Gearbox condition estimation using cyclo-stationary properties of

Key Engineering Materials Vols. 413-414 (2009) pp 471-478 Online available since 2009/Jun/24 at www.scientific.net © (2009) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.413-414.471

Gearbox condition estimation using cyclo-stationary properties of vibration signal Radoslaw Zimroz1, a, Walter Bartelmus1, b 1

Diagnostics and Vibro-Acoustics Science Laboratory, Wroclaw University of Technology, Pl Teatralny 2, 50 051 Wroclaw, Poland a

b

[email protected], [email protected]

Keywords: cyclo-stationarity analysis, condition monitoring, vibration, gearboxes

Abstract. The paper explores the cyclo-stationary properties of vibration signals for estimation of gearbox condition. The advantage of such approach may be clearly seen especially for so called multi-faults problem, i.e. for more than one faults that occurred in the system. In complex mechanical systems like multistage gearboxes, such situation may be often seen. Although this approach becomes more and more popular, it has been noticed that there is difficult to find examples highlighting its potential, especially for real industrial situations. In order to fill partially the gap, the paper deals with the multi fault detection in complex mechanical systems like multi-stage gearboxes: fixed axis and planetary. It has been discussed that during the operation in such machines many faults may appear simultaneously and the classical method like envelope analysis is difficult to use. The paper presents the use of cyclo-stationary properties of signals to identify and characterize sources of modulation. From Spectral Correlation Density Map or more precisely Spectral Coherence Map have been observed the number of sources with different properties of modulation. It is shown that the number of harmonics is important for a kind of fault extraction and interpretation. This approach has been applied to two, three and five stage gearboxes used in mining industry. Vibration signals received in industrial environment during normal operation of objects are considered. It has been also proposed the simple diagnostic feature to estimate the changes of condition with application to a planetary stage in a 5-stage gearbox. Introduction In the literature on the subject of using cyclo-stationary properties of vibration signals for condition monitoring (McCormick 1998[9], Capdessus 2000[10], Dalpiaz 2000[11], Antoniadis 2001[12], Bouillaut 2001[13], Li 2003[14], Lin 2004 [15], Antoni 2008,2007a,b,2004,2005 [4-8] , Zhu 2005 [16], Raad 2008 [17]) one can hardly find examples when potential of cyclo-stationarity for detecting multi faults are presented. It was mentioned in [1] that especially for such situation superiority of cyclo-stationary based approach may be clearly seen. In order to fill partially the gap, the paper deals with the multi fault detection in complex mechanical systems like multi-stage gearboxes. It has been shown that during the operation in such machines many faults may appear simultaneously [2,3] and the classical method like envelope analysis is difficult to use. To detect a fault one have to select carrier frequencies and a band-pass filter width to extract signal first and next perform enveloping. If many faults exist which may be detected at different carrier frequencies one has to select signal many times for each carrier separately that is rather inconvenient. Inspiring by Randall [1] and Antoni [4-8] we use cyclo-stationary properties of signal to identify and characterize sources of modulation. From Spectral Correlation Density Map or more precisely Spectral Coherence Map may observe number of sources with different properties of modulation (wideband, narrowband) and number of harmonics that may be important for the kind of a fault extraction and interpretation. This approach has been applied to two, three and five stage gearboxes used in mining industry. Vibration signals are received in industrial environment during normal operation of objects. Mining machines are quite difficult to diagnose so often advanced combination of techniques and reasoning need to be used (Bartelmus All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 156.17.73.88-05/08/11,10:47:56)

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2009a,b [18,19]). Finally, in this paper we propose simple diagnostic feature to estimate the change of condition with application to a planetary stage in a 5-stage gearbox. Theoretical background Due the rotating parts and modulation phenomena in considered mechanical systems, vibration signals reveal some cyclical behavior. The so-called property of cyclo-stationarity characterizes non-stationary random signals exhibiting some hidden periodicity (or more general cyclicity). This periodicity may be detected in the signal statistics. For instance, a gear meshing generates first-order cyclostationary vibration signals [4], i.e. signals whose a mean is periodic. Local faults in bearings or gears are likely to produce second-order cyclo-stationarity [4]. Signal x(t) is second-order cyclostationary when instantaneous autocorrelation function Rx(t,τ)=E{x(t)x(t-τ)*} has a Fourier Rαi (τ ) series expansion Rx (t ,τ ) = ∑ Rxα i (τ ) ⋅ e j 2πα i t where the Fourier coefficients x are known as the α i ∈Α

cyclic autocorrelation functions. The Fourier transform of the cyclic autocorrelation function at a αi αi given cyclic frequency αi then defines the cyclic power spectrum (CPS): S x ( f ) = F {Rx (τ )} . If one consider set of CPS’s for all αi∈A, then we can get Spectral Correlation Density Map:

S x (α , f ) =

S α ( f )δ (α − α ) ∑ α i

x

i

i ∈Α

(1) In order to get a normalized version of SCDM Antoni [8] proposed to use so called Spectral Coherence Map (SCM) α

γx =

SC xα ( f ) SC x0 ( f + α / 2) SC x0 ( f − α / 2)

(2)

Later we will give the proposal of using Spectral Coherence Map (SCM) as to construct the measure of machine condition. The idea can be referred to publications (Bartelmus [20] and [21]) when the ordinary coherence function components give the base for machine condition evaluation.

Emulation of multi fault problem The multi faults case is considered in this section. Four damages have been emulated which occur simultaneously in a gearbox. The faults are characterized as two local faults/damages generating wideband, cyclic excitations (with the shaft frequencies (4Hz and 16,5Hz)) and two faults characterizing two gear unbalances generating sine wave signals with the same (4Hz and 16,5Hz) shaft frequencies. As a carrier frequencies the second harmonic of a gear mesh frequencies equals to 300 (second stage) and 760Hz (first stage) were used. As carriers for local damage frequencies related to structural resonance (gear resonance) were used, namely ranges 3500- 5500Hz for the high speed shaft and 1500-3500Hz for a low speed shaft. For 2 shafts (4Hz and 16,5Hz) sources of modulation consist of a gear unbalance (sine wave) and a local (wideband excitation) fault. In Fig.1 one may clearly see 4 local maximums. From the Spectral Coherence Map one may easily find that: modulating (alpha) frequencies are 4Hz and 16,5Hz (X-axis, Fig.1a), and carriers are: 300Hz (narrowband) and 3500-5500 Hz (wideband) for the low speed shaft and 760Hz and 1500-3500Hz for the high speed shaft (carriers frequencies from Y axis, Fig1a). Definitely one may notice 4 modulations with 4 different carriers from one 2D (Fig.1a) or 3D (Fig.1b) plot instead of 4 filtering and demodulation procedures (please note that we really need to apply demodulation procedure 4 times because carriers are different for each modulation source).

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Fig.1. Spectral Coherence Map for simulated 4 modulations a) 2D plot, b) 3D plot: modulating (alpha) frequencies are 4Hz and 16,5Hz, and carriers are: 300Hz (narrowband) and 3500-5500 Hz (wideband) for the low speed shaft and 760Hz and 1500-3500Hz for the high speed shaft

Application to real vibration signals: Multi faults in 2 stage gearbox In this section a two stage gearbox used in belt conveyor systems will be considered. For more information regarding this object see [22-24] where behaviour of this gearbox has been studied. In Fig.2 a time frequency map (STFT) is presented. One may notice that some modulation is visible but it is difficult to detect precisely sources of modulation from STFT map.

Fig.2. STFT map for vibration signal from 2 stage gearbox

Fig.3. Spectral Coherence Map for 2 stage gearbox a) 2D plot, b) 3D plot By analysis of SCM plane (Fig.3) it is clear to identify sources and frequencies/bands that are carrying information. It is also possible to recognize a fault type (4Hz is narrowband source, 16.5 Hz is wideband) that leads to conclusion that two faults exist in the system, however one is related to a

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fault which is connected with generation what may be called distributed faults or unbalance as it was presented in emulation another one is local caused by cracking, tipping.

Multi faults in 3 stage gearboxes

Fig. 4. STFT map for vibration signal from 3 stage gearbox

Fig.5. Spectral Coherence Map for 3 stage gearbox a) 2D plot, b) 3D plot Now we repeat a diagnostic scheme from a previous section for more complicated case. On Fig 4 a time frequency map (STFT) of vibration signal is presented. One may notice very weak, wideband modulation. It is impossible to detect other sources of modulation from STFT map. First we analyze SCM map for a wide α-frequency range – up to 60Hz. One may notice again two phenomena that exist in a signal. First modulation related to α=16Hz is visible (3 harmonics around 1000Hz may be find), another phenomenon related to small α-frequency and many harmonics at frequency range 2000-3000Hz can be find. The detailed analysis Fig.6 in limited α-frequency range (0-20Hz) gives a chance to find other modulation phenomena. Apart from the modulation related to an input shaft (16Hz) and output shaft (1,7Hz) with many harmonics) one may detect the modulation caused by middle shaft (4,8Hz and harmonics) and a narrowband source of modulation related to second harmonic (and its multiple) of an output shaft (3,4Hz, 6,8Hz). We can conclude that many sources of modulations corresponding to shafts (that mean faults in the system) may be identified from one map. Again it is highlighting the potential of this approach in multi-faults case.

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Fig.6. Detailed Spectral Correlation Density Map for 3 stage gearbox with characteristic α frequencies and modulation ranges

Application of sum of SCM as a measure of planetary gearbox condition - a proposal of a new diagnostic feature. As it was mentioned, if signal is cyclo-stationary (AM modulated) on SCM plane some components will appear at some frequencies. Depends on a single or multi-fault problem, source(s) and carrier(s) frequencies/bands may be identified from SCM plane. Amplitude of component should give information about the intensity of a condition change and maybe use as a measure of cyclostationarity. To get general information one may use simple sum of amplitudes of components from SCM plane f α max

DF =

f max

∑ ∑ SCM ( fα , f )

f = fα min f = f min

(3) If multi-faults appear, one may sum amplitudes for particular frequencies “α” and also for specified frequency components/bands. In single fault results should be the same for both approaches. Before application of a new feature to real signals it is necessary to check its properties, namely properties of the feature-fault relation (if it is sensitive to fault development, monotonic, linear etc). Based on some simulations we can say that new feature is good, see Fig.7

Fig.7. Example of SCM for small and large modulation depth: a) m=0.1 and b) m=0.7

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SCM=f(m)

30 25

SCD

20

y = 18,903x + 12,735 R2 = 0,9973

15

Serie1 Liniowy (Serie1)

10 5 0 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

m

o Fig.8. Relation ΣSCM=f(m) As it was shown in Fig.8 the relation between a modulation depth (damage development stage) and the new feature is linear. Apart from identification of modulation sources there is a need to estimate the condition of a machine. In this section we will apply the proposed feature to a planetary gearbox used in a bucket wheel excavator [18]. In Fig.9 is seen a driving system for which the first stage is a planetary gearbox with a stationary rim. The planetary gearboxes considered consist of: a sun gear z1 , planetary gear z2 and rim gear z3. There are three cases of planetary gear element rotation. In the considered case where the sun gear is rotating, the planet gear makes planetary movement and the rim-gear is stationary as it is for the case given in Fig.9. Gears z4 – z9 give three stage cylindrical gearbox.

Z1

Z3

Z2 Z5 Z7

Z4

Z9

Z6 Z8

Fig.9. Part of driving system for a bucket wheel with planetary gearbox (gears: z1 – sun, z2 – planet, z3 – stationary rim, z4 – z9 three stage cylindrical gearbox We consider 2 measurement sessions at T1 and T2 for 2 gearboxes G1 and G3. The gearbox G1 at T1 is in bad condition. Between T1 and T2 G1 has been replaced with new one (Y1). The gearbox G3 was in the acceptable condition at T1. By comparison of DFAVE calculated as an average of DF estimated according to expression 3 for a few measurements, one may notice the significant decreasing of DF symptom due to gearbox replacement (Fig.10) . The value of the symptom for a bad condition gearbox is over twice bigger that for the new gearbox. DFAVE for gearbox G3 has increased due to condition change/deterioration of the gearbox G3.

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copmarison of results for 2 measurement sessions 80 70 sum of SCD

60

Replacement

50

Change of condition T1

40

T2

30 20

NEW Gear

10 0 G1

G3 gearbox

Fig.10. Averaged value of sum of SCM for gearboxes G1/Y1and G3

Summary The paper shows possibilities of using cyclo-stationarity for a gearbox condition evaluation. The possibilities and signal interpretations are based on signal emulation of multi faults problem. The signal evaluation is based on gearbox signal properties gained by the authors and presented in many publications on the gearbox condition monitoring. The new diagnostic measure of gearbox condition evaluation based on cyclo-stationarity properties of the signal is proposed.

Acknowledgements We would like to thanks Jerome Antoni, UTC Compiegne for his comments and support during our research related to cyclo-stationarity in gearboxes vibration analysis

References [1] R.B. Randall, J. Antoni and S. Chobsaard: Mechanical Systems and System Processing, Vol.15(2001), p.945 [2] W. Bartelmus, R. Boustany, R. Zimroz, J. Antoni: Application of Spectral Correlation Techniques on Mining Machines Signals: Identification of Faulty Components , 2nd World Congress on Engineering Asset Management and the 4th Int. Conf. on Condition Monitoring, 11-14 June 2007, Harrogate, UK. [3] R. Boustany, W. Bartelmus, J. Antoni, R. Zimroz: Application of Spectral Correlation Techniques on Mining Machines Signals: Extraction of Fault Signatures, 2nd World Congress on Engineering Asset Management and the 4th Int. Conf. on Condition Monitoring, 11-14 June 2007, Harrogate, UK. [4] J. Antoni, F. Bonnardot, A. Raad and M.El Badaoui: Mechanical Systems and Signal Processing, Vol.18 (2004), p.1285 [5] J. Antoni, R. B. Randall: Journal of Sound and Vibration, Vol.281(2005), p.463 [6] J. Antoni: Mechanical Systems and Signal Processing, Vol.21(2007), p.597 [7] J. Antoni: Journal of Sound and Vibration, Vol.304(2007), p.497 [8] J. Antoni: Mechanical Systems and Signal (2008), doi:10.1016/j.ymssp.2008.10.010

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[9] A.C. McCormick, A.C. Nandi: Mechanical Systems and Signal Processing, Vol.12(1998), p. 225 [10] C. Capdessus, M. Sidahmed and J.L. Lacoume: Mechanical Systems and Signal Processing, Vol.14(2000), p.371 [11] G. Dalpiaz, A. Rivola and R. Rubini: Mechanical Systems and Signal Processing, Vol. 14 (2000), p.387 [12] I. Antoniadis, G. Glossiotis: Journal of Sound and Vibration, Vol.248 (2001), p.829 [13] L. Bouillaut, M. Sidahmed: Mechanical Systems and Signal Processing, Vol.15 (2001), p.923 [14] L. Li, L. Qu: Journal of Sound and Vibration, Vol. 267 (2003), p.253. [15] J. Lin, M. Zuo: Journal of Vibration and Acoustics, Vol.126 (2004), p. 449 [16] Z. Zhu, F Kong: Mechanical Systems and Signal Processing, Vol. 19(2005), p.467 [17] A. Raad, J. Antoni and M. Sidahmed: Mechanical Systems and Signal Processing, Vol. 22(2008), p.574 [18] Bartelmus W. Zimroz R.: Mechanical Systems and Signal Processing, Vol. 23 (2009), p. 246 [19] Bartelmus W. Zimroz R.: Mechanical Systems and Signal Processing, (2009), in press [20] Bartelmus W: Application of some statistical estimators of vibration signal as the criterions of assessment of mesh state (in Polish) Zeszyty Naukowe Politechnika Slaska No.616 Gliwice 1979 pp 1-123 [21] Bartelmus W. Zimroz R.: Use of artificial intelligence for gear condition classification on the base of coherence parameters AI-METH Series, Gliwice, 2005 pp 17-20. [22] Bartelmus W.: Mechanical Systems and Signal Processing, Vol.15 (2001), p. 855 [23] Bartelmus W.: Mathematical Modelling of Gearbox Vibration for Fault Diagnosis, International Journal of COMADEM, Vol.3, no.4, 2000 [24] Bartelmus W.: Root cause analysis of vibration signals for gearbox condition monitoring. BINDT Inside Journal, Vol. 50, No. 4, April 2008

Damage Assessment of Structures VIII doi:10.4028/www.scientific.net/KEM.413-414 Gearbox Condition Estimation Using Cyclo-Stationary Properties of Vibration Signal doi:10.4028/www.scientific.net/KEM.413-414.471 References [1] R.B. Randall, J. Antoni and S. Chobsaard: Mechanical Systems and System Processing, Vol.15(2001), p.945 doi:10.1006/mssp.2001.1415 [2] W. Bartelmus, R. Boustany, R. Zimroz, J. Antoni: Application of Spectral Correlation Techniques on Mining Machines Signals: Identification of Faulty Components, 2nd World Congress on Engineering Asset Management and the 4th Int. Conf. on Condition Monitoring, 11-14 June 2007, Harrogate, UK. [3] R. Boustany, W. Bartelmus, J. Antoni, R. Zimroz: Application of Spectral Correlation Techniques on Mining Machines Signals: Extraction of Fault Signatures, 2nd World Congress on Engineering Asset Management and the 4th Int. Conf. on Condition Monitoring, 11-14 June 2007, Harrogate, UK. [4] J. Antoni, F. Bonnardot, A. Raad and M.El Badaoui: Mechanical Systems and Signal Processing, Vol.18 (2004), p.1285 doi:10.1016/S0888-3270(03)00088-8 [5] J. Antoni, R. B. Randall: Journal of Sound and Vibration, Vol.281(2005), p.463 doi:10.1016/j.jsv.2004.04.007 [6] J. Antoni: Mechanical Systems and Signal Processing, Vol.21(2007), p.597 doi:10.1016/j.ymssp.2006.08.007 [7] J. Antoni: Journal of Sound and Vibration, Vol.304(2007), p.497 doi:10.1016/j.jsv.2007.02.029 [8] J. Antoni: Mechanical Systems and Signal (2008), doi:10.1016/j.ymssp.2008.10.010 [9] A.C. McCormick, A.C. Nandi: Mechanical Systems and Signal Processing, Vol.12(1998), p. 225 doi:10.1006/mssp.1997.0148 [10] C. Capdessus, M. Sidahmed and J.L. Lacoume: Mechanical Systems and Signal Processing, Vol.14(2000), p.371 doi:10.1006/mssp.1999.1260 [11] G. Dalpiaz, A. Rivola and R. Rubini: Mechanical Systems and Signal Processing, Vol. 14 (2000), p.387

doi:10.1006/mssp.1999.1294 [12] I. Antoniadis, G. Glossiotis: Journal of Sound and Vibration, Vol.248 (2001), p.829 doi:10.1006/jsvi.2001.3815 [13] L. Bouillaut, M. Sidahmed: Mechanical Systems and Signal Processing, Vol.15 (2001), p.923 doi:10.1006/mssp.2001.1412 [14] L. Li, L. Qu: Journal of Sound and Vibration, Vol. 267 (2003), p.253. doi:10.1016/S0022-460X(02)01412-8 [15] J. Lin, M. Zuo: Journal of Vibration and Acoustics, Vol.126 (2004), p. 449 doi:10.1115/1.1760565 [16] Z. Zhu, F Kong: Mechanical Systems and Signal Processing, Vol. 19(2005), p.467 doi:10.1016/j.ymssp.2004.02.007 [17] A. Raad, J. Antoni and M. Sidahmed: Mechanical Systems and Signal Processing, Vol. 22(2008), p.574 doi:10.1016/j.ymssp.2007.09.011 [18] Bartelmus W. Zimroz R.: Mechanical Systems and Signal Processing, Vol. 23 (2009), p. 246 doi:10.1016/j.ymssp.2008.03.016 [19] Bartelmus W. Zimroz R.: Mechanical Systems and Signal Processing, (2009), in press [20] Bartelmus W: Application of some statistical estimators of vibration signal as the criterions of assessment of mesh state (in Polish) Zeszyty Naukowe Politechnika Slaska No.616 Gliwice 1979 pp 1-123 [21] Bartelmus W. Zimroz R.: Use of artificial intelligence for gear condition classification on the base of coherence parameters AI-METH Series, Gliwice, 2005 pp 17-20. [22] Bartelmus W.: Mechanical Systems and Signal Processing, Vol.15 (2001), p. 855 doi:10.1006/mssp.2001.1411 [23] Bartelmus W.: Mathematical Modelling of Gearbox Vibration for Fault Diagnosis, International Journal of COMADEM, Vol.3, no.4, 2000 [24] Bartelmus W.: Root cause analysis of vibration signals for gearbox condition monitoring. BINDT Inside Journal, Vol. 50, No. 4, April 2008