1、 STDmAGMA 97FTM3-ENGL 1997 U Ob87575 O005072 309 D 97FTM3 I Detection of Fatigue Cracks in Gears with the Continuous Wavelet Transform by: Djami1 Boulahbal, M. Farid Golnaraghi, Fathy Ismail, Mechanical Engineering Department, University of Waterloo I I TECHNICAL PAPER COPYRIGHT American Gear Manufa
2、cturers Association, Inc.Licensed by Information Handling Services- STD-AGHA 77FTH3-ENGL 1977 = b87575 0005073 ?Li5 = Detection of Fatigue Cracks in Gears with the Continuous Wavelet kansform Djami1 Boulahbal, M. Farid Golnaraghi, Fathy Ismail Mechanical Engineering Department, University of Waterlo
3、o The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract Under ideal operating conditions, gearboxes generate vibration signals with frequency components that are pure
4、harmonics of the gear meshing frequency. Developing tooth fatigue cracks introduce short-time amplitude and phase modulations of the gear meshing vibration signal. Traditional techniques for gear fault detection have focused on either the time domain or the frequency domain. The newly developed wave
5、let transform enables one to look at the evolution in time of a signals frequency content. This property is very well suited for the analysis of localized transients that are generated by the operation of faulty gears. In this study, magnitude wavelet maps of the vibration signal are calculated and
6、used to assess the condition of an instrumented gear test rig. A key finding is that the wavelet map of the residual vibration signal offers a better indicator to the presence of cracks than the map of the actual signal. The results obtained are also compared against those of the well-accepted phase
7、 demodulation approach. Copyright Q 1997 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 November, 1997 ISBN: 1-55589-697-9 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-AGMA 77FTM3-ENGL I777 1111
8、b87575 0005074 IL D INTRODUCTION There are significant financial rewards to industrial plants which can minimize equipment downtime as well as maintenance expenditure. Predictive condi- tion monitoring can effectively contribute towards both of these objectives. Among the many tools currently in use
9、 to achieve these goals are those based on processing of the machines vibration signal. These have gained a much wider acceptance because of the advantages they offer. The vibration and noise generated by a machine are directly related to its “health” condition. However, extracting from amongst all
10、components of the vibration signal that which is due to a specific fault is often not a trivial task. The complications arise because several faults generate similar vibration patterns. The task is then to properly process the vibration signal and display it in a form very suitable for reliable asse
11、ssment of a machines condition. At the heart of vibration based machinery diagnostics are the ideas and methods of discrettHme signal processing, where one is often confronted with the task of deciding on which method to use to reveal each specific machine fault or malfunction. This issue is still t
12、he subject of intensive analytical as well as experimental research activities. Gears are widely used in many mechanical systems. Their required accuracy spans a wide spectrum, from low accuracy in simple power transmission, to high accuracy in motion transmission. This paper is mainly concerned wit
13、h precision gears. Amongst the most dangerous failures observed in gears are fatigue induced teeth cracks. The appearance of cracks is often accompanied by changes in the gear meshing conditions and vibration patterns. GEAR VIBRATION The major source of vibration in a gearbox is the meshing action b
14、etween gears. For a mathematically perfect set of gears, operating under constant load and speed conditions, the vibration energy will be concentrated at the gear meshing frequency and its harmonics. Localized gear faults introduce short-time modulations of both the amplitude and phase of the vibrat
15、ion signal. Monitoring and processing of this vibration signal is thus the key to identifying devel- oping faults in the gear train. It is certainly true that imperfections in the gear train due to the geometry and surface finish, will also introduce some amount of modulation in both the amplitude a
16、nd phase of the vibration signal. These modulations are however “uniform” for all teeth, and they will show up as “slow” modulations of the vibration signal. In contrast, localized faults introduce abrupt changes in the vibration signal. CURRENT DIAGNOSTIC TECHNIQUES There are several techniques cur
17、rently available for vibration based diagnostics of gear faults in general and fatigue cracks in particular. The techniques can be divided into those based on analysis of the signal in the frequency domain, and those based in the time domain. Undoubtedly, each approach has its own merits and limitat
18、ions, and where one technique has difficulties, another could shed better light. While experience has shown that spectral analysis is very successful in pin-pointing “distributed” faults in geared systems, localized faults such as cracked and spalled teeth are extremely difficult to extract from the
19、 average spectrum of the vibration signal. For these localized faults, time domain based techniques often offer a better alternative. The time domain tech- niques range from the calculation of statistical indicators such as the kurtosis and crest factor, which assess the condition of a machine from
20、the “peaki- ness” of its vibration signal, to those based on the synchronous time averaging and its extensions, such as amplitude and phase demodulation via Hilbert transform. Synchronous Time Averaging was introduced by Weichbrodt and Smith (1970), but however did not gain much acceptance because i
21、t required the use of expensive hardware, such as phase-locked fre- quency multipliers and tracking filters. Nevertheless, the technique allows one to zero in on the vibration caused by a specific gear and enhance it while attenuating other frequency components that are not synchronous with the rota
22、tion of the gear of interest. The synchronous average signal x(t) is calculated from the raw vibration signal y( t) according to 1 N-1 k=O 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-AGHA 77FTH3-ENGL 1797 Ob87575 0005075 018 where N is the nu
23、mber of records to be averaged transient signals often associated with faults in and T is the period of rotation of the gear of interest. machinery. The Wavelet transform (WT) overcomes Viewed in the frequency domain, the above operation this short-coming, and in contrast with the FT, the correspond
24、s to feeding the signal y( t) through a building blocks are “little waves” that are localized in time and thus are very well adapted to the study of comb-filter whose frequency spacing corresponds transients. precisely to the period T. The basic idea behind the WT is to represent a signai While a di
25、rect examination of the synchronous x(t) in terms of a set of “basis functions” which are average signal often offers a good initial assessment of the meshing condition of a pair of gears, exten- all scaled (dilated) versions of a single wavelet sions of the techniques even enhance its fault functio
26、n w(t) . Mathematically speaking, the wavelet the modulation of the instantaneous phase of the convolution integral beween this same signal x(t) synchronous average signal could be effectively used to detect cracked teeth, while Dalpiaz and MenegheRi and dilated versions Of an analysis wavelet w(t)
27、(1991) used the kurtosis of the phase modulation derivative as an indicator for cracks. Stewart (1977) proposed a further enhancement to the signal averaging technique based on a residual signal that detection ability further. McFadden (1986) found that transform wx(t,a) of a signal x(t) is defined
28、as a w,( f, a) = LTx(T) (5)- dT (2) Ja, “measures” the departuFe in the vibration pattern of a gear from that of its mathematically perfect counter- where a = f / fo is the scale parameter -a normalized _. - part. frequency in some sense, and controls the dilation of the wavelet w(t) . The frequency
29、 fo is the lowest In more recent studies, Ismail et. al. (1995) ShOwed frequency of interest in the signal x(t) and in our Chaotic dynamics, Golnaraghi et al. (1 995) devel- oped indicators that “measure” the impact dynamics between meshing teeth and correlated their results to tooth crack size. fro
30、nt of the integral is required for proper energy normalization. We note here that the WT depends on both dual variables of time t and frequency f. Most current techniques are based uniquely in either the time or the frequency domains. The benefits of spectral and time domain analyses can be concilia
31、ted with the newly available Wavelet transform. This time-frequency approach allows one to look at the evolution in time of a signals frequency content. It thus appears to be the ideal tool for the detection of a wider spectrum of gear faults, and especially local- ized fatigue cracks. One crucial a
32、spect in calculating the wavelet trans- form is the choice of an appropriate wavelet basis. This choice is dictated by the signal itself and the purpose of the analysis. There are certainly many available and competing wavelet bases to choose from and yet, not a single procedure for doing so. For th
33、e vibration signal at hand -sinusoid-like, we found that the Morlet wavelet gives superior results. This wavelet is essentially a Gaussian enveloped complex exponential and is described by w(t) = exp - - . expiot (3) THE CONTINUOUS WAVELET TRANSFORM In the classical Fourier transform (FT), one expre
34、sses a given function as the superposition of sine and cosine waves that span the entire frequency axis and have constant amplitude along the time axis. These “building blocks” (sine and cosines) are not well adapted to the representation and analysis of Figure 1. i: where 6b = 5.336/ f is the “effe
35、ctive support” of the wavelet and 0 = 2zf The real and imaginary Parts Of this complex-valued Morlet wavelet are shown in 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-ALMA 77FTM3-ENGL 1777 b87575 000507b T54 Amplitude 1 = 0.5 3 h I Q, - 50 5
36、3 -0.5 If4 GMF = Number of Teeth x Shan RPM 60 -1 -4 -2 O 2 4 Time - t Figure 1 : Real (Thick) and Imaginary (Thin) Parts of the Complex Morlet Wavelet In the frequency domain, the wavelet w(t) is the impulse response .e., inverse Fourier Transform, of a band-pass filter with center frequency o, =ao
37、,. Calculating the Fourier transform W(W) of the basic wavelet w( t) , we obtain (4) The above expression represents a Gaussian shaped constant-Q band-pass filter. The integral given by equation 1 is now easily interpreted as a filtering of the signal x(t) with a bank of constant4 filters that span
38、the entire frequency axis. Lastly, we have to note that in the standard definition of the Wavelet Transform, the limits of integration respectively. The operator F-. denotes here inverse Fourier transformation. In the above imple- mentation, the Fourier transform X(o) is calculated only once, and W(
39、ao) is calculated for all values of the scale parameter a. RESIDUAL GEAR VIBRATION The residual signal approach proposed by Stewart (1 977) is based on looking at the difference between the actual vibration produced by a gear set, and that which would be generated by a perfect gear set. Figure 2 sho
40、ws the vibration spectra for both a perfect and a real set of meshing gears. 1xGMF 2xGMF 3xGMF Amplitude t 1xGMF 2xGMF 3xGMF quence of the synchronous averaging operation, the integration is calculated over one full revolution of the gear of interest only and the limits of integration thus become -T
41、/2 and +T/2. The convolution integral is then replaced by a circular convolution and then implemented very efficiently with the FFT algorithm in the frequency domain according to where X(o) and aW(ao) are the Fourier transforms of the signal x(t) and dilated wavelet w(t/a) In addition to the Gear Me
42、shing Frequency (GMF) component and its harmonics, the spectrum for a real gear contains side-bands that are centered around the GMF and its harmonics. These side-bands are responsible for the amplitude and phase modulations induced by imperfections in the gear train. The spectrum of the residual si
43、gnal is obtained by simply filtering-out the GMF and its harmonics from the vibration of the gear of interest. Inverse Fourier transformation is then performed on this residual spectrum and gives the residual signal. This residual signal, as we will see later on in our experimental results, is a mor
44、e sensitive diagnostic tool because it characterizes the amount of departure the gear being 3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STDaAGMA 77FTM3-ENGL 2977 Ob87575 0005077 790 = analyzed has from its ideal and mathematically perfect counte
45、rpart. EXPERIMENTAL APPARATUS The gear test rig used throughout this study is shown schematically in Figure 3. It consists of two 1.1 HP permanent magnet DC motors and a single stage gearbox with sixteen teeth on the input (driving) shaft and fourteen teeth on the output (driven) shaft. The gears us
46、ed are off-the-shelf and thus, very repre- sentative of most common and average precision applications. The motors and the gearbox are all mounted onto a stiffened I-beam, itself anchored to a massive concrete base. frequency was set sufficiently high to capture the third harmonic of the GMF. DATA P
47、ROCESSING The signal acquired from the accelerometer is resampled digitally at intervals that correspond to constant angular positions of the gear. The sampling positions are calculated from the one pulse per revolution reference signal obtained from the slotted disk. This procedure is described by
48、McFadden (1989) and is necessary to eliminate errors induced by fluctuations in the shafts rotation speed and also provide a uniform number of data samples per revolution. Synchronous averaging is then performed on the resampled signal to obtain x( t) . The resulting signal is then filtered if desir
49、ed and its wavelet transform is computed. EXPERIMENTAL RESULTS A pair of healthy gears was first installed in the test rig and the vibration data was collected and proc- essed. The resulting synchronously averaged signal, along with its instantaneous phase modulation and wavelet map, is shown in Figure 4. Figure 3: Schematic View of Gear Test Rig Synchronous Average The torque loading on the gears is provided by a network of power resistors that are connected to the driven motor. These power resistors offer a “quiet“ dissipation mechanism .e., they
