during a tool change does not produce an alert, even though it drives up the crest factor and kurtosis of that one window.

The level thresholds are starting values that we calibrate with test rig measurements and feedback from operation. The whole chain from measurement window to alert is described in the article From vibration to alert, and the comparison with fixed statistical thresholds in the article 3-sigma alarms or learned baseline. Whether your machines are suitable is shown on the CNC machining centres page.

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