Kurtosis monitoring

The Kurtosis meter as applied to vibration measurement was originally manufactured by CML Systems under licence from the then British Steel Corporation. Both companies have long been subsumed within larger industrial enterprises — CML by Rockwell Automation and British Steel by the Corus Group.

At the present time therefore the Kurtosis meter is not available. Because of its potential for producing a result which was not wholly dependent on trending, this is of some regret to the au­thor. He therefore felt that the following descriptive material de­served a permanent record:

What is Kurtosis?

It should firstly be recognised that Kurtosis is a statistical pa­rameter widely used in the analysis of distribution curves. If we have a number of measurements to plot, the value which oc­curs most frequently is called the mode. In a normal distribu­tion, a symmetrical bell-shaped curve can be drawn having its peak at the mode. Originally derived by Gauss, it is often called the Gaussian curve. The Kurtosis value P2 is defined in the equations below:

P2 = — xJ+0C(x-x)4P(x)dx Equ15.9

CT4

Where:

X = measurement

X = mean value of x

A = standard deviation or Root Mean Square for a

Zero mean signal

As an alternative we may say:

P2=^- Equ 15.10

Where:

|i4 = the fourth moment of the measurement distri­

Bution density function

L^22 = the second moment (variance) of the measure­

Ment distribution density function

Figure 15.25 Early bearing damage detector for Kurtosis measurements

The Kurtosis value of the normal or Gaussian distribution is 3. This level is used as a reference tojudge the “peakiness” of the distribution curve. Greater than 3 would be more peaky than Gaussian whilst less than 3 would indicate a flatter curve.

As mentioned before this work was introduced by ISVR (Insti­tute of Sound and Vibration Research) at Southampton Univer­sity whilst carrying out an investigative contract for the former British Steel Corporation. Kurtosis, when applied to the moni­toring of bearing condition, is protected by Patent Specification 1536 306 owned by the former British Steel Corporation and its successors.

Using the statistical theory outlined above, it was decided that peak acceleration values of vibration should be obtained over a frequency spectrum of 2.5 kHz to 80 kHz. Inserting these mea­surements in the formulae, it could be anticipated that the Kurtosis factor for a good bearing would equal 3. A deviation of more than ± 8% from this figure would indicate the presence of damage.

Further research showed that if Kurtosis measurements were taken in discrete frequency bands and used in conjunction with overall velocity and/or acceleration measurements of vibration, then a more detailed assessment could be made, together with a trend analysis.

It should be remembered that the system does not rely on ob­taining an absolute vibration measurement. The process of ob­taining a Kurtosis reading is a statistical one based upon accel­eration distribution. Thus although the variation in trans — missibility of the vibration signals over the frequency band will produce a wide dynamic range of signals, the Kurtosis value will hardly be affected.

The Kurtosis meter

In its commercial form the instrument was known as the K me­ter. It consisted of a battery powered portable instrument with its own inbuilt microcomputer, together with a transducer (acceler­ometer) and input cable. The batteries could be re-charged from the mains. A carrying case was also provided and the whole equipment is as shown in Figure 15.25.

Vibration signals were monitored either by using a probe fitted into the end of the transducer, or preferably by mounting the transducer using its hand nut to secure it to a stud fitted to the bearing housing under investigation. If the probe was used, then it had to be firmly held, and applied to a point on the ma­chine adjacent to the bearing race. The position selected should preferably have given the highest acceleration values of vibration in g RMS. The location should have been marked for future repeatability.

Grips could also be used where a hand probe might be danger­ous but sensitivity could have been reduced. Nevertheless, the meter adjusted itself to suit the strength of the vibration signal available and the operator did not have to range the instrument.

The method was also virtually unaffected by bearing size, a speed change or increase in bearing load.

Kurtosis values relative to frequency

The various stages of damage to a bearing are shown in Figure 15.26 together with the effect on the acceleration and Kurtosis value in each frequency band. It will be seen that these change significantly. The relative shape of the graphs will be true for a given amount of damage no matter where the bearing is in­stalled. These curve shapes can be recognised by the micro­computer within the meter and thus the degree of damage can be indicated on the display.

The meter was operated in three different modes:

• Assessment

• Enveloping Assessment

This was the most simple, and for many cases did suffice. Hav­ing fixed the transducer to the bearing housing and switched on the instrument, a battery check took place. The display panel I indicated if this was satisfactory or not, and whether re-charg­ing was necessary. When the panel indicated “READY” the bearing condition — LOW SPEED (less than 1000 rev/min) or HIGH SPEED (greater than 1000 rev/min) was pressed. Even this was not critical, as selection of the wrong button simply ex­tended the time taken to analyse the data and display the re­sults.

The meter in the meantime responded with “BUSY LS" (low speed) or “BUSY HS” (high speed) whilst the data signals from the transducer were gathered and the calculations carried out. If the data was unstable, or if the accelerometer was detached from the machine then “ERROR” appeared on the display, and the bearing condition button had to be pressed again. Once the instrument had accepted the data and carried out its internal calculations, it indicated bearing condition directly as “GOOD”, “LODAMAGE” (indicating early damage of the bearing) or “HIDAMAGE” (indicating a serious condition and imminent fail­ure).

New

Incipient damage — a/V_____________________________ /^/W- Intermediate damage _AAAn__ AAAn Extensive damage Rt>rAflAfr/lIV№A<iVvx ——— time
V.
/A
Damage Component
Combined damage and background	Combined forcing and structural response
Forcing waveforms

Kurtosis

Force spectra

Acceleration spectrum

Bearing details	K1	K2	K3	K4	K5	G1	G2	G3	G4	G5	V Mm/sec RMS
1 Pump bearing No. 4	Speed rev/min	Date	■J	Y	Y	Y	>/	-1 5.12	-1 6.24	-2 7.92	-2 2.98	-3 3.1	21
1500	7/10
Assessment Good
2	Speed rev/min	Date	J	J	Y	V	>/						23
1500	10/11
Assessment
3	Speed rev/min	Date	5.0	4.7	4.8	4.6	4.6	-1 8.7	-1 9.1	-1 21	-2 3.8	-3 4.6	2.2
1500	9/12
Assessment Early
4	Speed rev/min	Date	3.7	3.8	4.2	4.8	5.2	11.2	1.5	-1 7.1	-2 6.1	-2 27	25
1500	15/1
Assessment Advanced
5	Speed rev/min	Date	Y	•J	3.7	4.2	5.1	4.3	6.1	1.1	-1 2.1	-2 4.2	3.2
1500	12/2
Assessment Advanced
For further interpretation of results If no datamsvk* and operational details see K meter If K = < 3.5 mark V model 4100 handbook

Figure 15.26 Diagram showing value changes with increased bearing damage

Figure 15.27 Typical Kurtosis result sheet Analysis

V — Velocity RMS mm/sec E — enveloping function Both switches had a stepping function, for example the display might have shown: 03.87 KB3 which indicated a Kurtosis value of 3.87 in fre­quency band 3. By pressing the KgVE button the display could change to: 12.67 g B3 showing an acceleration level of 12.67g in fre­quency band 3. If a bearing was in a “GOOD” state, it suggested that gRMS val­ues were recorded for all frequency bands. When the bearing entered the “LODAMAGE” condition both gRMS and K should have been taken. Atrend in the readings then showed the prog­ress of damage, see Figure 15.27. g values increased whilst K factors will probably “peaked” at higher frequencies. By using this technique an experienced operator could predict the time to failure and thus the number of useful hours left in the bearing. Enveloping This was a facility used with an external analyser and provided an operator with the ability to identify damage repetition rate

In the assessment mode, whilst data was collected in five dis­crete frequency bands, the evaluation was automatically car­ried out to arrive at the final assessment. For analysis, the more proficient operator could use the switches at the right of the me­ter.

Switch “f BAND 1-5”, which selected the required frequency band:

Band	Frequency range kHz
1	2-5	To	5
2	5	To	10
3	10	To	20
4	20	To	40
5	40	To	80

And the “KgVE" switch, which selected either: K — Kurtosis value g — RMS acceleration

And thus that relating to machine speed. The resulting spec­trum analysis showed whether the vibration signal was random in phase and amplitude or whether there was a repetitive wave­form present.

The meter, once it had provided an assessment, stored indefi­nitely all the readings in its memory, and the last assessment, until power was switched off, the batteries run down or the speed buttons were pressed.

Conclusions

The intelligent use of condition monitoring techniques can as­sist greatly in the determination of necessary maintenance and the replacement of rolling element bearings. Systems are now available which have proved successful in giving warning of im­pending fatigue failure. Whilst often viewed with suspicion by the more conservative amongst us, it is believed that they will become widely accepted in the future. Only where there is the danger of imminent damage or malfunction should it be necessary to stop machinery.

Posted in Fans Ventilation A Practical Guide