ARTICLE IN PRESS. Mechanical Systems and Signal Processing

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1 Mechanical Systems and Signal Processing 23 (29) Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: Use of autocorrelation of wavelet coefficients for fault diagnosis J. Rafiee a,, P.W. Tse b a Department of Mechanical, Aerospace and Nuclear Engineering, Jonsson Engineering Center, 8th Street, Rensselaer Polytechnic Institute, Troy, NY , USA b Smart Engineering Asset Management Laboratory (SEAM), Department of Manufacturing Engineering and Engineering Management (MEEM), City University of Hong Kong, Kowloon, Hong Kong article info Article history: Received 9 January 28 Received in revised form February 29 Accepted 7 February 29 Available online 26 February 29 Keywords: Condition monitoring Fault detection and diagnosis Pattern recognition Wavelet Autocorrelation Sinusoidal approximation Mother wavelet Daubechies Gearbox db44 abstract This paper presents a novel time frequency-based feature recognition system for gear fault diagnosis using autocorrelation of continuous wavelet coefficients (CWC). Furthermore, it introduces an original mathematical approximation of gearbox vibration signals which approximates sinusoidal components of noisy vibration signals generated from gearboxes, including incipient and serious gear failures using autocorrelation of CWC. First, the drawbacks of the continuous wavelet transform (CWT) have been eliminated using autocorrelation function. Secondly, the autocorrelation of CWC is introduced as an original pattern for fault identification in machine condition monitoring. Thirdly, a sinusoidal summation function consisting of eight terms was used to approximate the periodic waveforms generated by autocorrelation of CWC for normal gearboxes (NGs) as well as occurrences of incipient and severe gear fault (e.g. slight-worn, medium-worn, and broken-tooth gears). In other words, the size of vibration signals can be reduced with minimal loss of significant frequency content by means of the sinusoidal approximation of generated autocorrelation waveforms of CWC as reported in this paper. & 29 Elsevier Ltd. All rights reserved.. Introduction Fault detection and diagnosis of gearboxes [,2] is one of the most common and intricate challenges in industries as a result of frequent gear defects in machines [3,4]. Vibration signal processing of gears [5] is categorized as a reliable method in condition monitoring. To analyze vibration signals, various techniques such as time (e.g. [6,7]), frequency (e.g. [8]), and time frequency domain (e.g. [9]) have been extensively studied. Among these, wavelet transform [ 3] has progressed in the last two decades, and outweighs the other time frequency methods, although it is lacking in a few aspects as well. The prime concern in machine fault diagnosis is to find a proper pattern with the characteristics of small-sized configuration and convincing classification ability. One of the influential approaches is based upon continuous wavelet transform (CWT) because of its minimal loss of information and maximal resolution of the signals. However, CWT generating continuous wavelet coefficients (CWC) suffers from a deficiency. The CWC encompasses too much data in each scale. When resampling for fault identification systems, the results may produce a loss of information. On the other hand, the obvious issue in rotating machines is that vibration signals contain a wide range of natural and defect frequencies because of periodic behaviors of the machine, and extracting significant frequencies within a small-sized pattern for fault diagnosis is still a challenge in signal processing. Moreover, finding a suitable feature applicable to a variety of datasets can be complicated. Since most of the vibration signals generated from machines are non-stationary Corresponding author. Tel.: ; fax: addresses: rafiee@rpi.edu, krafiee8@gmail.com (J. Rafiee) /$ - see front matter & 29 Elsevier Ltd. All rights reserved. doi:.6/j.ymssp

2 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Nomenclature a ARS b B D bi D bb E[] f G l bd MSE m n N O bd p P p P p P p scaling parameter adjusted R-Square shifting parameter/bearing number of bearings number of inner race defects number of rolling element defects expected value summation sinusoidal function number of gears vibrations of inner race defects mean squared error number of fitted coefficients number of response values number of sinusoidal terms vibrations of outer race defects interpolator piecewise cubic Hermite interpolation first derivative of piecewise cubic Hermite interpolation second derivative of piecewise cubic Hermite interpolation P s P s RMSE R xx R S SSE SSR SST t v v s v sg v r x x r bd t s s 2 r xx r xx piecewise cubic spline interpolation second derivative of piecewise cubic spline interpolation root mean squared error autocorrelation function number of epicyclic gear trains number of fixed-axis shafts sum of the squares of the error sum of squares of the regression total sum of squares of the total error time degrees of freedom vibrations of shaft vibrations of gear vibrations of epicyclic gear trains signal mean value of population vibrations of rolling element defects time delay standard deviation variance population autocorrelation coefficients population autocorrelation coefficients [4,5], it is critical to have a proper feature which adapts for different datasets. In previous research, several papers (e.g. [6 26]) have been documented in this area, particularly based on wavelet transform [27]. The improvement on timedomain analysis of wavelet transform is contributed by Halim et al. [28], who implemented time synchronous average and wavelet transform to extract the periodic waveforms of gear vibration signals at different scales. To minimize the above-mentioned deficiencies, an original technique based on time-series analysis of CWT was designed and tested in this research. Vital features were obtained from the autocorrelation of CWC of gearbox vibration signals. The autocorrelation of CWC is able to reduce the size of data without information loss in significant frequency content. The down-sampling improved upon the work done by Halim et al. A sinusoidal summation function is presented approximating the periodic trends of autocorrelation of CWC with satisfactory preciseness. The simple sinusoidal summation function could approximate the behaviors of vibration signals for different incipient and serious faults. In wavelet analysis, signal decomposition (scale) is another issue which needs to be considered. In experiments, the high-noise vibration signals were divided into 2 4 sub-signals (2 4 scales) in fourth level of decomposition by CWT. In such a way, the complex signals are converted into simplified signals with more resolution in time and frequency domains. Then, autocorrelation is applied to reduce the length of the sub-signals (series of wavelet coefficients) containing significant frequencies. These frequencies were found to be different from one condition of the gearbox to another. For example, different classes of faulty signals produced different amplitudes in their dominant frequencies and harmonics as well as related sidebands... Nature of gearbox vibrations In machine construction, there is a frequent need to change the rotational speed between the motor and the working machine. Hence, geared systems are extensively used. Gearboxes are essential sources of vibrations because of discrete transfer of load by the successive meshing teeth. Simulated models of gearbox vibrations use the sum of vibrating components (e.g. gear, bearings, shafts) modified by the transmission path effects. These include the sum of the vibration for fixed-axis shafts, meshing points of their mounted gears, epicyclic gear trains and bearing defects. In general, the gearbox vibration is defined as follows [29]:!! vðtþ ¼ XR v r ðtþþ XS v s ðtþþ XGs v sg ðtþ þ XB X D bi l bd ðtþþ XD bo O bd ðtþþ XD bb r bd ðtþ r¼ s¼ g¼ b¼ d¼ where R is the number of epicyclic gear trains, S is the number of fixed-axis shafts, G s is the number of gears on the shaft s, B is the number of bearings, D bi is the number of inner race defects on bearing b, D bo is the number of outer race defects on bearing b, D bb is the number of rolling element defects on bearing b, v s (t) is the vibration of shaft s, v sg (t) is the vibration of gear g on shaft s, v r (t) is the vibration of epicyclic gear train r, l bd (t), O bd (t), and r bd (t) are the vibrations of inner race, outer race, and rolling element defects d on bearing b, respectively. d¼ d¼

3 556 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Periodically variable numbers of gear teeth in the unit cases is one of the important causes of the parametric vibrations with characteristics modulating effects. The structure of gear vibrations is complex because it includes vibrational effects caused by manufacturing errors and assembly faults. A typical spectrum of gearboxes includes shaft rotational frequency, gear natural and mesh frequencies as well as sidebands. 2. A novel methodology for feature extraction The development of our algorithm is outlined in the following steps:. Raw vibration signals were recorded from a motorcycle gearbox system. Three types of gear defects were selected and tested. The conditions of the gearbox consisted of slight-worn gear (SW), medium-worn gear (MW), broken-tooth gear (BT), and normal gearbox (NG). 2. Piecewise cubic spline interpolation (PCSI) [3] was used to synchronize the vibration signals. Note that a sample signal is defined as a segmented signal with the length of one complete revolution of the input shaft as shown in Fig.. 3. Continuous wavelet coefficients of synchronized vibration signals (CWC-SVS) were calculated at the fourth level of decomposition (2 4 scales for each sample). These were calculated with 324 selected mother wavelet functions from diverse families. 4. At the fourth level of decomposition, variance of CWC-SVS was calculated for 324 mother wavelets. Among them, the most similar one to gear vibration signals, the Daubechies 44 (db44) [3], was selected. The db44 has the highest values for wavelet coefficients compared to the other 323 mother functions, and subsequently provided the proper similarity to the gear vibration signals. 5. Autocorrelation of CWC-SVS was determined. Then, the power spectrum density (PSD) of CWC-SVS and autocorrelated CWC-SVS were compared to observe the advantage of applying autocorrelation to CWC-SVS. 6. Frequency attributes of autocorrelated CWC-SVS were calculated using the PSD to identify proper features for classifying gearboxes operating under normal and faulty conditions. 7. The sinusoidal summation function was used to approximate the autocorrelated CWC-SVS to verify the preciseness of reconstructing the original signal patterns to both normal and faulty gear patterns. Normal Slight-worn 4 Medium-worn Broken-tooth Fig.. Raw vibration signals of four gearbox conditions.

4 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Experimental set-up To record vibration signals, a gearbox of a four-speed motor-driven system was running during data recording. The system consists of a driving motor with a constant nominal rotation speed of 42 rpm, a load mechanism including a friction wheel to make inconsistent rotations, four shock absorbers under the bases of the test-bed, and the gearbox including 24 teeth for the pair driving gear and 29 teeth (tested gear) for the driven gear. A schematic diagram of the geared system in the neutral state is shown in Fig. 2 [25]. In this figure, gears A4 and B4 are a pair of driving and driven gears. Gears A2 and A4 mounted on the output shaft and B and B3 mounted on the input shaft were fixed in the gearbox. The rest of the gears move axially across the shafts depending on the specific speed. As the rotation speed of the motor (input shaft) is 24.5 Hz (fr); according to Fig. 2, the rotation speed of the output shaft is 29.6 Hz and the meshing frequency is Hz. For collecting vibration data, a multi-channel Pulse analyzer system, a triaxial accelerometer and a tachometer were used. The vibration signals were recorded by mounting the accelerometer on the outer surface of the gearbox s case near the input shaft of the gearbox. Three different fault conditions were selected as slight-worn, medium-worn, and brokentooth of a spur gear. To evaluate the precision of the technique, two very similar models of worn gear were taken into account with partial difference. Also, a serious failure of a broken-tooth gear was considered to show the reliability of the technique for different faulty signals. The real rotational speed of the motor was measured by the tachometer. The sampling rate was set at 6,384 Hz as well. More detail is addressed to Rafiee et al. [32] Synchronization of raw vibration signals The number of data-points per each shaft revolution change in our gearbox because the shaft speed fluctuates (e.g. see Fig. 3). To overcome this flaw, the PCSI was exploited to resample the data to a regular time base before signal analysis. For interpolation purpose, on each sub-interval x(t), kptpk+, suppose P(x) be the interpolant of the given values having certain slopes at the two end points. Between each two adjacent data sites x(k) and x(k+), x(t) is a polynomial. For piecewise cubic Hermite interpolation (PCHI), P p (x) indicates the interpolator. The first derivative, P p(x), is continuous, but P p(x) is not necessarily continuous, which is a drawback of PCHI. The function P s (x) supplied by the PCSI is constructed so that the slopes at the x(k) are chosen to make P s(x) continuous. Therefore, this process makes P s (x) smoother and more accurate. Thus, PCSI was considered to synchronize the vibration signals. The length of sample signals which were not equal in gear dataset was synchronized by PCSI with minimal loss of information Continuous wavelet transform Basic theory of CWT as well as potential applications in machine condition monitoring can be found in several papers (e.g. [27]). The result of the CWT, wavelet coefficients, shows how well a mother wavelet function correlates with a particular signal. If the signal has a major frequency component corresponding to a particular scale, then the mother wavelet at that scale (daughter wavelet) is similar or close to the signal at a particular location where the frequency component occurs. As a result, the CWC have a large value at that location and scale. CWT may process the gearbox vibration signals better than discrete wavelet transform (DWT) because down-sampling of the signals using DWT would lead to the loss of significant information. Wavelet decomposition is divided into two main branches: pyramid and packet decompositions. In both methods, signals are divided into approximation (low frequency) and detail (high frequency) in the first level. In pyramid decomposition, after the first level only approximations are Fig. 2. Schematic of the gearbox in neutral state [25].

5 558 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) No. of datapoints in sample signal No. of revolution of the driven shaft Fig. 3. Number of data-points vs. the revolution of the input shaft for 5 segmented signals. decomposed into higher levels. However, in packet decomposition both approximation and detail are decomposed into further levels. Therefore, packet decomposition offers richer contents of the signals. However, what is commonly known as wavelet packet transform (WPT) is the discrete transform with packet decomposition. That means down-sampling of the wavelet coefficients while increasing the decomposition level and consequently the loss of information. CWT, which means continuous shifting through the time, was used with packet decomposition through the scales in this research. Based on this idea, better resolution on frequency domain is achieved by means of packet decomposition as well as no loss of information throughout the time-domain signals. Therefore, after synchronizing the raw vibration signals, the CWT and autocorrelation function were applied to the synchronized signals and generated continuous wavelet coefficients of synchronized vibration signals as mentioned above. To find the most suitable mother wavelet, 324 candidate mother wavelet functions were studied form various families including Haar, Daubechies (db), Symlet, Coiflet, Gaussian, Morlet, complex Morlet, Mexican hat, bio-orthogonal, reverse bio-orthogonal, Meyer, discrete approximation of Meyer, complex Gaussian, Shannon, and frequency B-spline. The most similar mother wavelet for analyzing the gear vibration signal was selected based on the following steps:. Raw vibration signals were recorded and synchronized. The feature vector is: the variance of CWC for each of the 2 4 scales calculated by each of the 5 segmented signals in each gearbox condition. The average of the feature vector in the 5 segmented signals was computed for each gearbox condition. 2. Variances of the mentioned average of the four gearbox conditions were determined for each scale (2 4 elements). The five highest values of the calculated vector were selected as the feature because the more variance we have, the greater the ability to properly classify failures. 3. The summation of the five elements, called SUMVAR for simplicity, was compared with those obtained from the other 323 candidate mother wavelets (a total of 324 mother wavelets). The one that had the highest SUMVAR was selected as the most similar function to our vibration signals. Fig. 4a shows the decision-making flow chart for selecting the most similar mother wavelet. Among the 324 mother wavelets, the SUMVAR of Daubechies 44 was the highest. The db44 is the most similar function to gear vibration signals in this research. As illustrated in Fig. 4b, the shape of db44 has a near-symmetric characteristic that the shapes of other high order db do not have Autocorrelation The autocorrelation function is an important diagnostic tool for analyzing time series in the time domain. Autocorrelation plots, called correlograms, present a better understanding of the evolution of a process through time using the probability of the relationship between data values separated by a specific number of time steps (lags). The correlogram plots autocorrelation coefficients on the vertical axis, and lag values on the horizontal axis. For a signal x(t), the autocorrelation function [33] is the average value of the product xðtþxðt þ tþ, where t is time delay. Formally, the autocorrelation function, R x (t), is defined as R x ðtþ E½xðtÞxðt þ tþš ¼ Lim T! Z T xðtþxðt þ tþ dt () Mean and variance of this function are independent of time. Therefore, E½xðtÞŠ ¼ E½xðt þ tþš ¼ x (2)

6 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Raw vibration signals Segmentation of signals into sample signals 324 mother wavelet functions 6 scales (in 4 th level) 5 segmented & synchronized signals by PCSI (NG) 5 Segmented & synchronized signals by PCSI (BT) 5 segmented & synchronized signals by PCSI (SW) 5 segmented & synchronized signals by PCSI (MW ) Selecting one mother wavelet, one scale, and one sample signal in a gearbox condition A= Variance of CWC-SVS in each scale for each 5 segmented signals (for 4 gearbox conditions) Calculated A Average of A in 5 samples (NG) Average of A in 5 samples (BT) Average of A in 5 samples (SW) Average of A in 5 samples (MW) F = Variance of the calculated average in gearbox conditions in each scale SUMVAR = Sum of five F elements with higher values (out of 6) Max SUMVAR Fig. 4. (a) Algorithm of the most similar mother wavelet function and (b) Daubechies 44. and s 2 xðtþ ¼ s2 xðtþtþ ¼ s2 x ¼ E½x2 ðtþš x 2 (3) The autocorrelation coefficients can be defined as r xx ðtþ Ef½xðtÞ x Š½xðt þ tþ x Šg s 2 x (4)

7 56 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) which can be expanded as follows: r xx ðtþ Ef½xðtÞxðt þ tþš x E½xðt þ tþš x E½xðtÞŠ þ x 2 g s 2 x (5) Substitution of Eqs. () and (2) into Eq. (5) leads to an expression that relates the autocorrelation function to its coefficient: r xx ðtþ R xðtþ x 2 s 2 x (6) or R x ðtþ ¼r xx ðtþs 2 x þ x2 (7) Some limits can be placed on the value of R x (t). Because r xx ðtþ; R x ðtþ is bounded as s 2 x þ x2 R x ðtþs 2 x þ x2 (8) The variance of x can be expressed in terms of the expectation of x 2, E[x 2 ], and the square of the mean of x, x 2 : s 2 ¼ E½x 2 Š x 2 (9) Based upon Eq. (), in Eq. (8) with regard to Eq. (9) we have R x ðþ ¼E½x 2 Š¼s 2 þ x 2 () Therefore, the maximum value that R x (t) can have is E[x 2 ]. That is, the maximum value of R x (t) occurs at t ¼. Using Eq. (9) and the definition of the autocorrelation coefficients as in Eq. (6), r xx ðþ ¼ () Further, as t-n there is a lesser correlation between x(t) and x(t+t) because x(t) is the signal of a random variable. That is, r xx ðt!þ¼ (2) which indicates that R x ðt!þ¼x 2 (3) a value equal to the limiting value which signifies that there is no correlation. Autocorrelated CWC-SVS will help us to classify different types of gear health conditions in a small-size structure with acceptable performance. Loss of information in preprocessing of non-stationary signals is a challenging problem and autocorrelated CWC-SVS allows for reduction of size with minimum information loss. This point distinguishes this research from prior proposed techniques Sinusoidal approximation For approximation purposes, we used a summation sinusoidal function defined as follows: f ðxþ XN i¼ a i sinðb i x þ c i Þ where a i, b i, and c i are constant coefficients. In this paper, we used trial and error to determine an N value of 8. Fitted coefficients were obtained based on the nonlinear least-square method. It is important to note that the main purpose in vibration machine monitoring is to present a small-structure pattern for different conditions. It is obvious that the higher the N value, the better the approximation will be. However, f(x) consisting of eight terms approximates all gearbox conditions with high accuracy and, with consideration to three constant coefficients this function can compress the subsignals (series of wavelet coefficients for each scale) to a meaningful and reliable approximation with 3 8 elements. This would reduce the data to coefficients for each gearbox condition. To verify approximation accuracy, the following common statistical criteria were used in this research: SSE: The sum of the squares of the error measures the total deviation of the response values from the fit to the response values: SSE XN ðx i x ci Þ 2 (5) i¼ A value closer to indicates that the approximation has a smaller random error component. (4)

8 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Autocorrelation of SVS (Normal-Gearbox) Autocorrelation of SVS (Medium-Worn) Lags Lags Autocorrelation of SVS (Slight-Worn) Autocorrelation of SVS (Broken-Tooth) Lags Lags Fig. 5. (a) Autocorrelation plot of a synchronized vibration signal for normal gearbox, (b) autocorrelation plot of a synchronized vibration signal for slightworn gear, (c) autocorrelation plot of a synchronized vibration signal for medium-worn gear and (d) autocorrelation plot of a synchronized vibration signal for broken-tooth gear. Synchronized segmented signal v Level,, Level 2 2, 2, 2,2 2,3 Level 3 3, 3, 3,2 3,3 3,4 3,5 3,6 3,7 Level 4 4, 4, 4,2 4,3 4,4 4,5 4,6 4,7 4,8 4,9 4, 4, 4,2 4,3 4,4 4,5 Fig. 6. Decomposition tree of the wavelet transform.

9 562 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) (4,) (4,) (4,8) (4,9) (4,2) (4,3) (4,) (4,) (4,4) (4,5) (4,2) (4,3) (4,6) (4,7) (4,4) (4,5) (4,) (4,) (4,8) (4,9) (4,2) (4,3) (4,) (4,) (4,4) (4,5) (4,2) (4,3) (4,6) (4,7) (4,4) (4,5) Fig. 7. (a) Autocorrelation of a CWC-SVS for normal gearbox condition (X-axis: lags ¼ 25, Y-axis: to, scale in title), (b) autocorrelation of a CWC-SVS for slight-worn gear (X-axis: lags ¼ 25, Y-axis: to, scale in title), (c) autocorrelation of a CWC-SVS for medium-worn gear (X-axis: lags ¼ 25, Y-axis: to, scale in title) and (d) autocorrelation of CWC-SVS for broken-tooth gear (X-axis: lags ¼ 25, Y-axis: to, scale in title).

10 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) (4,) (4,) (4,8) (4,9) (4,2) (4,3) (4,) (4,) (4,4) (4,5) (4,2) (4,3) (4,6) (4,7) (4,4) (4,5) (4,) (4,) (4,8) (4,9) (4,2) (4,3) (4,) (4,) (4,4) (4,5) (4,2) (4,3) (4,6) (4,7) (4,4) (4,5) Fig. 7. (Continued)

11 564 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) RS: R-square is defined as the ratio of the sum of squares of the regression (SSR) and the total sum of squares (SST): SSR XN i¼ w i ð^y i ȳþ 2 (6) SST is also called the sum of squares about the mean, and is defined as SST XN i¼ w i ðy i ȳþ 2 (7) where SST ¼ SSR+SSE. Given these definitions, R-square is expressed as RS ¼ SSR SSE ¼ (8) SST SST R-square can take on any value between and, with a value closer to indicating that a greater proportion of variance is accounted for by the approximation. ARS: Degrees of freedom adjusted R-square uses the R-square and adjusts it based on the residual degrees of freedom. The residual degree of freedom is determined as v ¼ n m (9) where n is the number of response values and m is the number of fitted coefficients. The residual degree of freedom indicates the number of independent pieces of information including the n data-points that are required to determine the sum of squares. The degrees of freedom are increased by the number of such parameters SSEðn Þ ARS ¼ (2) SSTðvÞ ARS can take on any value less than or equal to, with a value closer to indicating a better fit. Negative values can occur when the model contains terms that do not help to predict the response Power spectral density (Normal-Gearbox) 7f 6f 5f 4f 8f 2 f 2f 3f 9f f Frequency (Hz) 2 8 Sidebands f Sidebands PSD of NG Frequency (Hz) Fig. 8. (a) The PSD of raw vibration signal with f as tooth meshing frequency and 2f, 3f, etc., as its harmonics (normal gearbox) and (b) the zoomed PSD at the tooth meshing frequency, f, and its sidebands (normal gearbox).

12 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) RMSE: Root mean squared error is also known as the fit standard error, which can estimate the standard deviation of the random components in the data: p RMSE ¼ ffiffiffiffiffiffiffiffiffi MSE (2) where MSE is the mean square error. By using the above statistical evaluation methods, one can determine whether the employed approximation function is suitable for estimating a particular signal. 4 2 Power spectral density (Slight-Worn) 4f 7f 8 6 5f 4 2 f 2f 3f 6f 8f Frequency (Hz) Power spectral density (Medium-Worn) 4f 8 7f 6 3f 4 2 f 2f 5f 6f 8f 9f f Frequency (Hz) Power spectral density (Broken-Tooth) 4f 7f 6f 9f 8f 2 2f 3f f Frequency (Hz) Fig. 9. (a) The PSD of raw vibration signals (slight-worn), (b) the PSD of raw vibration signals (medium-worn) and (c) the PSD of raw vibration signals (broken-tooth).

13 566 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Results and discussion 3.. Use of autocorrelated CWC-SVS for gear fault classification In autocorrelation function of the signal, which literally correlates the signal with itself, essential information can be gathered by examining how the amplitude of the signal s time history record at one point compares to its amplitude at.5.5 (4,) PSD of Autocorrelation 6f 5f 4f 7f 8f 9f (4,) PSD of CWC-SVS 4f 5f 6f 7f 8f 9f f (4,4) PSD of Autocorrelation 3f 4f 5f (4,4) PSD of CWC-SVS 3f 4f 5f (4,2) PSD of Autocorrelation f f'-3fr f (4,2) PSD of CWC-SVS f'-3fr 2f (4,8) PSD of Autocorrelation 2f other normal frequency components (4,8) PSD of CWC-SVS 2f other normal frequency components (f-fr) (4,) PSD of Autocorrelation 5(f-fr) 6(f-fr) 9f-fr 8f-9fr 7(f-fr) f-fr 3 2 (4,) PSD of CWC-SVS 7(f-fr) 4(f-fr) 6(f-fr) 5(f-fr) 8f-9fr 9f-fr f-fr (4,2) PSD of Autocorrelation 5(f-fr) 6(f-fr) 7(f-fr) 8f-9fr (4,2) PSD of CWC-SVS 5(f-fr) 6(f-fr) 7(f-fr) 8f-9fr (4,3) PSD of Autocorrelation 4(f-fr) 5(f-fr) 6(f-fr) 5 (4,3) PSD of CWC-SVS 4(f-fr) 5(f-fr) 6(f-fr) (4,6) PSD of Autocorrelation 2(f-fr) 3(f-fr) 4(f-fr) 5 5 (4,6) PSD of CWC-SVS 2(f-fr) 8f-9fr 3(f-fr) 4(f-fr) Fig.. (a) A comparison of the PSDs generated by autocorrelated CWC-SVS and those generated by CWC-SVS for normal gearbox and (b) a comparison of the PSDs generated by autocorrelated CWC-SVS and those generated by CWC-SVS for broken-tooth gear. Fig.. Autocorrelated CWC-SVS in 5 revolutions of the shaft [slight-worn gear/scale (4,)].

14 ARTICLE IN PRESS J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) another point in time. Hence, the periodic behaviors of the signals can be revealed by autocorrelation. The autocorrelation plots of synchronized vibration signals show that the periodic trends exist in all gearbox conditions (see Figs. 5a d). If the autocorrelation dies out quickly the series is considered stationary. If the autocorrelation dies out slowly this indicates that the process is non-stationary, as with the worn gears shown in Figs. 5b and c. From the viewpoint of time-domain signals, it is difficult to spot the periodic impacts of the defect gears, particularly worn ones. Recognizing any meaningful features from the autocorrelation of raw signals in all conditions seems to be a demanding effort. Hence, the signals needed to be divided into aforementioned sub-signals using CWC-SVS. Afterward, the autocorrelations of each scale of CWC-SVS were determined for all four gearbox conditions, including normal, slight-worn, medium-worn and broken-tooth. The calculation of autocorrelation was extended to the fourth level of decomposition as illustrated in the decomposition tree of Fig. 6. In the fourth level, the 6 decomposition plots representing the 6 scales from (4,) to (4,5) of the calculated autocorrelation of CWC-SVS under the gear normal condition are displayed in Fig. 7a. Similarly, the 6 decomposition plots of the gears in the slight-worn, medium-worn, and broken-tooth conditions are shown in Figs. 7b d, respectively. From these decomposition plots, one can clearly recognize the variation in the decomposed components of the signals under different gear health conditions. Hence, the difference between each condition is easily obvious in these sub-signals and can be used for the classification of different gear health conditions. In applying vibration-based gear fault diagnosis, traditional frequency domain methods, such as determination of tooth meshing frequency, its harmonics, and sidebands are usually used to identify the gear faults. The PSD of raw signals recorded from the gearbox operating under normal condition is depicted in Fig. 8a with f as tooth meshing frequency and 2f, 3f, etc., as its harmonics. Fig. 8b shows the zoomed PSD at the tooth meshing frequency and its sidebands. The PSDs of slight-worn, medium-worn and broken-tooth gears are shown in Figs. 9a c, respectively. To verify that there is no Fig. 2. Automatic frequency extraction of vibration signals using PSD of autocorrelated CWC-SVS distributed on 25 lags (X-axis) for segmented signals of each condition (Y-axis).

15 568 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) significant loss of information after applying autocorrelation to the CWC-SVS, the PSDs generated at every quarterly scale of the fourth level are shown in Figs. a and b to compare to those generated by CWC-SVS in normal and broken-tooth conditions. In machine fault diagnosis, one of the central questions is: Is the pattern used for recognition applicable to different signals extracted at various times from the machine given that the signals are embedded with non-stationary attributes? The autocorrelation of CWC-SVS is a powerful tool for pattern recognition because the results generated have minimal fluctuations from one sample signal to another even though the raw signals are non-stationary. Fig. shows the results obtained from 5 revolutions of rotating when the gearbox was under slight-worn condition. Note that the variation is small. Hence, the proposed method is robust even when applied to non-stationary signals. Autocorrelation function has proven its reliability for checking the randomness of data. For the gearbox vibration data, the number of lags can be limited to less than 3. Usually, such a lag value is sufficient to verify the randomness contained in the data. In our experiments, we set the number of lags to 25. This number was obtained by trial and error. We selected such a large value so that it is capable of not only checking the randomness of the data, but also reducing the size of CWC- SVS by almost one-sixth. The observation is that the larger the number of lags, the better the accuracy. Nevertheless, as mentioned above, our goal is to maintain a small-size feature pattern for machine fault diagnosis. The large value of lags will lead to large-size feature. Fig. 2 shows the 6 PSD plots generated by the 6 scales of the fourth level of decomposition of the autocorrelated CWC-SVS s results. Note that the X-axis is the number of lags (25). It is mandatory to further explain that autocorrelation of CWC-SVS will present 25-element vector for each scale. Therefore, it is more logical that each PSD plot is presented for.5 (4,) (4,2) (4,5) (4,5) Fig. 3. The approximation of autocorrelated CWC-SVS for medium-worn gearbox (original values of the autocorrelation of CWC-SVS data points; the approximated values-continuous curves).

16 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) frequency points rather than up to 8 khz. Instead of distributing the frequency from to 8 Hz as in Figs. 8a and 9a d, the X-axis has been distributed evenly from to 25 (the number of lags) units. In other words, the X-axis is indirectly proportional to time. The Y-axis represents different health conditions of the gearbox including sample signals for each of the four conditions. The Z-axis is the magnitude of autocorrelations of CWC-SVS shown as different color scales. As observed in Fig. 2, at high scale values (low frequency) from (4,8) to (4,5), the dominant frequency as well as its Table Statistical criteria to determine the appropriateness of approximation for normal gearbox. (a) Statistical criteria (4,) (4,) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) SSE RS ARS RMSE (b) Statistical criteria (4,8) (4,9) (4,) (4,) (4,2) (4,3) (4,4) (4,5) SSE RS ARS RMSE Table 2 Statistical criteria to determine the appropriateness of approximation for slight-worn gearbox. (a) Statistical criteria (4,) (4,) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) SSE RS ARS RMSE (b) Statistical criteria (4,8) (4,9) (4,) (4,) (4,2) (4,3) (4,4) (4,5) SSE RS ARS RMSE Table 3 Statistical criteria to determine the appropriateness of approximation for medium-worn gearbox. (a) Statistical criteria (4,) (4,) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) SSE RS ARS RMSE (b) Statistical criteria (4,8) (4,9) (4,) (4,) (4,2) (4,3) (4,4) (4,5) SSE RS ARS RMSE

17 57 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Table 4 Statistical criteria to determine the appropriateness of approximation for broken-tooth gearbox. (a) Statistical criteria (4,) (4,) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) SSE RS ARS RMSE (b) Statistical criteria (4,8) (4,9) (4,) (4,) (4,2) (4,3) (4,4) (4,5) SSE RS ARS RMSE sidebands of each health condition can be revealed. However, classifying the four health conditions using these scales, particularly the scales (4,) to (4,5), are difficult as their features are located adjacently. Although, in low scales (high frequency) such as (4,2), the features of each of the four health conditions are easily distinguished. The ability to identify the characteristic frequencies will be useful in the future for making an automatic feature extraction algorithm so that the process of fault detection and classification can be automated The sinusoidal approximation for reconstructing the gearbox vibration signals Using the aforementioned statistical evaluation methods, the preciseness of the sinusoidal approximation was verified by all four conditions of the gearbox. The original values of the autocorrelation of CWC-SVS (displayed as continuous curves) as compared to the approximated values (displayed as data points) for some arbitrary selected scales under the medium-worn health condition are shown in Fig. 3. Note that the approximation can follow the waveforms of autocorrelated CWC-SVS closely, particularly at higher scales. The results of the statistical evaluation are tabulated in Tables 4. From the results shown in Fig. 3 and Tables 4, the sinusoidal summation function with eight terms approximates the waveforms generated by autocorrelation of CWC-SVS for all the four health conditions. By observing the results in the tables, the statistical methods show much better fitness in comparison with low scales. The reason is that autocorrelation of CWC-SVS in high scales possesses a greater variety of frequency contents compared to those in low scales (see Figs. 7a d). In these figures, the frequency components of autocorrelation of CWC-SVS in high scales are higher than those in low scales. Therefore, the assessment of preciseness of sinusoidal approximation in low scales is better observed as stated in Tables 4. Although the fit is satisfactory when using a sinusoidal summation function with eight terms, the number of sinusoidal terms (N) in Eq. (4) can be increased for more complex waveforms. The proposed approximation function can also be applicable to other defects, such as bearing defects, because the bearing faulty signals are impulsive in nature, similarly to gear faulty signals. 4. Conclusion Based on our proposed algorithms and the experimental results used in evaluating the effectiveness of the algorithms, we can summarize our findings as follows:. Autocorrelation of CWC-SVS has been introduced as a suitable feature for non-stationary signals in machine condition monitoring. 2. A simple sinusoidal summation function can approximate the waveforms generated by autocorrelation of CWC-SVS for normal gearboxes as well as other defective gears with satisfactory performance. The function achieved proper approximation even though the waveforms are different from one condition to another as they possess different frequency contents of vibration signals. The proposed simple algorithm can be the base of feature extraction in machine condition monitoring such that the meaningful approximation coefficients with the small-size attribute can be realized. 3. The authors believe that the proposed techniques can be applied to other faulty vibration signals, even bearing faulty signals. Further research could be conducted to confirm the effectiveness of the proposed techniques using a variety of signals collected from industrial machines. Further research on the mother wavelet function could be conducted to optimize db44 for specific purposes.

18 J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (29) Acknowledgements The research was supported by the Research Grants Council of Hong Kong SAR, China (project no. CityU 256), the Vibration and Modal Analysis Lab at University of Tabriz, Iran, and the Department of Mechanical, Aerospace & Nuclear Engineering at Rensselaer Polytechnic Institute, USA. The authors would like to write in memoriam of a dedicated mentor, Professor James Li at RPI, who devoted his life to enlightening a myriad of students as well as making noteworthy contributions to several aspects of machine condition monitoring. The authors appreciate the very constructive comments of the anonymous reviewers and would also like to offer special thanks to them for spending their valuable time to review the current research. They also extend their appreciation to Diane V. Michaelsen, for assistance in editing and preparing this paper. References [] D. Boulahbal, M.F. Golnaraghi, F. 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