Extracting periodically repeating shocks in a gearbox from simultaneously occurring random vibration

Abstract

Periodically repeating shocks are a quite common indication of certain defect in machinery. Detecting these shocks in early stage, before the defect is severe enough to cause failure, can provide a huge advantage in maintenance planning. The earliest possible warning of a defect may be highly important, especially in targets where failure can lead to a vast loss of production or a safety risk. Shock-like vibrations are, however, usually rather faint when the defect is a minor one. This simply means that the shocks may often be too low in magnitude to be easily detected. In this paper, we use different techniques based on real order derivative to detect gear defects. Higher real order derivative, discrete Fourier transform, and Hilbert transform are discussed.

Full text

Extracting periodically repeating shocks in a
gearbox from simultaneously occurring random
vibration
Konsta Karioja
Mechatronics and machine diagnostics, Faculty of technology,
P.O. Box 4200, FI-90014 University of Oulu, Finland
E-mail: konsta.karioja@oulu.fi
Sulo Lahdelma
Tirriäisentie 11, Oulu, Finland
E-mail: sulo.lahdelma@gmail.com
Grzegorz Litak, Bartłomiej Ambrożkiewicz
Lublin University of Technology, Faculty of Mechanical Engineering,
Nadbystrzycka 36, 20-618 Lublin, Poland
E-mail: g.litak@pollub.pl
Abstract
Periodically repeating shocks are a quite common indication of certain defect in
machinery. Detecting these shocks in early stage, before the defect is severe enough
to cause failure, can provide a huge advantage in maintenance planning. The earliest
possible warning of a defect may be highly important, especially in targets where
failure can lead to a vast loss of production or a safety risk. Shock-like vibrations
are, however, usually rather faint when the defect is a minor one. This simply
means that the shocks may often be too low in magnitude to be easily detected.
In this paper, we use different techniques based on real order derivative to detect
gear defects. Higher real order derivative, discrete Fourier transform, and Hilbert
transform are discussed.
Keywords: condition monitoring, envelope spectrum, feature extraction, fractional
derivative, Hilbert transform, real order derivative
1. Introduction
The example case here is a gearbox with an induced fault in a single gear tooth of
the secondary gear. The gearbox was coupled to a water pump. The pump was run

in a normal state, and in a state where the intake valve of the pump was closed, and
this induced cavitation in the pump. In this study, we experiment some techniques
to extract the periodically repetitive shocks out of the random vibrations due to
cavitation. Enveloping is commonly utilised in this type of cases. In our study
we point out that the technique to obtain the envelope may have some effect to
the result. In addition, we show that using an envelope of some other signal than
acceleration or velocity may be useful.
Previous research on different methods to apply on gear fault cases also exists.
Techniques applied include e.g. acoustic emission(1,2) and spectral kurtosis of vi-
bration(3,4). Faults of rotating elements such as gears, bearings or shafts can be
also detected by application of Hilbert transform to obtain envelope signal(5,6) in a
different way than usually in condition monitoring (7,8) .
In this paper, we study testing arrangement using fractional derivatives and different
techniques for enveloping. Brief analysis and concluding remarks are presented.
2. Signal processing
Signal processing methods utilised here include Hilbert transform, real order deriva-
tive and ideal filtering. The two latter ones can be achieved by discrete Fourier
transform (DFT).
2.1 Real order derivative and ideal filtering
Several signal processing methods, e.g. the real order derivatives applied here are
widely discussed in(9,10). In fact these papers discuss the methodology more exten-
sively and thoroughly than applied in this paper.
Here we apply the definition in(11) for an αorder derivative of an exponential func-
tion where α∈R. If we consider displacement x(t), which is usually represented by
discrete series {xr}in computer applications(12), the αorder derivative of displace-
ment can be calculated in three following steps(13) :
1. Compute the DFT of {xr}, to obtain {Xk}, k = 0,1,2, ..., (N−1), which is a
sequence of complex numbers.
2. Multiply each term of {Xk}by factor (iωk)α, where ωkis the kth angular
frequency and iis the imaginary unit.
3. Compute the sequence {x(α)
r}representing the αorder time derivative of dis-
placement x(α)(t)by utilising the inverse DFT.
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The so called ideal filter is applied in this study as well. It is a simple operation,
where the DFT is performed first and frequency components in the stop band of the
filter are replaced with zeros, and the inverse DFT is applied after. The techniques
presented above have been applied before for example in(14,15,16).
2.2 Hilbert transform
The Hilbert transform (HT) of the signal x(t), denoted by ˆx(t)is defined by an
integral transform(17,5) :
ˆx(t) = 1
π
∞
Z
−∞
x(τ)
t−τdτ. (1)
The Hilbert transform can be utilised in order to obtain the envelope signal A(t).
This is done by calculating the absolute value or modulus of a so called analytic
signal z(t)defined by:
F z(t) = x(t) + iˆx(t).(2)
The envelope signal results from the modulus of Equation (2). This is to say the
envelope signal obtained via Hilbert transform can be written: A(t) = |z(t)|.(5,6,18)
3. Testing arrangement
The test rig used here, and which is shown in Figure 1, was originally manufactured
by G.U.N.T. Gerätebau GmbH, and was later modified in the Otto von Guericke
University, Magdeburg. The tests in study were conducted when a single tooth of
the secondary gear of the gearbox was intentionally damaged.
The test equipment consisted of:
•Bodywork of G.U.N.T. PT 500 test rig
•1.1 kW electric motor manufactured by EMK
•Nordac 700E frequency converter
•Mädler 41200102 bevel gearbox with transmission ratio 1:2 (z1= 54, z2= 27)
•Centrifugal pump with 3 blades (i.e. B= 3) on impeller for cavitation testing,
manufactured by G.U.N.T.
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Figure 1. The testing equipment consisting of electric motor, bevel gear
and pump
•2 KTR claw clutches, with 4 claws on flexible elements
On these tests the motor was run at 2000 rpm. This is to say the rotational speed
of the pump was 4000 rpm. The respective frequencies are n1= 33.33 Hz and
n2= 66.67 Hz. The blade pass frequency (BPF) is B·n2= 200 Hz.
4. Analysis
The measurements were conducted by using accelerometers of type IMI 621B51 and
the signals were sampled using data acquisition card NI 9233. The sampling rate
was set to 50 kHz.
4.1 Acceleration signals
The signals presented in Figures 2 and 3 show the signals with no cavitation on the
left (blue) and the signal with cavitation on the right (black). The frequency range
of the signals is from 1 Hz to 19530 Hz. It can be stated that the difference between
the situation with cavitation to the one with no cavitation is clear when considering
the measurement from the pump, but no clear difference can be seen in the signals
from the gearbox.
4

0 2 4 6 8 10
Time(s)
-300
-200
-100
0
100
200
300
x(2) (m/s 2)
0 2 4 6 8 10
Time(s)
-300
-200
-100
0
100
200
300
x(2) (m/s 2)
Figure 2. Acceleration signals from the gearbox, the signal on the left is
measured when there is no cavitation and the signal on the right is from
the situation when cavitation occurs
0 2 4 6 8 10
Time(s)
-300
-200
-100
0
100
200
300
x(2) (m/s 2)
0 2 4 6 8 10
Time(s)
-300
-200
-100
0
100
200
300
x(2) (m/s 2)
Figure 3. Acceleration signals from the pump, the signal on the left is
measured when there is no cavitation and the signal on the right is from
the situation when cavitation occurs
4.2 Enveloping by rectifying
Envelope techniques(8) are often utilised when aiming to detect periodically repeat-
ing shocks from vibration measurements. The envelope signal is quite commonly
created by first band pass filtering the signal, then rectifying it, and finally low pass
filtering the resulting signal.
Enveloping is a common method in condition monitoring. However, it is rarely ap-
plied to any other signals than acceleration or velocity. These signals might some-
times be adequate, but e.g. derivatives of fractional order where order of derivative
is any real number, can be used.
Here the envelope signals are generated utilising the ideal band bass filter (see Sec-
tion 2) and taking absolute values of the resulting signal. Here we do not utilise a
separate low pass filter.
It is a generally accepted fact that a cracked tooth in a gear causes shocks in a time
interval, equal to time of revolution of the gear. In this case it means the shocks are
expected to occur in time interval which correspond to frequency n2= 66.67 Hz.
Distinguishable peaks in envelope spectrum at that frequency and its multiples are
considered a sign of this type of fault.
In Figure 4 are the envelope spectra up to 400 Hz from the gearbox. The envelope
spectra of acceleration and snap (x(4))show that in this case, the both signals seem
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to be adequate detect the fault. However, when comparing the peaks near the
rotational frequency (66.67 Hz) of the second stage of the gearbox, the envelope
spectrum of snap Figure 4 has a more distinctive peak than the envelope spectrum
of acceleration, this is to say the snap signal seems to be more sensitive in this case.
0 100 200 300 400
Frequency (Hz)
0
0.1
0.2
0.3
0.4
x(2) (m/s 2)
Envelope spectrum
33.28 Hz
66.57 Hz
133.2 Hz
199.9 Hz
266.5 Hz
333.1 Hz
299.8 Hz
231.4 Hz
168.3 Hz
0 100 200 300 400
Frequency (Hz)
0
10
20
30
40
50
60
x(4) (Mm/s 4)
Envelope spectrum
33.38 Hz
66.57 Hz
133.2 Hz
99.85 Hz
164.8 Hz
199.9 Hz
231.4 Hz
266.5 Hz
299.7 Hz
Figure 4. Envelope spectra of acceleration and snap from the gearbox
when cavitation occurs, band pass filtering from 1000 Hz to 2000 Hz
In Figures 5 and 6 are the envelope spectra from the pump. It can be stated here
as well that utilising fractional derivatives x(2.67),x(3.33) and the snap signal instead
of the acceleration, the indication of the fault seems to be more distinguishable. In
addition Figures 5 and 6 show a peak at 54.65 Hz. This suggests that there is some
mechanical phenomenon on this frequency, but this has no obvious explanation. A
bearing fault in the pump could be one possible explanation, but seems unlikely as
this can not be seen in the envelope spectrum when pump operates normally without
cavitation. This can be seen in Figure 7 where the envelope spectra of acceleration
and snap are presented from the situation when no cavitation occurs.
0 100 200 300 400
Frequency (Hz)
0
0.1
x(2) (m/s 2)
Envelope spectrum
54.74 Hz
66.57 Hz
133.2 Hz
0 100 200 300 400
Frequency (Hz)
0
10
20
30
40
50
60
x(2.67) (m/s 2.67 )
Envelope spectrum
54.65 Hz
66.57 Hz
133.2 Hz
Figure 5. Envelope spectra of acceleration and x(2.67) from the pump when
cavitation occurs, band pass filtering from 1000 Hz to 2000 Hz
Moreover, comparing Figures 5, 6 and 7 shows that enveloping as a technique is,
as expected, rather insensitive to random vibrations. The peaks in the envelope
spectra indicate the periodically repeating shocks, are clearly distinguishable even
when the random vibration occurs.
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0 100 200 300 400
Frequency (Hz)
0
5
10
15
20
25
30
x(3.33) (km/s 3.33 )
Envelope spectrum
54.65 Hz
66.66 Hz
133.2 Hz
0 100 200 300 400
Frequency (Hz)
0
5
10
15
x(4) (Mm/s 4)
Envelope spectrum
66.66 Hz
133.2 Hz
54.65 Hz
Figure 6. Envelope spectra of x(3.33) and snap from the pump when cavi-
tation occurs, band pass filtering from 1000 Hz to 2000 Hz
0 100 200 300 400
Frequency (Hz)
0
0.1
x(2) (m/s 2)
Envelope spectrum
66.66 Hz
133.4 Hz
200.1 Hz
0 100 200 300 400
Frequency (Hz)
0
5
10
15
20
x(4) (Mm/s 4)
Envelope spectrum
66.76 Hz
200.1 Hz
Figure 7. Envelope spectra of acceleration and snap from the pump
without cavitation, band pass filtering from 1000 Hz to 2000 Hz
4.3 Enveloping by Hilbert transform
The envelope spectra in Figure 8 are obtained as discussed in Section 2.2. In this case
the envelope spectra obtained through HT the peaks indicating the fault seem to be a
bit more clear than in envelope spectra obtained through rectifying. Nonetheless, all
the other components seem to be higher as well, so it is at least debatable whether
or not the HT produces better indications of the shocks than the enveloping by
rectifying technique.
0 100 200 300 400
Frequency (Hz)
0
0.1
x(2) (m/s 2)
Envelope spectrum via HT
54.65 Hz
66.57 Hz
133.2 Hz
0 100 200 300 400
Frequency (Hz)
0
10
20
30
40
x(3.33) (m/s 3.33 )
Envelope spectrum via HT
54.65 Hz
66.66 Hz
133.2 Hz
Figure 8. Envelope spectra of acceleration and x(3.33) obtained via Hilbert
transform from the pump when cavitation occurs, band pass filtering from
1000 Hz to 2000 Hz
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5. Conclusions
Results show, that envelope spectrum of fractional derivatives and snap signal have
potential to detect shocks when random vibration occurs simultaneously. In this
case, it seems to provide better results than using the envelope of acceleration.
Because the indication of the fault is greater in the envelope spectrum of the snap
signal here, it seems probable that a less severe fault of this type, which cannot be
detected using mere acceleration, can be detected when using the snap signal.
The snap signal was the most sensitive one studied here, but some other real order
derivative might produce better results, and even higher order of derivative than
snap could be the optimal choice(10). Creating a signal of any order of derivative
is effortless with modern technology, so it would be quite simple to implement in
several condition monitoring systems.
For future work, the differences between the two enveloping techniques discussed
here might be worth some study, because in this study we have concluded that
these two techniques produce slightly different results.
6. Acknowledgements
The authors express gratitude to Otto von Guericke University for cooperation,
which made the experimental part of this study possible.
References
1. B Eftekharnejad and D Mba, ‘Seeded fault detection on helical gears with
acoustic emission’, Applied Acoustics, Vol 70, No 4, pp 547 – 555, 2009, ISSN
0003-682X, doi: 10.1016/j.apacoust.2008.07.006.
2. D Mba and R B Rao, ‘Development of Acoustic Emission Technology for Con-
dition Monitoring and Diagnosis of Rotating Machines; Bearings, Pumps, Gear-
boxes, Engines and Rotating Structures’, The Shock and Vibration Digest, Vol
38, No 1, pp 3–16, 2006.
3. T Barszcz and R B Randall, ‘Application of spectral kurtosis for detection
of a tooth crack in the planetary gear of a wind turbine’, Mechanical Sys-
tems and Signal Processing, Vol 23, No 4, pp 1352 – 1365, 2009, ISSN
0888-3270, doi: https://doi.org/10.1016/j.ymssp.2008.07.019, URL http:
//www.sciencedirect.com/science/article/pii/S0888327008002239.
8

4. F Combet and L Gelman, ‘Optimal filtering of gear signals for early damage
detection based on the spectral kurtosis’, Mechanical Systems and Signal Pro-
cessing, Vol 23, No 3, pp 652 – 668, 2009, ISSN 0888-3270, doi: https://
doi.org/10.1016/j.ymssp.2008.08.002, URL http://www.sciencedirect.com/
science/article/pii/S0888327008002033.
5. J S Bendant, ‘The Hilbert Transform and Applications to Correlation Measure-
ments’, 1985, Brüel & Kjær.
6. N Thrane, ‘The Hilbert Transform’, Technical Review No. 3, pp 3–15, Brüel &
Kjær, 1984.
7. M Angelo, ‘Vibration Monitoring of Machines’, Technical Review No. 1, Brüel
& Kjær, 1987.
8. U Klein, ‘Schwingungsdiagnostische Beuerteilung von Maschinen und Anlagen,
2. überarbeitete Auflage’, Verlag Stahleisen GmbH, Düsseldorf, 2000, ISBN
3-514-00663-6.
9. S Lahdelma and E Juuso, ‘Signal processing and feature extraction by using
real order derivatives and generalised norms. Part 1: Methodology’, The Inter-
national Journal of Condition Monitoring, Vol 1, No 2, pp 46–53, 2011, ISSN
2047-6426, doi: 10.1784/204764211798303805.
10. S Lahdelma and E Juuso, ‘Signal processing and feature extraction by using
real order derivatives and generalised norms. Part 2: Applications’, The Inter-
national Journal of Condition Monitoring, Vol 1, No 2, pp 54–66, 2011, ISSN
2047-6426, doi: 10.1784/204764211798303814.
11. S Lahdelma, ‘On the Derivative of Real Number Order and its Application to
Condition Monitoring’, Kunnossapito, Vol 11, No 4, pp 25–28, 1997.
12. D E Newland, ‘Random vibrations, spectral & wavelet analysis’, Harlow :
Longman Scientific and Technical, 3rd ed. edition, 1994, ISBN: 0-582-21584-6.
13. J Laurila and S Lahdelma, ‘Condition monitoring by means of vibration and
sound measurements’, In Proceedings of CM 2013/MFPT 2013, The Tent Inter-
national Conference on Condition Monitoring and Machinery Failure Prevention
Technologies, Vol 1, pp 227–245, Kraków, Poland, 2013.
14. S Lahdelma and E Juuso, ‘Generalised Moments and lpNorms in Vibration
Analysis’, In Proceedings of The Sixth International Conference on Condition
Monitoring and Machinery Failure Prevention Technologies, Vol 1, pp 606–618,
Dublin, Ireland, 2009.
15. S Lahdelma, J Laurila, J Strackeljan and R Hein, ‘Separating Different Vi-
bration Sources in Complex Fault Detection’, In Proceedings of The Eighth
International Conference on Condition Monitoring and Machinery Failure Pre-
vention Technologies, Vol 2, pp 938–956, Cardiff, UK, 2011.
9

16. K Karioja and S Lahdelma, ‘Detecting cavitation and simultaneously occurring
mechanical faults’, In Proceedings of CM 2014/MFPT 2014, The Eleventh Inter-
national Conference on Condition Monitoring and Machinery Failure Prevention
Technologies, pp 427 – 442, Manchester, UK, 2014.
17. M Feldman, ‘Hilbert transform in vibration analysis’, Mechanical Systems
and Signal Processing, Vol 25, No 3, pp 735 – 802, 2011, ISSN 0888-
3270, doi: https://doi.org/10.1016/j.ymssp.2010.07.018, URL http://www.
sciencedirect.com/science/article/pii/S0888327010002542.
18. N Thrane, J Wismer, H Konstantin-Hansen and S Gade, ‘Application Note:
Practical use of the "Hilbert transform"’, URL https://www.bksv.com/media/
doc/bo0437.pdf.
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