GEAR FAULT MONITORING: COMPARISON OF VIBRATION ANALYSIS TECHNIQUES

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1 GEAR FAULT MONITORING: COMPARISON OF VIBRATION ANALYSIS TECHNIQUES G. DALPIAZ, A. RIVOLA and R. RUBINI Dipartimento di Ingegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia - University of Bologna Viale Risorgimento, - I-46 Bologna - Italy giorgio.dalpiaz@mail.ing.unibo.it ABSTRACT This paper deals with gear condition monitoring based on vibration analysis techniques. The detection and diagnostic capability of some of the most effective techniques are discussed and compared on the basis of experimental results, concerning a gear pair affected by a fatigue crack. In particular, the results of new approaches based on time-frequency and cyclostationarity analysis are compared against those obtained by means of the well accepted cepstrum analysis and amplitude and phase demodulation of meshing harmonics. Moreover, the sensitivity to fault severity is assessed by considering two different depths of the crack. The effect of choosing different transducer locations and different processing options are also shown. In the case of the experimental results considered in this paper, the power cepstrum is practically insensitive to the crack evolution. Conversely, the Spectral Correlation Density function is able to monitor the fault development and does not seem to be significantly influenced by the transducer position. The demodulation techniques are able to localise the damaged tooth; however, their sensitivity is strongly dependent on the proper choice of the filtering band and is affected by the transducer location. The Wavelet transform seems to be a good tool for crack detection; it is particularly effective if the residual part of the time synchronous averaged signal is processed. - INTRODUCTION Most modern techniques for gear diagnostics are based on the analysis of vibration signals picked up from the gearbox casing. The common target is to detect the presence and the type of fault at an early stage of development and to monitor its evolution, in order to estimate the machine s residual life and choose an adequate plan of maintenance. It is well known that the most important components in gear vibration spectra are the tooth meshing frequency and its harmonics, together with sidebands due to modulation phenomena. The increment in the number and amplitude of such sidebands may indicate a fault condition. Moreover, the spacing of the sidebands is related to their source []. The simple spectral analysis is generally unable to detect gear failures at an early stage; for this reason, many researchers have proposed the application of other vibration analysis techniques for the early detection of fault symptoms. The aim of this paper is to assess and compare the detection and diagnostic capability of some of the most effective techniques, on the basis of experimental results. Cepstral analysis has been widely applied to gear monitoring. The cepstrum is well suited for the detection of sidebands in vibration spectra and for the estimation of their evolution during gear life. In addition, since the cepstrum estimates the average sideband spacing over a wide frequency range, it allows very accurate measurement of the sideband periodicity. It is therefore applicable to both detection and diagnosis of gear faults [-]. The amplitude and phase demodulation of one of the tooth meshing harmonics is a well known gear monitoring technique [4, ]. This technique requires the time synchronous averaging of the vibration signal in order to remove any periodic events not exactly synchronous with the gear of interest and to reduce the effects of noise and vibration sources other than gear pairs [, 4]. The averaged signal is then bandpass filtered around one of the larger meshing harmonics and the amplitude and phase modulation are obtained by means of a procedure based on the Hilbert transform. The cyclostationary process theory has been recently applied to gear monitoring. In particular, the Spectral Correlation Density (SCD) function of gear vibration indicates the correlations between meshing harmonics 6

2 and their sidebands [6, 7], that are just the spectral characteristics primarily affected by gear faults. Thus the SCD function permits gear fault detection and identification of the damaged element. Local faults in gears (e.g. crack in a gear tooth) produce impacts []. As a result of this excitation, transient modifications of vibration signals may be observed. Therefore, the vibration signal can be considered as non-stationary. However, most of the widely-used signal processing techniques, are based on the assumption of stationarity and globally characterise signals. Thus they are not fully suitable for detecting short-duration dynamic phenomena; as a matter of fact, the time-localisation of transient events is impossible. On the other hand, the application of time-frequency distribution techniques - such as Wavelet Transform (WT) - is highly suitable [8-]. By means of this time-variant method it is possible to detect and localise the presence of cracks in gears. In this paper, the above-mentioned analysis techniques are applied to experimental vibration data, concerning a gear pair affected by a fatigue crack at the root of one of the teeth. The capability of fault detection and diagnosis are discussed and compared; in particular, the sensitivity to fault severity is assessed by considering two different depths of the crack. The effect of choosing different transducer locations and different processing options are also shown. - TESTS Tests were performed on a power circulating gear testing machine composed of two identical single-stage gear units mounted back to back, with a locked-in torque []. Each gear unit contains a carburised spur gear pair of module mm; the pinion and the wheel have respectively 8 and teeth. Further data about the gears can be found in []. Tests were carried out after the introduction of a crack at the root of one tooth of the wheel mounted in one of the gear units. The tooth was precracked before mounting the gear on the testing machine, by using an appropriate device in order to apply a fatigue load to the tooth in a similar way to when the tooth is loaded during meshing. The crack affected the whole tooth flank, extending between the two wheel faces. In order to investigate the influence on the vibration signal of both the transducer location and the development stage of the crack, the signals were picked up on a gearbox casing in two different positions and two crack dimensions were examined. In particular, two vibration signals were simultaneously measured from the casing of the damaged gearbox unit by means of two Brüel & Kjær 469 piezoelectric accelerometers: one was mounted with the sensitivity axis parallel to the shaft axis ("axial" accelerometer); the other one was placed on the wheel bearing casing in a radial direction ("radial" accelerometer) []. The signals were integrated in Brüel & Kjær 6 charge preamplifiers to obtain velocity signals. In addition, a one-per-wheel revolution tachometer signal was taken using an inductive proximity probe. Tests were subsequently performed with two crack lengths, corresponding to about % and 4% of whole fracture surface after breakage. The results presented in this paper are relative to a nominal pinion torque of 8 Nm and nominal pinion speed of rpm (6.67 Hz); thus the meshing frequency is Hz. It is worth noting that the actual operating conditions were slightly different in the tests carried out with the two crack dimensions. The signals were recorded on magnetic tape and converted to a digital time-series using a DIFA Scadas unit having a -bit ADC with variable sampling frequency, which was driven by the LMS Cada-X software. The analysis was limited to the frequency range - Hz which includes the most meaningful meshing harmonics. The digital signals were processed and analysed by means of the LMS Cada-X and MATLAB softwares. Whenever it was required, the signals were pre-processed by means of a synchronising technique in order to compensate slight - but quite important - variations of the machine angular speed []. For this purpose, a particular synchronising methodology was used [, 4]. The synchronised signal records have 4 points per wheel revolution. Some techniques were also applied to the signals, after a process of time synchronous averaging over 8 revolutions of the damaged gear (i.e. the wheel); as mentioned, this averaging reduces the effects of vibration sources other than the wheel. - EXPERIMENTAL RESULTS AND DISCUSSION. - Cepstrum analysis As mentioned in the introduction, the gear vibration spectra commonly show families of sidebands of the meshing harmonics. For gearboxes in good condition the sideband level generally remains constant with time. Changes in the number and strength of the sidebands normally indicate a deterioration condition. Such sidebands typically arise from the modulation of the gear vibration by the rotational frequency, as in the case of a crack in one of the teeth (there is a fault once per revolution). Therefore, a family of sidebands has the same spacing, that is the fundamental modulating frequency. Consequently, the sideband spacing contains diagnostic information, since it is related to the modulation source []. However, it can be difficult to distinguish and evaluate the sideband spacing by means of spectral analysis, due to its poor resolution. In order to overcome this problem, cepstral analysis can be employed. Various forms of cepstrum exist, but all can be considered as a spectrum of a logarithmic (amplitude) spectrum []. The cepstrum is 64

3 therefore useful in interpreting the spectrum structure and in detecting the spectrum periodicity [-]. The quefrency of the cepstrum peaks represents the modulation period and its reciprocal the modulating frequency. The power cepstrum of the experimental signals was computed from a 48-line power spectrum in the range -444 Hz, obtained with spectral averages. The results reported in Fig. (where the amplitude of the cepstrum is plotted), are relative to the case of the radial vibrations. Peaks at about.6 s (6.67 Hz) and about. s, are present; they correspond to the first rahmonic of the pinion and the wheel, respectively. Although the signals are relative to different crack dimensions, their cepstra are very similar (see Fig. ). Moreover, the cepstral component corresponding to the first rahmonic of the damaged gear (. s), is practically insensitive to the crack dimension. In other words, for this case, the cepstrum analysis gives unclear monitoring information and is not able to detect the presence of the damage. The cepstrum approach was also employed for the axial vibration signals, but the results were not comforting and have not been reported for the sake of brevity. It was suspected that variations of the machine angular speed were the cause of such results. As a matter of fact, such variations could alter the sidebands during the FFT computation. Therefore, the signals were pre-processed by means of a synchronising technique and the cepstral analysis was repeated, employing a 48-line power spectrum in the range Hz (7 averages were used). [db] [db] Quefrency [s] Quefrency [s] Fig. - Cepstrum of the radial vibration: small and large crack. [db] [db] Quefrency () Quefrency () Fig. - Cepstrum of the radial vibration (synchronised signal): small and large crack. Fig. shows the results for the radial vibration. In comparison to the non-synchronised signals, the first rahmonic of the wheel is enhanced, whilst that of the pinion almost disappears. Although the first rahmonic of the damaged gear now dominates the cepstrum, the fault monitoring cannot still be performed, since the cepstrum is insensitive to the crack increment. The cause of the cepstrum inefficiency could be due to the fact that speed and torque were slightly different in the tests carried out with the two crack dimensions. As a matter of fact, it was found that the different operating conditions altered the vibration spectra; further investigation is thus needed.. - Demodulation analysis The effects of local faults, such as cracked teeth, on the amplitude and phase modulation of the meshing harmonics are widely discussed in []. In [4], the author proposed a signal processing methodology based on the demodulation of the amplitude and phase of one of the meshing harmonics of the signal, as a gear monitoring technique. That method was further developed in [] in order to increase its sensitivity. The time synchronous average (TSA) has to be performed before applying the demodulation approach. As a matter of fact, the TSA process enhances the signal components due to the gear of interest and strongly reduces the effects of all other sources, and the noise. In addition, as the TSA is exactly 6

4 periodic, several operations such as filtering and signal component elimination can be easily carried out in the frequency domain. As a drawback, it is necessary to repeat the analysis for each gear, for a complete machine diagnosis. The demodulation was applied to the experimental vibration signals by filtering about the third meshing harmonic (at 6 wheel shaft orders), since this was the most important and showed the strongest modulation sidebands. In particular, upper sidebands of the third meshing were extracted for the analysis (c) 6. (d) 6 Fig. - modulation (6-9 orders); axial vibration: small and large crack; radial vibration: small (c) and large (d) crack (c) (d) - 6 Fig. 4 - Phase modulation derivative (6-9 orders); axial vibration: small and large crack; radial vibration: small (c) and large (d) crack. 66

5 Fig. presents the results of the amplitude demodulation process applied on both the axial and radial velocity signals after the time synchronous averaging over 8 revolutions of the wheel. Both the crack dimensions are examined. The amplitude modulation is able to localise the damaged tooth and shows a good sensitivity to the crack severity; however, in the case of the radial signal, it is not possible to the detect the small crack. The crack also affects the phase modulation which is characterised by a lag due to the deflection of the affected tooth as it meshes with the other gear. In particular, as established in other tests performed on the same gear system, the earlier phase modulation symptom is its sudden variation; thus, the phase modulation derivative (PMD) was introduced to make crack detection easier []. For the same signals as in Fig., the PMD is reported in Fig. 4. The peaks in the PMD clearly reveal the presence and position of the crack. The PMD approach is more sensitive than the amplitude one; however, the small crack in the case of radial vibration is hardly detected by this tool either. Therefore one can say that the transducer location affects the results of the demodulation analysis. In order to give an indication about the damage severity, a statistical parameter of the PMD can be employed; here the kurtosis is used. For the axial signal, the ratio of the kurtosis value in the case of large crack to the value for the small crack is.7; this ratio is.8 in the case of the radial vibration. The results of the demodulation approach are strongly dependent on the filtering band, as discussed in []. As an example, Fig. shows the phase modulation derivative of the radial signal obtained by bandpass filtering the TSA in the range to 8 wheel shaft orders (right sidebands of the first meshing harmonic). Thus, this technique gives good results on the condition that a proper frequency band is chosen Fig. - Phase modulation derivative (-8 orders); radial vibration: small and large crack.. - Cyclostationary analysis Since the spectral structure of a gear vibration signal is mainly characterised by the interaction between the meshing harmonics and their sidebands, the cyclostationary process theory has been recently applied to gear monitoring [6,7]. As a matter of fact, by means of the cyclostationary approach it is possible to investigate the correlation degree between different frequency components of the spectrum or, in other words, to establish whether they are related each other or not []. In particular, the SCD function of a signal (which can be defined as the Fourier transform of the Fourier coefficient of the signal autocorrelation function) can be employed to analyse the correlation between a meshing harmonic and one of its modulating sidebands. Such a sideband is associated with a specific gear element, being a multiple of its revolution frequency. Therefore, the analysis of the gear spectrum evolution by means of the SCD function allows both the detection and diagnosis of gear fault. The SCD function of the experimental signals was computed and the relationship between the third meshing harmonic and the pertinent first upper modulating sideband, relative to both the gear elements, were investigated. The synchronised signal records were used; they included 8 wheel revolutions in order to ensure an adequate cyclic frequency resolution (. Hz). The FFT length employed during the SCD computation was points (i.e. resolution frequency is 6.97 Hz). The results are shown in Fig. 6, which plots the SCD magnitude over the bifrequency plane (f, α). The correlation between the third meshing harmonic (f m =4 Hz) and the revolution frequency of the pinion (α P =6.67 Hz) is located at the bifrequency (f m +α P /, α P ); the SCD value at the point (f m +α W /, α W ) is a measure of the correlation between the meshing harmonic and the wheel revolution frequency (α W =8.48 Hz). Further details about the interpretation of the SCD plot can be found in [7]. 67

6 It is noteworthy that peaks are present also for the gear in sound conditions (the pinion); thus, the SCD function can only be employed to monitor the fault evolution. However, by observing the amplitude modification of the correlation peaks, one can conclude that the damage affects the wheel. As a matter of fact, the peak that corresponds to the wheel shows a strong increment with the crack severity, whilst the pinion correlation peak only exhibits slight changes, that might be due to the mentioned small differences in the operating conditions. In addition, the transducer position does not seem to significantly influence the results. (c) (d) Fig. 6 - Spectral Correlation Density between the third meshing harmonic and its upper modulating sideband; axial vibration: small and large crack; radial vibration: small (c) and large (d) crack..4 - Wavelet transform The presence of a crack in one tooth introduces short-duration changes in the vibration signal. On the contrary, more distributed faults (e.g. geometrical imperfections in the gear train and uniform wear), introduce "slow" modifications of the signals over the revolution period. For advanced local faults, time domain techniques may be sufficient to detect the damage, but the early detection of defects requires more sophisticated signal process methods. The non-stationary nature of the signal suggests the use of time-frequency techniques, which make it possible to look at the time evolution of the signal s frequency content. Many such time-variant methodologies exist; in [8-], the authors successfully applied the WT technique to the detection of cracked teeth in gears. In order to examine the sensitivity of the method, the WT was applied to the radial vibration in the case of the small crack dimension; as a matter of fact, this signal seems to be scarcely affected by the fault, as has been established by means of the demodulation technique [see Fig. (c) and 4(c)]. Fig. 7 reports the WT amplitude map corresponding to one wheel revolution; the analysed frequency range is - Hz, which includes the most important meshing harmonics. The transient event should cause an increment of a WT amplitude of short duration over a wide frequency range. Unfortunately, that does not happen, making the result useless. The signal was probably affected by noise or other vibration sources; its TSA was therefore subsequently considered. The result does in fact improve, as is shown in Fig. 8, where a localised increment of the WT amplitude is visible at 68

7 about degrees of the wheel rotation. The presence and the angular position of the crack can be detected only by carefully inspecting the WT cross-sections [see Fig. 8(c)]; however, the crack effects are evident only in some frequency ranges [], thus a fault detection procedure would be strongly dependent on the choice of the cross-section frequency. In order to overcome this limit, the residual approach [6-8] was applied to the radial TSA signal before computing the WT. The residual signal can be obtained by removing the harmonic part of the original signal, and it represents the amount of the original signal departure from its ideal counterpart. Fig. 7 - Wavelet Transform map of the raw radial vibration for the small crack dimension (one wheel revolution).. WT cross-section at Hz (c) sum of the WT cross-sections (d). 6 6 Fig. 8 - WT of the TSA radial vibration for the small crack dimension: original signal (left); residual signal (right). and WT map; (c) WT cross-section at Hz; (d) sum of the WT cross-sections (- Hz). 69

8 Fig. 8 shows that the residual signal is a more sensitive diagnostic tool; in fact, by observing its WT map, it is possible to clearly distinguish the transient effects introduced by the cracked tooth; conversely, in the original signal they were obscured by the harmonic components. Moreover, such a procedure makes it possible to localise the damage whatever the frequency section of the WT map. In particular, the sum of the WT cross-sections [Fig. 8(d)] can be used in order to make a feature extraction procedure easy. Fig. 9 shows the results obtained by computing the WT of the residual of TSA signals: both the transducer locations are examined (axial and radial vibration) for the two crack dimensions. As a measure of the crack severity, the fourth statistical central moment was evaluated for the four cases of Fig. 9: in the case of the axial vibration, the ratio of this statistical indicator value in the case of the large crack to the value for the small crack is.7; the ratio is 6.7 in the case of the radial vibration (c) (d) 6 6 Fig. 9 - Sum of the WT cross-sections (- Hz) of the residual of TSA signal; axial vibration: small and large crack; radial vibration: small (c) and large (d) crack. 4 - CONCLUSIONS This paper compares the effectiveness of some vibration analysis techniques for detection and diagnostics of cracks in gear teeth, on the basis of experimental results. In particular, the capability of new approaches based on time-frequency and cyclostationarity analysis are compared against those obtained by means of the well accepted cepstrum analysis and amplitude and phase demodulation of meshing harmonics; in addition, different processing options are considered in some cases. For each technique, the sensitivity to crack depth and the influence of the transducer position are assessed. Applying cepstrum analysis and Spectrum Correlation Density, it is possible to monitor each gear element in a gearbox at the same time, by observing amplitude modifications of the correspondent quefrency and correlation peak. On the other hand, these techniques give only information about spectrum evolution; in other words, they are unable to detect faults from the analysis of only one vibration measurement, as peaks are present also for gears in sound conditions; thus the evolution during machine operation must be considered. In the case of the experimental results considered in this paper, the power cepstrum of both the raw vibration signals and the synchronised signals was considered; in both cases this technique is practically insensitive to the crack dimension, i.e. to the fault evolution. On the other hand, the SCD of the synchronised signals is very sensitive to the fault evolution, as the peak corresponding to the wheel shows a strong increment with the crack dimension, whilst the pinion correlation peak only exhibits slight changes. In addition, the SCD does not seem to be significantly influenced by the transducer position. 6

9 For the evaluation of the amplitude and phase modulations of one of the tooth meshing harmonics, the time synchronous average - relative to the monitored gear - must be previously computed; thus, it is necessary to repeat the analysis for each gear, in order to obtain a complete machine diagnosis. According to the presented results, both amplitude and phase modulation are sensitive to the presence and dimension of the crack. In particular, the phase modulation derivative exhibits high sensitivity, as a sudden phase variation is the earlier modulation symptom. Moreover, this technique is able to localise the damaged tooth. However some drawbacks were found: the effectiveness and the sensitivity of this technique strongly depend on the proper choice of the processed frequency band and, in addition, the sensitivity is influenced by the transducer position. For these reasons, the practical application of this technique may be sometimes ineffective. As faults localised in one or a few teeth produce transient dynamic effects, the application of the Wavelet Transform appears well suited. This technique makes it possible to localise the damaged tooth. The presented results show that the WT of the raw signals is practically insensitive to cracks, while the sensitivity of the WT of TSA signals is quite satisfactory. Thus, for a complete diagnosis, it is necessary to repeat the analysis with reference to each gear in the machine. Moreover, as the crack effects are evident only in some frequency range, a fault detection procedure would be strongly dependent on the choice of a proper cross-section in the WT map. This limit is overcome if the residual of the TSA signal is processed by the WT. The expounded approach proves itself indispensable in monitoring small crack effects - which are often covered by harmonic components of the signal - and makes it possible to localise the damage over a wide range of frequency in the WT map. Consequently, the sum of the WT cross-section is a useful tool for crack detection: as a matter of fact, it is very sensitive to both the presence and the dimension of the crack, even if its sensitivity fairly changes for different transducer positions. A successful diagnostic tool should undoubtedly integrate several of the currently available techniques; in this context, the SCD approach and the WT of the residual part of the TSA signal appear to be very suitable techniques for crack detection in gear. ACKNOWLEDGEMENTS This work was partially supported by a grant from the CNR - Italian National Research Council. REFERENCES. RANDALL R.B. 98 Journal of Mechanical Design 4, A new method of modelling gear faults.. RANDALL R.B., HEE J. 98 Brüel & Kjær Technical Review, No., -4. Cepstrum analysis.. SIDAHMED M. 99 Bulletin S.F.M. Revue Française de Mécanique 4, 4-4. Détection précoce de défauts dans les engrenages par analyse vibratoire. 4. MCFADDEN P.D. 986 ASME J.of Vibration, Acoustics, Stress, and Reliability in Design 8, 6-7. Detecting fatigue cracks in gear by amplitude and phase demodulation of the meshing vibration.. DALPIAZ G. 99 Österreichische Ingenieur - und Architekten - Zeitschrift (ÖIAZ), -7. Early detection of fatigue cracks in gears by vibration analysis techniques. 6. CAPDESSUS C., SIDAHMED M., LACOUME J.L. 99 Proceedings of the nd International Symposium on Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques, Senlis, France, 9-4. Apport de la theorie des processus cyclostationnaires a l'analyse et au diagnostic des engrenages. 7. RUBINI R., SIDAHMED M. 997 Proceedings of the Symposium on Fault Detection, Supervision, and Safety for Technical Processes, Hull, UK. Diagnostics of gear systems using the Spectral Correlation Density of the vibration signal. 8. STASZEWSKI W.J., TOMLINSON G.R. 994 Mechanical Systems and Signal Processing 8(), Application of the Wavelet transform to fault detection in a spur gear. 9. MCFADDEN P.D. 994 Proceedings of an Internation Conference on Condition Monitoring, Swansea, UK, 7-8. Application of the wavelet transform to early detection of gear failure by vibration analysis.. DALPIAZ G., RIVOLA A., RUBINI R. 996 Proceedings of the Congress of Technical Diagnostics, Gdansk, Poland, 8-9. Dynamic modelling of gear system for condition monitoring and diagnostics.. DALPIAZ G., U. MENEGHETTI 99 Proceedings of rd International Conference on Condition Monitoring, Windsor, UK. Ed. McEwan J.R., London & N.Y.: Elsevier Applied Science, 7-8. Detection and modelling of fatigue cracks in gears.. MCFADDEN P.D. 989 Mechanical Systems and Signal Processing (), Interpolation technique for time domain averaging of gear vibration.. RUBINI R. 998 Pub. DIEM, University of Bologna, No. 9. Tecnica di sincronizzazione di segnali periodici per il calcolo della media temporale sincrona in presenza di fluttuazioni della frequenza fondamentale - Parte prima: Presentazione del metodo (in italian). 6

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