A Review of Feature Extraction Methods in Vibration-Based Condition Monitoring and Its Application for Degradation Trend Estimation of Low-Speed Slew Bearing

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Review A Review of Feature Extraction Methods in Vibration-Based Condition Monitoring and Its Application for Degradation Trend Estimation of Low-Speed Slew Bearing

Wahyu Caesarendra 1,2 ID and Tegoeh Tjahjowidodo 3,* ID

1 Mechanical Engineering Department, Diponegoro University, Semarang 50275, Indonesia; [email protected] 2 School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Wollongong, NSW 2522, Australia 3 School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore * Correspondence: [email protected]

Received: 9 August 2017; Accepted: 19 September 2017; Published: 27 September 2017

Abstract: This paper presents an empirical study of feature extraction methods for the application of low-speed slew bearing condition monitoring. The aim of the study is to ﬁnd the proper features that represent the degradation condition of slew bearing rotating at very low speed (≈1 r/min) with naturally defect. The literature study of existing research, related to feature extraction methods or algorithms in a wide range of applications such as vibration analysis, time series analysis and bio-medical signal processing, is discussed. Some features are applied in vibration slew bearing data acquired from laboratory tests. The selected features such as impulse factor, margin factor, approximate entropy and largest Lyapunov exponent (LLE) show obvious changes in bearing condition from normal condition to ﬁnal failure.

Keywords: vibration-based condition monitoring; feature extraction; low-speed slew bearing

1. Introduction Slew bearings are large roller element bearings that are used in heavy industries such as mining and steel milling. They are particularly used in turntables, cranes, cranes, rotatable trolleys, excavators, reclaimers, stackers, swing shovels and ladle cars [1]. In this paper, the slew bearing being analyzed is the one typically used in BlueScope Steel Australia; speciﬁcally for the coal loading into the steel ladle turret, coal stacker and reclaimer. Slew bearing is different to typical rolling element bearing in terms of geometry, load and speed application. Slew bearings are usually operated at very low rotational speed, from 0.5 to 15 rpm in either continuous or reversible rotation and are designed with three axes to support high axial and radial loads. The bearings have large diameter (e.g., approximately from 200 mm to 5000 mm). Due to the high geometric and cost issue, industries purchase these bearings for replacement in a regular maintenance period. However, in some cases, these bearings may fail before the predicted time. To prevent catastrophic failure and to effectively schedule the bearing purchase for minimizing the replacement cost, a condition monitoring method for early failure detection is required. Analyzing the vibration signatures of slew bearings that operate at a very low speed is more complex than that of typical rolling element bearings. Slew bearing signals have low energy. They are also non-stationary, non-linear, chaotic and practically hard to analyze. Due to their low energy, the signal is masked by a higher background noise level. Therefore, it is difﬁcult to extract the pertinent

Machines 2017, 5, 21; doi:10.3390/machines5040021 www.mdpi.com/journal/machines Machines 2017, 5, 21 2 of 28 bearing signal. It is necessary to have reliable condition monitoring parameters or features that could extract accurate information about the bearing condition from the signal. The number of studies on slew bearing condition monitoring and early failure detection based on vibration analysis is lower than for rolling element bearing. Studies of slew bearing condition monitoring and life prediction utilizing vibration signal are presented in [2–7]. Žvokelj et al. [2] presents the condition monitoring method utilizing vibration signal based on ensemble empirical mode decomposition (EEMD) and principal component analysis (PCA). An extension work of [2] is presented in [3] which used combined EEMD with kernel PCA. Recent studies of the slew bearing fault detection and diagnosis is presented by Žvokelj et al. [4]. The paper presents combined EEMG and multiscale independent component analysis (ICA). Nevertheless, studies of slew bearing failure prediction and prognosis also increased recently [5–7]. Feng et al. [5] develops a reliability model for residual life prediction based on Weibull analysis. A method for degradation trend estimation of slew bearings was presented in [6]. The method employed PCA, particle swarm optimization (PSO) and least squares support vector machine (LSSVM). Another residual life prediction model study of slewing bearing can be found in [7]. The authors used PCA and support vector regression (SVR) to construct the model. In order to monitor the condition of slew bearings, a feature that represents the trends related to degradation condition from normal to failure is necessary. Due to the unique vibration signal generated by the roller and raceway contact in slew bearing, a suitable and sensitive feature extraction method is mandatory in slew bearing fault diagnosis and prediction. There are numerous existing methods, which are divided into different categories. For instance, Henao et al. [8] presented diagnostic methods in a non-stationary condition. The methods are classiﬁed into four approaches: (1) frequency domain approach; (2) time domain analysis; (3) slip-frequency; (4) diagnosis in the time-frequency domain. This paper particularly presents an empirical study of feature extraction methods for vibration signals acquired from naturally degraded slew bearings operating at very low rotational speed. In terms of feature extraction methods, six categories are reviewed, namely (i) time-domain features extraction; (ii) frequency-domain features extraction; (iii) time-frequency representation; (iv) phase-space dissimilarity measurement; (v) complexity measurement; and (vi) other features. Among the six categories, the time-domain features extraction method is found to be the most dominant method in typical rolling element bearings. In addition, the third and fourth methods are suitable for non-stationary, non-linear and chaotic signals.

2. Laboratory Slew Bearing Experiment

2.1. Slew Bearing Test-Rig and Sensor Location The vibration data employed in this paper were acquired from a slew bearing test rig [1]. The test rig was designed to replicate real working conditions of a slew bearing in a steel company, such as a high-applied load and low rotational speed. The slew bearing used was a large thrust bearing with an inner and outer diameter of 260 mm and 385 mm, respectively. The rotational speed of the lab test rig can be operated from 1 to 12 r/min. A photo of the test rig and sensor location on the slew bearing is shown in Figure1. Machines 2017, 5, 21 3 of 28 Machines 2017, 5, 21 3 of 28 Machines 2017, 5, 21 3 of 28

FigureFigure 1. 1.Photo Photoof ofslew slew bearingbearing lab test rig rig and and sensor sensor location. location. Figure 1. Photo of slew bearing lab test rig and sensor location. 2.2.2.2. Data Data Acquisition Acquisition Procedure Procedure 2.2. Data Acquisition Procedure The vibration data for continuous rotation was collected from four accelerometers that were The vibration data for continuous rotation was collected from four accelerometers that installedThe vibrationon the inner data diameter for continuous surface rotation at 90° wasto each collected other fromwith four4880 accelerometers Hz sampling thatrate. wereThe were installed on the inner diameter surface at 90◦ to each other with 4880 Hz sampling accelerometersinstalled on the were inner IMI diameter 608A11 ICP surface (IMI Sensors,at 90° to PCB each Piezotronics other with Division, 4880 Hz New sampling York, NY,rate. USA) The rate. The accelerometers were IMI 608A11 ICP (IMI Sensors, PCB Piezotronics Division, New York, typeaccelerometers sensors. The were accelerometers IMI 608A11 ICP were (IMI connected Sensors, PCBto a Piezotronicshigh-speed PicoScope Division, NewDAQ York, (PS3424) NY, USA)(Pico NY, USA) type sensors. The accelerometers were connected to a high-speed PicoScope DAQ (PS3424) Technology,type sensors. St The Neots, accelerometers United Kingdom). were connected The bearing to wasa high-speed subjected PicoScopeto an axial DAQload of(PS3424) 15 tons. (Pico The (PicobearingTechnology, Technology, began St runningNeots, St Neots, United in UK2006). Kingdom). Thewith bearing rotational The was bearing sp subjectedeed wasof 1 subjectedr/min. to an However, axial to an load axial to of provideload 15 tons. of 15continuous The tons. bearing The beganmonitoringbearing running began and in running 2006produce with in ru rotational2006n-to-failure with speedrotational bearing of 1 data, r/min.speed the of However,bearing 1 r/min. data However, to providewas collected to continuous provide on a continuousdaily monitoring basis andfrommonitoring produce February and run-to-failure to produce August ru2007n-to-failure bearing with total data, bearing number the bearingdata, of 139 the days. databearing wasTo data accelerate collected was collected the on bearing a dailyon a service daily basis basis life, from Februarycoalfrom dust February to was August injected to August 2007 intowith 2007 the total bearingwith number total in numbermid-April of 139 days. of 2007139 To days. (58 accelerate days To accelerate from the the bearing beginning).the bearing service Inservice life, practice, coal life, dust wasespeciallycoal injected dust was intoin steel-making injected the bearing into companie inthe mid-April bearings, the in 2007 mid-Aprilslew (58 bearing days 2007 from is (58locatedthe days beginning). infrom open the air beginning). In where practice, the In especiallybearing practice, is in steel-makingexposedespecially to in companies,a steel-makingdusty environment the slewcompanie bearing ands, thefor is locatedslewthis reasonbearing in open the is airlocatedcoal where dust inthe wasopen bearing inserted air where is exposedin themid-April bearing to a dusty tois environmentsimulateexposed tothe anda realdusty for working this environment reason conditions. the and coal Thefor dust thisschematic was reason inserted ofthe the incoal mid-Aprilvibration dust was data to inserted simulate acquisition in the mid-April realfrom working slew to conditions.bearingsimulate test-rig Thethe real schematic is presentedworking of theconditions.in Figure vibration 2. The data schematic acquisition of the from vibration slew bearing data acquisition test-rig is presentedfrom slew in Figurebearing2. test-rig is presented in Figure 2. Speed sensor Speed sensor

Belt Gear configuration Belt Gear configuration Drive ring DriveSlewing ring bearing Slewing bearing Hidraulic pressure Belt Hidraulic pressure Belt

Motor Motor

Figure 2. Schematic diagram of slew bearing test-rig and data acquisition flow process. FigureFigure 2. 2.Schematic Schematic diagram diagram of ofslew slew bearingbearing test-rig and data data acquisition acquisition flow ﬂow process. process.

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3. Features Extraction Methods and Its Application on Slew Bearing Vibration Signal

3.1. Category 1: Time-Domain Features Extraction

3.1.1. Statistical Time-Domain Features Statistical time-domain features such as mean, root mean square (RMS), standard deviation and variance were usually used in past studies to identify the differences between one vibration signal and another. More advanced statistical-based features such as skewness and kurtosis can be applied to the signal which is not purely stationary. These features examine the probability density function (PDF) of the signal. It is a well-known fact that if the condition of the bearing changes, the PDF also changes, thus the skewness and kurtosis might also be affected. In particular, skewness is used to measure whether the signal is negatively or positively skewed, while kurtosis measures the peak value of the PDF and indicates if the signal is impulse in nature. For a signal with a normal distribution i.e., normal bearing signal has a skewness value of zero. In addition, kurtosis is obtained from the peak of the PDF of the vibration signal, while skewness is obtained from the mean value of the PDF of the vibration signal. It is well known that the kurtosis value of vibration signal from normal bearing is approximately three [9] and the skewness value of approximately zero. Then, when the PDF of vibration signal changes due to faults, the kurtosis value will increase to greater than three and the skewness value will shift to either negative or positive. Other features which can be calculated from PDF of the vibration signal includes: entropy, which calculates the histogram of the PDF and measures the degree of randomness of the vibration signal; lower bound and upper bound histogram, which measure the lower and upper values of the PDF respectively. Aside from the statistical time-domain features mentioned above, there are other non-dimensional features such as shape factor (deﬁnes as RMS divided by mean) and crest factor (deﬁnes as standard deviation divided by RMS). Both features will change as the mean, RMS and standard deviation change. The summary of the statistical time-domain features is presented in Table1.

Table 1. Brief review of statistical time-domain features extraction applied in condition monitoring of typical rolling element bearings [10].

Description Feature Name Brief Deﬁnition Formula

The RMS value increase gradually as fault developed. However, RMS is unable q RMS to provide the information of incipient fault stage while it increases with the fault 1 N 2 RMS = N ∑i=1 xi development [11]. N ( − )2 Variance Variance measures the dispersion of a signal around their reference mean value. = ∑i=1 xi m Var (N−1)σ2

Skewness quantiﬁes the asymmetry behavior of vibration signal through its N ( − )3 Skewness Sk = ∑i=1 xi m probability density function (PDF). (N−1)σ3

Kurtosis quantiﬁes the peak value of the PDF. The kurtosis value for normal N ( − )4 Kurtosis Ku = ∑i=1 xi m rolling element bearing is well-recognized as 3. (N−1)σ4 q Shape factor is a value that is affected by an object’s shape but is independent of 1 N x2 Shape factor N ∑i=1 i SF = 1 N its dimensions [12]. N ∑i=1|xi |

max |xi | Crest factor (CF) calculates how much impact occur during the rolling element CF = q Crest factor 1 N 2 and raceway contact. CF is appropriate for “spiky signals” [12]. N ∑i=1 xi

Entropy, e(p), is a calculation of the uncertainty and randomness of a sampled n Entropy vibration data. Given a set of probabilities, (p1, p2,..., pn), the entropy can be e(p) = − ∑ p(zi) log2 p(zi) calculated using the formulas as shown in the right column. i=1 Machines 2017, 5, 21 5 of 28 Machines 2017, 5, 21 5 of 28

3.1.2.3.1.2. Upper Upper and and Lower Lower Bound Bound of of Histogram Histogram TheThe application application of theof histogramsthe histograms or a discreteor a discre PDFte forPDF time-domain for time-domain feature feature extraction extraction is presented is presented in [13]. Let d be the number of division that group the ranges, and h with 0≤

hU = =+Δmax(xi) + ∆/2 (3) hxUimax( ) / 2 (3)

hL = =−Δmax(xi) − ∆/2 (4) hxLimax( ) / 2 (4) where ∆ =Δ= (max(max((xxxni) )− − min(min(x ))i)) / (/( −n 1)− 1). where ii . According to the features extraction results presented in Figure3, some features, such as RMS, According to the features extraction results presented in Figure 3, some features, such as RMS, kurtosis, histogram upper bound and histogram lower bound, show the ﬂuctuation in the last kurtosis, histogram upper bound and histogram lower bound, show the fluctuation in the last measurement day, approximately from the 90th day to the 139th day. These indicate that some measurement day, approximately from the 90th day to the 139th day. These indicate that some features features are sensitive to the slew bearing condition while others are less. The incipient damage are sensitive to the slew bearing condition while others are less. The incipient damage occurred in day occurred90th that in is dayclearly 90th indicated that is by clearly RMS, kurtosis indicated, histogram by RMS, upper kurtosis, bound histogram and histogram upper lower bound bound and histogramfeature, lowerwhere boundthe fluctuation feature, whereis associated the ﬂuctuation with roller is associated and outer with race roller defects. and However, outer race more defects. However,appropriate more and appropriate reliable features and reliable are necessary features for arethe necessarycondition monitoring for the condition of slew monitoring bearing to detect of slew bearingan impending to detect damage an impending and to es damagetimate the and degradation to estimate index. thedegradation index.

40 1

20 0 RMS Skewness

0 -1 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

200 2000

100 0 Kurtosis Shape factor 0 -2000 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

4 x 10 2 5

0 0 Entropy Crest factor -2 -5 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

0 400

-200 200 Histogram lower Histogram upper -400 0 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 t, days t, days

FigureFigure 3. 3.Statistical Statistical time-domain time-domain featuresfeatures calculatedcalculated from the the vi vibrationbration slew slew bearing bearing signal. signal.

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3.1.3. Autoregressive (AR) Coefficients 3.1.3. Autoregressive (AR) Coefficients In other studies [13], autoregressive (AR) coefficients of the vibration signal are calculated as In other studies [13], autoregressive (AR) coefficients of the vibration signal are calculated bearing vibration features. It is known that a faulty vibration signal of typical rolling element bearing as bearing vibration features. It is known that a faulty vibration signal of typical rolling element produces different autoregressive coefficients compared to that of normal vibration signal. The bearing produces different autoregressive coefficients compared to that of normal vibration signal. following is a common univariate time series model to calculate the AR coefficients: The following is a common univariate time series model to calculate the AR coefficients: n =+ +++=+εεn yayaytt11−− 2 t 2... ay ntntitt − ay − 1 (5) = yt = a1yt−1 + a2yt−2 + ... + anyt−n + εt i=1 ∑ aiyt−1 + εt (5) i=1 where a1 to a n are the autoregressive coefficients, yt is the digitized vibration signal, n is the orderwhere ofa 1theto ARan are model the autoregressive( n=8) and ε iscoefficients, the residualyt (Gaussianis the digitized white vibration noise). signal, n is the order of the ARThe model RMS, (nskewness,= 8) and εkurtosis,is the residual entropy (Gaussian and AR white coefficients noise). have been applied in vibration datasetsThe from RMS, induction skewness, motor kurtosis, [14] entropy and rolling and ARelement coefficients bearing have [15]. been These applied papers in vibrationhave made datasets great contributionsfrom induction to motor the [research14] and rolling areas elementon the bearingbearing [15 condition]. These papers monitoring have made and great fault contributions diagnosis. However,to the research the features areas on were the bearing applied condition for artificial monitoringly bearing and faultdamage. diagnosis. In practice, However, the thedamage features is typicallywere applied developed for artificially naturally bearing rather damage.than artificially. In practice, the damage is typically developed naturally ratherThe than AR artificially. coefficients results are presented in Figure 4. It is shown that most of the features are less relevantThe AR coefficientsto represent results the slew are presentedbearing condit in Figureion 4except. It is shown for AR that coefficient most of the4 which features is areshown less fluctuatingrelevant to in represent the last themeasurement. slew bearing condition except for AR coefficient 4 which is shown fluctuating in the last measurement.

1 0.5

0 0 AR coef 1 AR coef 2

-1 -0.5 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

1 0.2

0 0 AR coef 3 AR coef 4

-1 -0.2 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

0.5 0.5

0 0 AR coef 5 AR coef 6

-0.5 -0.5 0 50 100 150 0 50 100 150

0.2 0.5

0 0 AR coef7 AR coef8

-0.2 -0.5 0 50 100 150 0 50 100 150 t, days t, days

Figure 4. AR features (up to 8 coefﬁcients) calculated from the vibration slew bearing signal. Figure 4. AR features (up to 8 coefficients) calculated from the vibration slew bearing signal.

Other time domain features such as impulse factor (IF) and margin factor (MF) have been recently used [12,16,17]. Similar to CF, IF is used to measure how much impact is generated from the bearing defect. The CF measures the impact by dividing the maximum absolute value of vibration

MachinesMachines 20172017, ,55, ,21 21 77 of of 28 28 signal and RMS value, while IF divides the maximum absolute value by the mean of absolute value. The formulaOther time is presented domain features as follows: such as impulse factor (IF) and margin factor (MF) have been recently used [12,16,17]. Similar to CF, IF is used to measure how much impact is generated from the bearing defect. The CF measures the impact by dividing themax maximumxi absolute value of vibration signal and IF = 1 N (6) RMS value, while IF divides the maximum absolute valuex by the mean of absolute value. The formula  i=1 i is presented as follows: N max|x | Margin factor (MF) measures the levelIF of= impact betweeni rolling element and raceway. The MF(6) 1 N |x | is calculated by dividing the maximum absoluteN ∑vai=lue1 ofi vibration signal to the RMS of absolute valueMargin of vibration factor signal (MF) measures(7): the level of impact between rolling element and raceway. The MF is calculated by dividing the maximum absolute value of vibration signal to the RMS of absolute value max x of vibration signal (7): = i MF 2 1 maxN |xi| (7) MF = x (7)   i=1 i 2 N1 N p N ∑i=1 |xi| TheThe progression ofof slew slew bearing bearing deterioration deterioration of IF of and IF MFand features MF features are more are apparent more comparedapparent comparedto RMS, kurtosis to RMS, and kurtosis histogram and histogram lower features lower as features presented as presented in Figure5 .in Figure 5.

Impulse factor 20

0 0 20 40 60 80 100 120 140

150

100

50 Margin factor Margin

0 0 20 40 60 80 100 120 140 t, days

FigureFigure 5. 5. ImpulseImpulse factor factor (IF) (IF) and and margin margin factor factor (MF) (MF) feature feature calculated calculated from from the the vibration vibration slew slew bearingbearing signal. signal.

3.1.4.3.1.4. Hjorts’ Hjorts’ Parameters Parameters InIn addition addition to to the the time-domain time-domain features features mentione mentionedd in in previous previous subsections, subsections, Hjorts’ Hjorts’ parameters parameters alsoalso falls falls into into this this category category [18]. [18]. These These features features are are calculated calculated based based on on the the first ﬁrst and and the the second second derivativesderivatives of of the the vibration vibration signal. signal. In In the the time time seri serieses context, context, the the numerical numerical values values for for the the derivatives derivatives areare obtained obtained as as the the differences differences between between the the current current value value and and the the prior prior value. value. The The first ﬁrst Hjorth’s Hjorth’s parameterparameter is is called called ‘activity’. ‘activity’. The The activity activity of of vibration vibration signal signal can can be be computed computed by by simply simply calculating calculating 2 σ 2 σ thethe variance ofof thethe vibration vibration signal signal amplitude amplitudeσx , wherex , whereσx is thex standardis the standard deviation deviation of the sampled of the = sampledvibration vibration signal x =signal (x1, xx2, x3(x,...,123 ,xx ,xN ,). ..., xN ) . The ‘mobility’ parameter of vibration signal is calculated as the square root of the ratio of the activity of the first derivative and the activity of the vibration signal.

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Machines 2017, 5, 21 8 of 28 The ‘mobility’ parameter of vibration signal is calculated as the square root of the ratio of the activity of the ﬁrst derivative and the activity of the vibration signal. σ mobility = x' σx0 (8) mobility = x (8) σx σ where ' is the standard deviation of the first derivative of the vibration signal. The numerical where σx0 xis the standard deviation of the ﬁrst derivative of the vibration signal. The numerical values forvalues the derivativesfor the derivatives can be obtainedcan be obtained as the differences as the differences between between samples samples as follows. as follows.

0 ' =− =− xx= xxx((1)iii+1) − x i i =iN1,1, ...... , , N − 1 (9) (9)

The lastlast parameterparameter called ‘complexity’ which is calculated as the ratio of mobility of the ﬁrstfirst derivative andand thethe mobilitymobility ofof thethe vibrationvibration signal.signal. σσ/ σ x00"'/σx 0 complexitycomplexity== x x (10)(10) σσσ 0 /σ x' x

These parametersparameters have have been been used used in electroencephalography in electroencephalography (EEG) (EEG) signals signals to detect to the detect epileptic the seizuresepileptic [ 19seizures]. They [19]. have neverThey have been usednever in been vibration used bearing in vibration signal exceptbearing for signal activity except feature, for whichactivity is similarfeature, to which the variance is similar feature to the in variance the statistical feature time-domain in the statistical features time-domain extraction. features It can beextraction. seen from It Figurecan be 6seen that from the progression Figure 6 that of the bearing progression condition of bearing is difﬁcult condition to track is usingdifficult mobility to track and using complexity mobility feature.and complexity In contrast, feature. the activityIn contrast, or variance the activity feature or shows variance one feature highest shows peak atone approximately highest peak the at 90thapproximately day and continue the 90th ﬂuctuationday and continue on the lastfluctuation measurement on the days.last measurement days.

600

400

200 Activity

0 0 20 40 60 80 100 120 140 1.5

1 Mobility

0.5 0 20 40 60 80 100 120 140 4

2 Complexity 1 0 20 40 60 80 100 120 140 t, days

Figure 6. Hjorts’ features calculated from the vibration slew bearing signal. Figure 6. Hjorts’ features calculated from the vibration slew bearing signal. 3.1.5. Mathematical Morphology (MM) Operators 3.1.5. Mathematical Morphology (MM) Operators In image processing, mathematical morphology (MM) was introduced as a non-linear method to In image processing, mathematical morphology (MM) was introduced as a non-linear method analyze two dimensional (2D) image data including binary images and grey-level images based on set to analyze two dimensional (2D) image data including binary images and grey-level images based theory [20]. The fundamental principle of the MM method is to modify the shape of the original signal on set theory [20]. The fundamental principle of the MM method is to modify the shape of the original by transforming it through its intersection with another object called the ‘structuring element’ (SE). signal by transforming it through its intersection with another object called the ‘structuring element’ Four operators including erosion, dilation, closing and opening are used to achieve the transformation. (SE). Four operators including erosion, dilation, closing and opening are used to achieve the transformation. After having been applied successfully in 2D image processing, the MM method has been used in biomedical research field to analyze one dimensional (1D) EEG signals [21–23] and electrocardiography (ECG) signals [24]. The first known study of MM in 1D vibration signals was

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After having been applied successfully in 2D image processing, the MM method has been used in biomedical research ﬁeld to analyze one dimensional (1D) EEG signals [21–23] and electrocardiography (ECG) signals [24]. The ﬁrst known study of MM in 1D vibration signals was presented by Nikolaou and Antoniadis [25]. The authors used the MM method for vibration data with inner race and outer race faults. The effects of four MM operators on the enveloping impulse-type signals were examined. The result shows that the ‘closing’ operator is more suitable for extracting the impulse signal corresponding to the bearing fault type. The SE value was empirically investigated within the range of 0.2 T to 1.2 T. The optimum SE for a signal with additive noise was selected as 0.6 T (0.6 times the pulse repetition period). In addition, T is the period of each bearing component frequency such as BPFO, BPFI and BSF. The original discrete one dimensional (1D) signal and structuring element (SE) must be first determined to use the four basic operators in the MM method. Suppose the discrete 1D vibration signal x = {x(1), x(2),..., x(N)}, which functions over a domain x(n) = (1, 2, . . . , N). Let S = {S(1), S(2),..., S(M)} be the SE, which is the discrete function over a domain S(m) = (1, 2, . . . , M), where N and M are integers and N ≥ M. Then, the four morphology operators for the discrete 1D vibration signal can be defined as follows:

• Erosion: also refer to as min ﬁlter.

(xΘS)(n) = min(x(n + m) − S(m)) (11)

• Dilation: also refer to as max ﬁlter.

(x ⊕ S)(n) = max(x(n − m) + S(m)) (12)

• Closing: Dilates 1D signal and then erodes the dilated signal using the similar structuring element for both operations. (x•S)(n) = ((x ⊕ S)ΘS)(n) (13)

• Opening: Erodes 1D signal and then dilates the eroded signal using the similar structuring element for both operations. (x ◦ S)(n) = ((xΘS) ⊕ S)(n) (14) where n ∈ (1, 2, ... , N) and m ∈ (1, 2, ... , M), the notations Θ, ⊕, •, ◦ denote the erosion operator or Minkowski subtraction, dilation operator or Minkowski addition, closing operator and opening operator in the MM method, respectively. The original simulated 1D signal is shown in the first row of Figure7. The various effects of the basic MM operators application in the simulated 1D vibration signal are presented in the second to fifth row of Figure7. The second row of Figure7 to the fifth row of Figure7 illustrate the effect of erosion, dilation, closing and opening, respectively. Please note that the solid blue line represents the original signal and the dashed red line is the signal after being processed or transformed by the MM operators. The simulated signal has zero mean value, which is typical to 1D vibration signals. The work principle of the basic MM operators can be summarized as follows: erosion of x(n) by the structuring element A(m) reduces the positive part and enlarges the negative part of x(n), while dilation of x(n) by the structuring element A(m) reduces the negative part and enlarges the positive part of x(n). In addition, closing of x(n) smoothens the signal x(n) by cutting down any impulses in the negative part and opening x(n) smoothens the signal x(n) by cutting down its impulses in the positive part. In rolling element bearing, the MM method has been studied. The multi-scale morphological analysis was developed by Zhang et al. [26]. The method shows independency of empirical rules in terms of SE selection, but the technique is rather complex. Wang et al. [27] proposed an improved morphological filter as features of periodic impulse signals. An average weighted combination of open-closing and close-opening morphological operators was employed to eliminate the statistical Machines 2017, 5, 21 10 of 28 terms of SE selection, but the technique is rather complex. Wang et al. [27] proposed an improved Machines 2017, 5, 21 10 of 28 morphological filter as features of periodic impulse signals. An average weighted combination of open-closing and close-opening morphological operators was employed to eliminate the statistical deflectiondeflection of amplitude and to extract the impulse component from the original signal. To optimize the SE,SE, theythey usedused a a new new criterion criterion called called impulse impulse attenuation attenuation (IMA). (IMA). Dong Dong et al. et [ 28al.] presented[28] presented the fault the faultdiagnosis diagnosis of rolling of elementrolling bearingselement usingbearings the modifiedusing the morphological modified morphological method. The morphologicalmethod. The morphologicaloperator used in operator their method used in was their the method average was weight thecombination average weight of the combination closing and of opening the closing operator. and Theyopening proposed operator. a new They criterion proposed called a new SNR criterion criterion called to optimize SNR criterion the length to optimize of SE. Anotherthe length recently of SE. publishedAnother recently research published on MM method research was on presented MM method by Raj was and presented Murali [29 by]. TheRaj authorsand Murali combined [29]. The the authorsMM operators combined with the fuzzy MM inference operators to with classify fuzzy the infere artificialnce bearingto classify defects. the artificial The vibration bearing data defects. used Thein [29 vibration] were similar data used to the in data [29] usedwere insimila Zhangr to et the al. data [26]. used in Zhang et al. [26].

10 5 0

Amplitude -5 0 5 10 15 20 25 30 35 40 45 50 55 10 5 0

Amplitude -5 0 5 10 15 20 25 30 35 40 45 50 55 Data number (n)

Figure 7. Illustration of four MM operators (second to ﬁfth row). Figure 7. Illustration of four MM operators (second to fifth row).

The application of the MM method in detectingdetecting the bearing fault frequencies from among impulsive-type signals signals have have been been demonstrated demonstrated [26–29]. [26– 29These]. These signals signals are generated are generated from known from bearingknown bearingconditions conditions such as inner such and/or as inner outer and/or race faults. outer raceIt is acknowledged faults. It is acknowledged that the signals that from the artificialsignals fromfaults artificial are easier faults to be are identified easier tothan be those identified from natural than those bearing from faults. natural Besides, bearing typically, faults. theBesides, impulse-type typically, signals the impulse-type are easily identified signals visually are easily without identified any addi visuallytional without tool or processing any additional step [28].tool orThe processing application step of [ 28MM]. The method application to the natura of MMlly method damaged to the slew naturally bearing damaged is presented slew in bearing Figures is presented8–10. in Figures8–10. The MMMM methodmethod has has been been used used effectively effectively for thefor damagethe damage detection detection of high of speed high rollingspeed bearing.rolling bearing.It is because It is whenbecause the when damage the occurred,damage occurred the contact, the between contact between the damage the spotdamage and spot rolling and element rolling elementgenerates generates detectable detectable impulse dueimpulse to the due contact to the between contact the between defect spotthe defect and race spot way and or roller.race way In theor roller.case of In naturally the case degradedof naturally low-speed degraded slew low-speed bearing, slew the bearing, multiple the damage multiple e.g., damage outer race,e.g., outer inner race, race innerand roller race areand typicallyroller are occurred typicallyat occurred close sequence at close timesequence and thetime magnitude and the magnitude of impulse of is impulse very low. is Thus,very low. it is Thus, difficult it is to difficult identify to the identify origin the of impulseorigin of fromimpulse outer from race, outer inner race, race inner or roller race or damage. roller Therefore,damage. Therefore, the feature the extraction feature extraction result of MM result for of the MM three for conditions: the three conditions: BPFO, BPFI BPFO, and BSF BPFI present and BSF an presentidentical an figure. identical figure.

Machines 2017, 5, 21 11 of 28 Machines 2017, 5, 21 11 of 28 Machines 2017, 5, 21 11 of 28

Figure 8. RMS feature calculated from MMMM operatorsoperators forfor 0.90.9 TT (T(T == 1/BPFI):1/BPFI): ( (aa)) erosion; erosion; ( (b)) dilation; Figure 8. RMS feature calculated from MM operators for 0.9 T (T = 1/BPFI): (a) erosion; (b) dilation; (c)) closingclosing andand ((d)) opening.opening. (c) closing and (d) opening.

Figure 9. RMS feature calculated from MM operators for 0.9 T (T = 1/BPFO): (a) erosion; (b) dilation; Figure(c) closing 9. RMS and feature (d) opening. calculated from MM operators for 0.9 TT (T(T == 1/BPFO): ( (aa)) erosion; erosion; ( (bb)) dilation; dilation; (c) closing and (d) opening.

Machines 2017, 5, 21 12 of 28 Machines 2017, 5, 21 12 of 28

Figure 10. RMS feature calculated from MMMM operatorsoperators forfor 0.90.9 TT (T(T == 1/BSF):1/BSF): ( (aa)) erosion; erosion; ( (b)) dilation; (c)) closingclosing andand ((d)) opening.opening.

3.2. Category 2: Frequency-Domain Features Extraction 3.2. Category 2: Frequency-Domain Features Extraction To apply frequency-domain features, the time-domain vibration signals must initially be To apply frequency-domain features, the time-domain vibration signals must initially be transformed into frequency-domain signals using fast Fourier transform (FFT). FFT is a common transformed into frequency-domain signals using fast Fourier transform (FFT). FFT is a common method in vibration analysis for bearing fault detection. FFT measures the dominant frequency of the method in vibration analysis for bearing fault detection. FFT measures the dominant frequency of the repetitive impulse period of certain faults due to the contact between rolling elements and defective repetitive impulse period of certain faults due to the contact between rolling elements and defective spot. Therefore, the type of bearing defects such as outer race, inner race or ball defect can be detected. spot. Therefore, the type of bearing defects such as outer race, inner race or ball defect can be detected. FFT performed effectively in stationary periodic signals; however, it is less effective for non- FFT performed effectively in stationary periodic signals; however, it is less effective for non-stationary stationary signals that arise from time-dependent events [30]. signals that arise from time-dependent events [30].

3.2.1. Statistical Frequency-Domain Features Other frequency-domain features such as freque frequencyncy centre (FC), root mean square frequency (RMSF) and root variance frequency (RVF)(RVF) havehave beenbeen studiedstudied inin [[13].13]. When fa faultult exists, the frequency element changes,changes, andand thethe values values of of the the FC, FC, RMSF RMSF and and RVF RVF also also change. change. The The FC FC and and RMSF RMSF indicate indicate the positionthe position changes changes of main of main frequencies, frequencies, while thewhile RVF the shows RVF theshows convergence the convergence of the power of the spectrum. power Anotherspectrum. feature Another called feature median called frequency median is frequency sometimes is used.sometimes The median used. The frequency median is measuresfrequency byis dividingmeasures the by areadividing of the the amplitude area of the spectrum amplitude into sp twoectrum equal into part. two In addition,equal part. the In mean addition, frequency the mean (FC) canfrequency be deﬁned (FC) ascan the be ﬁrst-order defined as moment the first-order of the mo Fourierment spectrum, of the Fourier RVF spectrum, as the second-order RVF as the moment, second- SSorder as themoment, third-order SS as the moment third-order and SK moment as the fourth-orderand SK as the moment fourth-order of the moment Fourier spectrumof the Fourier [18]. Thesespectrum features [18]. canThese be features calculated can as be follows: calculated as follows: N N xx'0 ∑i=i=2 xiixi FCFC== i (15) NN 2 (15) 2π ∑i= x  i=1 ii

N ()x'2 =  i=2 i MSF N (16) 4π 22x  i=1 i

Machines 2017, 5, 21 13 of 28

2 ∑N (x0) MSF = i=2 i (16) 2 N 2 Machines 2017, 5, 21 4π ∑i=1 xi 13 of 28 √ RMSFRMSF== MSFMSF (17) (17) p 2 RVFRVF==−MSF − FCFC2 (18) (18) The resultThe result of FC, of RMSF FC, RMSF and and RVF RVF feature feature extraction extraction is is presented presented in Figure in Figure 11. 11.

1000

500 FC

0 0 20 40 60 80 100 120 140 1500

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400 0 20 40 60 80 100 120 140 t, days

FigureFigure 11. 11.Frequency-domain Frequency-domain features: features: FC, RMSF FC, and RMSF RVF andcalculated RVF from calculated the vibration from slew the bearing vibration signal. slew bearing signal. 3.2.2. Spectral Skewness, Spectral Kurtosis, Spectral Entropy and Shannon Entropy Feature 3.2.2. SpectralRecently, Skewness, new methods Spectral such Kurtosis, as spectral Spectral skewne Entropyss, spectral and Shannonkurtosis, spectral Entropy entropy Feature and Shannon entropy have been developed. Spectral skewness (SS) and spectral kurtosis (SK) are the Recently,advanced new statistical methods measures such asapplied spectral to the skewness, magnitude spectral spectrum. kurtosis, The SS measures spectral the entropy symmetry and of Shannon entropythe have distribution been developed. of the spectral Spectral magnitude skewness values around (SS) andits mean spectral [31]. It kurtosis is defined (SK) by are the advanced Β− statistical measures applied to the magnitudeL /2 1 spectrum. The SS measures the symmetry of the −μ 3 2(|(,)|) Xkn ||X distribution of the spectral magnitudeSS( valuesn ) = aroundk=0 its mean [ 31]. It is deﬁned by (19) Β⋅σ 3 LX|| BL/2−1 3

The SK measures the distribution of2 the∑ spectr( Xal (magnitudek, n) −µ| Xvalues|) and compare to a Gaussian distribution. It is defined by ( ) = k=0 SS n 3 (19) Β−/2 1 B · σ L L |X| −μ 4 2(|(,)|) Xkn ||X SK(n )=−k=0 3 (20) The SK measures the distribution of the spectralΒ⋅σ 4 magnitude values and compare to a Gaussian LX|| distribution. It is deﬁned by Antoni and Randall [32] presents the development of classical kurtosis analysis called spectral kurtosis (SK) to deal with the transient behaviour in a signal. SK can be used to detect incipient faults BL/2−1 4

even in the presence of resilient masking2 ∑noise.( ThX(ek ,conceptn) −µ |ofX |spectral) statistic is adopted in this = method to supplement the SKclassical(n) = powerk spectral0 density (PSD). PSD− 3ideally gives zeros values at (20) · 4 those frequencies when the signal has stationary BGauL ssianσ|X| noise, and it gives high positive values at those frequencies during the occurrence of the transients. When the noise-to-signal ratio is high, the Antonitransient and signal Randall will [be32 buried] presents in the the background development noise, ofthus classical the incipient kurtosis fault analysis cannot be called clearly spectral kurtosisdetected. (SK) to dealSK can with overcome the transient this problem behaviour by analyz ining a signal. the entire SK frequency can be used band to and detect to select incipient the faults sensitive frequency band that correspond to the bearing condition. even in the presence of resilient masking noise. The concept of spectral statistic is adopted in this method3.3. to Category supplement 3: Time-Frequency the classical Representation power spectral density (PSD). PSD ideally gives zeros values at those frequencies when the signal has stationary Gaussian noise, and it gives high positive values Time-frequency domain representation methods such as short-time Fourier transform (STFT), at thosewavelet frequencies transform during and Wigner-Ville the occurrence distribution of the (WVD) transients. are commonly When the used noise-to-signal for non-stationary ratio or is high, the transient signal will be buried in the background noise, thus the incipient fault cannot be clearly detected. SK can overcome this problem by analyzing the entire frequency band and to select the sensitive frequency band that correspond to the bearing condition. Machines 2017, 5, 21 14 of 28

3.3. Category 3: Time-Frequency Representation Time-frequency domain representation methods such as short-time Fourier transform (STFT), wavelet transform and Wigner-Ville distribution (WVD) are commonly used for non-stationary or transient signal. These methods implement a mapping of one-dimensional time-domain signals to a two-dimensional function of time and frequency. The objective is to provide a true time-frequency representationMachines 2017 of a, 5, signal. 21 A review of time-frequency analysis methods for machinery fault14 of diagnosis 28 has beentransient presented signal. by These Feng methods et al. [33 implement]. a mapping of one-dimensional time-domain signals to a two-dimensional function of time and frequency. The objective is to provide a true time-frequency 3.3.1. Short-Timerepresentation Fourier of a Transformsignal. A review (STFT) of time-frequency analysis methods for machinery fault Duediagnosis to its low has computationalbeen presented by complexity Feng et al. [33]. and definite physical meaning, STFT is often used as an initial pre-processing3.3.1. Short-Time tool Fourier to analyzeTransform non-stationary (STFT) vibration signals. The STFT maps the vibration signal into a two-dimensional (2D) function of time and frequency [34]. STFT divides a non-stationary Due to its low computational complexity and definite physical meaning, STFT is often used as signal into small windows of equal time frame. The Fourier transformation is then applied to the time an initial pre-processing tool to analyze non-stationary vibration signals. The STFT maps the segment.vibration Unlike signal FFT whereinto a two-dimensional the frequency (2D) transformation function of time includes and frequency the whole [34]. STFT original divides signal, a non- in STFT originalstationary signal is signal decomposed into small using windows FFT of at equal pre-defined time frame. time The intervals.Fourier transformation By doing this, is then the applied defect signal frequencyto the can time be identifiedsegment. Unlike in a particular FFT where windowthe freque ofncy time. transformation The key to includes this method the whole is the original selection of the timesignal, interval. in STFT The original STFT cansignal be is expressed decomposed as usin followsg FFT [at35 pre-defined]: time intervals. By doing this, the defect signal frequency can be identified in a particular window of time. The key to this method is the selection of the time interval. The STFTt+T/2 can be expressed as follows [35]: Z tT+ /2 j2π f τ STFT( f , t) = w(t − τ)x(τjf2πτ)e dτ (21) STFT(f ,twtxed )=− (ττ ) ( ) τ (21) t−T/2tT− /2 STFT (f ,t ) t f wt()−τ where STFTwhere( f , t) is short-term is short-term frequency frequency spectrum, spectrum,t is the is the time, time,f is frequency, is frequency,w (t − τ) isis thethe shifted shifted window, x()τ denotes the input vibration signal, exp(jf 2πτ ) is the complex exponential, window, x(τ) denotes the input vibration signal, exp(j2π f τ) is the complex exponential, T is the T is the window interval and w()τ is the windowing function which satisfies w()τ = 0 for window interval and w(τ) is the windowing function which satisfies w(τ) = 0 for |τ| => T/2. τ => T /2 By taking the. square magnitude of the STFT in (21), an energy density using the spectrogram S(t, f ) methodBy taking can be the obtained square magnitude of the STFT in (21), an energy density using the spectrogram S(tf , ) method can be obtained S(t, f ) = |STFT(t, f )|2 (22) 2 S(tf , )= STFT( tf , ) Equation (22) represents the energy density of the vibration signal x(τ) windowed by(22)w (τ). The visualizationEquation (22) of represents window the length energy and density hop of size the isvibration presented signal in x Figure()τ windowed 12. The by vibration w()τ . signal with the sampleThe visualization index i is split of window into overlapping length and hop blocks size is of presented length B inL andFigure block 12. The index vibrationn. While, signal the hop Β n size Hs iswith the the distance sample fromindex thei is start split ofinto one overlapping particular blocks block of and length its subsequentL and block block. index The. While, STFT result Η of one daythe hop slew size bearing s is the signal distance measurement from the start isof shownone particular in Figure block 13and for its subsequent 10 s vibration block. signal The with windowSTFT length result of 2048,of one hop day sizeslew ofbearing 256 and signal number measurement of FFT is points shown of in 16,384. Figure It13 canfor 10 be s seen vibration that during 10 s sampledsignal with signal, window the length amplitudes of 2048, athop lower size of frequencies256 and number were of FFT higher points than of 16,384. the amplitudeIt can be seen at high that during 10 s sampled signal, the amplitudes at lower frequencies were higher than the amplitude frequencies (approximately at 1300 Hz and 1750 Hz). This indicates that slew bearing frequencies were at high frequencies (approximately at 1300 Hz and 1750 Hz). This indicates that slew bearing dominatedfrequencies by low were frequency dominated contents. by low frequency contents.

Figure 12.FigureSchematic 12. Schematic visualization visualization of of hop hop size size andand overlapping block block length length in STFT in STFTalgorithm. algorithm.

Machines 2017, 5, 21 15 of 28 Machines 2017, 5, 21 15 of 28

Figure 13. STFT of the vibration slew bearing signal (data on 3 May).

3.3.2. Wavelet Transform andand WaveletWavelet DecompositionDecomposition Wavelet transformtransform is well-knownwell-known as anan appropriateappropriate methodmethod toto analyzeanalyze non-stationarynon-stationary and transient signal [[14,36].14,36]. WaveletWavelet transformationtransformation employsemploys waveletwavelet function instead of sinusoidalsinusoidal function as as the the basis basis function. function. It Itenhances enhances a scale a scale variable variable in addition in addition to the to time the timevariable variable in the in inner the 2 innerproduct product transformation transformation [33]. [33For]. Forany any limited limited energy energy signal signal xx()(tLRt)∈∈ L (2( )R, )the, the wavelet wavelet function function is defineddefined as [[26]26] 1 Z +∞ τ − t WTx(t, a) = √ +∞ x(τ)ψ τ − dτ (23) = a1 −∞ τ ψ ta τ WTx (ta , ) x ( ) d (23) −∞ a [( − ) ] a  ( ) where wavelet ψ τ t /a is derived by dilating and√ translating the wavelet basis ψ t , a is thewhere scale wavelet parameter ψ [(τ −(ata ) /> ]0 ) is, derivedt is the by time dilating shift and and 1translating/ a is a normalization the wavelet basis factor ψ () tot , maintaina is the energy preservation. scale parameter (0)a > , t is the time shift and 1/ a is a normalization factor to maintain energy A review of the wavelet transform applications in machine condition monitoring and preservation. fault diagnostics is presented by Peng et al. [37]. As a non-stationary signal analysis method, A review of the wavelet transform applications in machine condition monitoring and fault wavelet transform can be applied in machine fault diagnostics as a feature extraction method [37]. diagnostics is presented by Peng et al. [37]. As a non-stationary signal analysis method, wavelet Since the wavelet transformations can provide an excellent energy concentration properties due to transform can be applied in machine fault diagnostics as a feature extraction method [37]. Since the the uses of the wavelet basis function; the wavelet transform can represent the signal with a certain wavelet transformations can provide an excellent energy concentration properties due to the uses of number of coefficients. The process is usually called wavelet decomposition. the wavelet basis function; the wavelet transform can represent the signal with a certain number of Wavelet decomposition technique decomposes a non-stationary vibration signal into linear form coefficients. The process is usually called wavelet decomposition. of time-scale units. It decomposes and organizes original signal into several signal components Wavelet decomposition technique decomposes a non-stationary vibration signal into linear form according to the translation of the wavelet function. This changes the scale and shows the transition of time-scale units. It decomposes and organizes original signal into several signal components of each frequency component [38]. Through this decomposition, a selected coefficient can be used according to the translation of the wavelet function. This changes the scale and shows the transition directly as the fault features. In wavelet decomposition, the input signal is basically decomposed of each frequency component [38]. Through this decomposition, a selected coefficient can be used into two coefficients. The input signal that is passed through the low-pass filter becomes the directly as the fault features. In wavelet decomposition, the input signal is basically decomposed into ‘approximation coefficient’. The input signal which is passed through the high-pass filter becomes the two coefficients. The input signal that is passed through the low-pass filter becomes the ‘detail coefficient.’ Since the low-frequency component is the most sensitive part to extract the useful ‘approximation coefficient’. The input signal which is passed through the high-pass filter becomes information related to the bearing condition, the ‘approximate coefficient’ is passed to the next wavelet the ‘detail coefficient.’ Since the low-frequency component is the most sensitive part to extract the decomposition loop in multilevel wavelet decomposition process. Figure 14 shows three level wavelet useful information related to the bearing condition, the ‘approximate coefficient’ is passed to the next decomposition structures. The structure illustrates the decomposition of slew bearing vibration signal wavelet decomposition loop in multilevel wavelet decomposition process. Figure 14 shows three into approximate coefficient (A) and detailed coefficient (D) at each level. For further processing, level wavelet decomposition structures. The structure illustrates the decomposition of slew bearing features such as mean, variance, skewness and kurtosis are calculated from the detailed coefficients of vibration signal into approximate coefficient (A) and detailed coefficient (D) at each level. For further level 3 (D3) [39]. The features extraction results are presented in Figure 15, it can be seen that kurtosis processing, features such as mean, variance, skewness and kurtosis are calculated from the detailed coefficients of level 3 (D3) [39]. The features extraction results are presented in Figure 15, it can be seen that kurtosis can predict the damage compared to mean, variance and skewness calculated from

Machines 2017, 5, 21 16 of 28 Machines 2017, 5, 21 16 of 28 cansignal predict D3 of the the damage wavelet compared decomposition to mean, due variance to the significant and skewness changes calculated after day from 90 of signal the bearing D3 of the life wavelettime. decomposition due to the significant changes after day 90 of the bearing life time. TheThe application application ofof wavelet wavelet decomposition decomposition toto thethe transient transient stator stator current current signal signal acquired acquired from from inductioninduction motor motor is is presented presented by by Niu Niu et et al. al. [ 40[40].]. Statistical Statistical features features have have also also been been used used to to reveal reveal the the faultfault information information from from the the selected selected coefficients. coefficients. Another Another study study that that uses uses the waveletthe wavelet coefficients coefficients as the as faultthe fault features features for ball for bearingball bearing is presented is presented by Liu by et Liu al. et [41 al.]. [41].

200

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FigureFigure 14. 14.Three Three level level wavelet wavelet decompositions decompositions of of the the vibration vibration slew slew bearing bearing signal. signal.

Machines 2017,, 5,, 21 21 1717 of of 28 28

-5 x 10 5

0 Mean -5 0 20 40 60 80 100 120 140 200

100

Variance 0 0 20 40 60 80 100 120 140 0.2

Skewness -0.2 0 20 40 60 80 100 120 140 200

100 Kurtosis 0 0 20 40 60 80 100 120 140 t, days

Figure 15. Statistical-based features: mean, variance, skewness and kurtosis calculated from D3 signal Figureof wavelet 15. Statistical-based decomposition. features: mean, variance, skewness and kurtosis calculated from D3 signal of wavelet decomposition. 3.3.3. Empirical Mode Decomposition-Based Hilbert Huang Transform 3.3.3. Empirical Mode Decomposition-Based Hilbert Huang Transform Empirical mode decomposition (EMD) [42] has been used in some applications that involve Empirical mode decomposition (EMD) [42] has been used in some applications that involve non- non-stationary signals (see, e.g., Braun and Feldman [43]). A review of EMD applications in fault stationary signals (see, e.g., Braun and Feldman [43]). A review of EMD applications in fault diagnosis diagnosis of rotating machinery is discussed in [44]. The review gives a detailed introduction to EMD of rotating machinery is discussed in [44]. The review gives a detailed introduction to EMD application in various areas such as bearings, gears and rotors etc. application in various areas such as bearings, gears and rotors etc. The EMD decomposes the vibration signal into several signals from high to low The EMD decomposes the vibration signal into several signals from high to low frequencies frequencies content, namely intrinsic mode functions (IMFs) based on the enveloping technique, content, namely intrinsic mode functions (IMFs) based on the enveloping technique, i.e. Hilbert- i.e., Hilbert-Huang transform (HHT). From the speciﬁc frequencies content information, the bearing Huang transform (HHT). From the specific frequencies content information, the bearing fault fault frequency can be identiﬁed. In some preliminary studies on condition monitoring [39,45], frequency can be identified. In some preliminary studies on condition monitoring [39,45], the original the original EMD [46] is used for non-stationary slew bearing data. A bearing fault such as outer race, EMD [46] is used for non-stationary slew bearing data. A bearing fault such as outer race, inner race inner race or rolling element fault has occurred when one of the IMF frequencies is identical to one of or rolling element fault has occurred when one of the IMF frequencies is identical to one of the bearing the bearing fault frequencies (as shown in Table2)[ 45]. The EMD results for slew bearing data are fault frequencies (as shown in Table 2) [45]. The EMD results for slew bearing data are presented in presented in Table3. A detail EMD result is presented in [39,45]. Table 3. A detail EMD result is presented in [39,45]. Table 2. Fault frequencies of slew bearing test-rig (run at 1 r/min). Table 2. Fault frequencies of slew bearing test-rig (run at 1 r/min).

Fault FrequenciesFault Frequencies (Hz) (Hz) (Calculati (Calculationon is isGiven Given in in Appendix Appendix AA)) DefectDefect Mode Mode AxialAxial Radial Radial OuterOuter raceway raceway (BPFO) (BPFO) 1.32 1.32 0.55 0.55 InnerInner raceway raceway (BPFI) (BPFI) 1.37 1.37 0.55 0.55 Rolling element (BSF) 0.43 0.54 Rolling element (BSF) 0.43 0.54

Table 3. Decomposition result of EMD method for the slew bearing data. Table 3. Decomposition result of EMD method for the slew bearing data.

IMFs of EMD (Hz) Vibration Data IMFs of EMD (Hz) Vibration Data IMF2IMF2 IMF3 IMF3 IMF4 IMF4 ... … IMF11IMF11 IMF12IMF12 IMF13 IMF13 IMF14IMF14 IMF15 IMF15 24 February 641.81 694.52 390.35 . . . 2.47 1.44 0.69 0.33 1.99 243 May February 702.88641.81 684.34 694.52 346.23 390.35 . . . … 2.04 2.47 1.231.44 0.640.69 0.430.33 1.99 0.19 330 May August 651.72702.88 679.86 684.34 245.92 346.23 . . . … 1.37 2.04 0.681.23 0.330.64 0.100.43 0.19 30 August 651.72 679.86 245.92 … 1.37 0.68 0.33 0.10

MachinesMachines2017 2017,,5 5,, 21 21 1818 ofof 2828 Machines 2017, 5, 21 18 of 28 According to Table 3, the frequencies that correspond to the bearing fault condition did not appearAccordingAccording on 24 February. to to Table Table3 ,In the3, contrast, the frequencies frequencies the fault that that correspondfrequenc correspondy of to rolling the to bearing the element bearing fault (BSF conditionfault axial) condition didis identified not did appear not on on3appear May 24 February. at on 0.43 24 Hz.February. In This contrast, frequency In contrast, the fault is associatedthe frequency fault frequenc to of the rolling rollery of element rolling defects element (BSF as shown axial) (BSF in is identiﬁedFigureaxial) is 16a. identified on Interesting 3 May on at 0.43result3 May Hz. was at This 0.43 obtained frequency Hz. This from frequency is associateddata on is 30 associated to August, the roller whereto defectsthe roller the as BSF defects shown axial inas is Figureshownhidden 16in bya. Figure InterestingBPFI 16a.axial Interesting of result 1.37 was Hz. obtainedTheresult reason was from obtained could data onbe from 30that August,data the on amplitude where30 August, the of BSF where signal axial the is0.43 BSF hidden Hz axial is by istoo BPFIhidden weak axial by and ofBPFI 1.37 is axial covered Hz. of The 1.37 reasonby Hz. the couldbackgroundThe reason be that noise.could the amplitude Thebe that photo the of of signalamplitude the outer 0.43 Hzracewayof issignal too damage weak0.43 Hz and is is isshown too covered weak in Figure by and the is16b. background covered The roller by noise. theand Theouterbackground photo race damage of thenoise. outer photo The raceway photo was taken of damage the after outer final is raceway shown failure in duringdamage Figure the is16 bearingshownb. The in rollerdismantle Figure and 16b. on outer 2 The September. race roller damage and The photodetailedouter wasrace vibration takendamage after data photo ﬁnal used was failure from taken duringthe after slew final the bearing bearing failure test during dismantle rig is the presented bearing on 2 September. dismantlein [39]. Statistical The on detailed2 September. features vibration Thefrom detailed vibration data used from the slew bearing test rig is presented in [39]. Statistical features from datathe decomposition used from the slewresults bearing of EMD test method rig is presented is shown in in [39 Figure]. Statistical 17. The features mean, fromvariance, the decomposition skewness and the decomposition results of EMD method is shown in Figure 17. The mean, variance, skewness and resultskurtosis of features EMD method are calculated is shown from in Figure the summati 17. Theon mean, of the variance, low IMF skewness frequencies and (from kurtosis IMF features9 to the kurtosis features are calculated from the summation of the low IMF frequencies (from IMF 9 to the arelowest calculated IMF) offrom the EMD thesummation results. The ofselection the low of IMF these frequencies IMFs is based (from on IMFthe characteristic 9 to the lowest of slew IMF) bearing of the lowest IMF) of the EMD results. The selection of these IMFs is based on the characteristic of slew bearing EMDsignal results. being dominated The selection by low of these frequency IMFs componen is based onts theas presented characteristic previously of slew in bearing STFT result. signal being dominatedsignal being by dominated low frequency by low components frequency componen as presentedts as previously presented inpreviously STFT result. in STFT result.

FigureFigureFigure 16.16. ( ((aa)) AA photophoto ofof damageddamaged axialaxialaxial rollers;rollers; ((bb)) A A photo photo of ofof outer outerouter raceway racewayraceway damage. damage.damage.

0.1

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5 Variance Variance 0 0 20 40 6060 8080 100100 120120 140140 2

0 Skewness Skewness -2-2 0 20 40 6060 8080 100100 120120 140140 40

20 Kurtosis Kurtosis 0 0 20 40 6060 8080 100100 120120 140140 tt,, daysdays

FigureFigure 17.17. Statistical-basedStatistical-based featuresfeatures calculatedcalculated from from EMD EMD decomposition. decomposition.

3.3.4. Wigner-Ville Distribution (WVD) The Wigner distribution (WD) is derived from the relationship between the power spectrum and the autocorrelation function for time-variant and non-stationary processes [47]. The correlation

Machines 2017, 5, 21 19 of 28 function is a product of the function at a posterior time with the function at a subsequent time [48]. The original WD for real signal x(t) is deﬁned as [49]

Z +∞ ∗ τ τ −j2π f τ Wx(t, f ) = x t − x t + e dτ (24) −∞ 2 2

The symbol (∗) in Equation (24) denotes the complex conjugate. When the WD is applied for the analytical signal xA(t), it is called the Wigner-Ville distribution (WVD). The analytical signal is deﬁned as

xA(t) = x(t) + jxh(t) (25) where xh(t) is the Hilbert transform of x(t):

1 Z +∞ 1 H[x(t)] = xh(t) = x(τ) dτ (26) π −∞ t − τ

The WVD has been used for gear fault detection [47,48] and it is recently used for rolling element bearing to represent the time-frequency features of vibration signals [50].

3.4. Category 4: Phase-Space Dissimilarity Measurement Phase-space dissimilarity measurements, such as fractal dimension, correlation dimension approximate entropy and largest Lyapunov exponent (LLE), are developed based on the chaos theory. These methods have been applied in biomedical engineering studies [19,51] and vibration signals [52,53]. Present studies have conducted investigations into these methods in the slew bearing condition monitoring for features extraction. The motivation is that multiple defects occur more frequently than single defects in slew bearings, therefore the phase-space dissimilarity measurements can be used to analyze the irregular or chaotic vibration signals. The phase-space can be computed as follows:

  y1 y1+J y1+2J ... y1+(m−1)J  y y y ... y   2 2+J 2+2J 2+(m−1)J    X =  y3 y3+J y3+2J ... y3+(m−1)J  (27)    . . . .   . . . .  yM yM+J yM+2J ... yM+(m−1)J where J is reconstruction delay that can be calculated by time lag/∆t, m is the embedding dimension and M is the number of reconstructed vectors. The relation between N, J, m and M can be deﬁned as: N = M + (m − 1)J or M = N − (m − 1)J. Thus, the dimension of phase-space is a M × m matrix.

3.4.1. Fractal Dimension Fractal dimension is a method to quantify the signal complexity by characterizing fractal patterns. The complexity can be quantified by examining the dynamical behaviours of phase-space matrix. The frequently used algorithm of fractal dimension was proposed by Higuchi [54]. The field is rapidly growing as estimated fractal dimensions for statistically self-similar phenomena. This method has been applied in many fields including image analysis [55], neuroscience [56,57], medicine [58], physics [59] and acoustics [60]. The application of fractal dimension in vibration signal of rolling element bearing has been presented in [10,61]. The algorithm of fractal dimension is shown in (28) [10]. Once the Machines 2017, 5, 21 20 of 28 phase-space matrix, X is obtained (27), the mean absolute length between Jth and (Jth − 1) element of X is defined as follows: f or a = 1 : J f or b = 2 : m L (a, b − 1) = |X(a, b) − X(a, (b − 1))| 1 (28) Lm(a, b − 1) = mean(L1) end end where J and m is reconstruction delay and embedding dimension, respectively. The fractal dimension D can be calculated using standard least-square fitting method of the logarithmic natural of (1/m) and Lm.

3.4.2. Correlation Dimension Correlation dimension can be used to quantify the self-similarity of vibration signal. A large value of correlation dimension corresponds to less-similarity of vibration signal. An example application of correlation dimension in vibration signal for bearing fault diagnosis was proposed by Logan and Mathew [52]. The correlation dimension was applied for vibration signals acquired from four different conditions: (1) normal (new bearing); (2) outer race fault; (3) inner race fault; and (4) roller fault. The damages were artiﬁcially introduced to the bearing parts. The method is able to identify four different bearing conditions. The commonly used procedure to calculate the correlation dimension was introduced by Grassberger and Procaccia [62]. Similar to fractal dimension, the phase-space matrix (27) is fed to the correlation dimension algorithm as an input matrix. The formula for correlation dimensionis given by: " M−k M # 2 C(l) = lim Θ l Xi − Xj (29) x→∞ 2 ∑ ∑ M i=1 j=i+k where Xi and Xj are the position vectors on attractor, l is the distance under consideration, Θ(x) is the Heaviside step function, Θ(x) = 0 if X ≤ 0, Θ(x) = 1 if X > 0, k is the summation offset, M is the number of reconstructed vectors from the original vibration signal and C(l) is the correlation dimension.

3.4.3. Approximate Entropy The approximate entropy quantiﬁes the degree of regularity in the vibration signal. The smaller the value of the approximate entropy, the more the regularity of the behavior of the signal is. An application example of approximate entropy in vibration signal was proposed by Yan et al. [63] for high speed rolling element bearing. Moreover, Yan et al. [63] mentioned that the deterioration of bearing condition corresponds to the increase of number of frequency components. In addition, when the bearing starts to deteriorate the regularity will decrease and the corresponding approximate entropy value will increase. Similar to fractal dimension and correlation dimension, approximate entropy also uses phase-space matrix, X in (27) as an input. The initial step analysis of the approximate entropy algorithm is to measure the distance between two vectors X(i) and X(j) which can be deﬁned as the maximum difference in their respective corresponding elements [63]:

d(X(i), X(j)) = max (|x(i + k − 1) − x(j + k − 1)|) (30) k=1,2,...,m Machines 2017, 5, 21 21 of 28 where i = 1, 2, ... , N − m + 1, j = 1, 2, ... , N − m + 1 and N is the number of data points in the vibration signal. A calculation that describes the similarity between the vector X(i) and all other vectors X(j), where j 6= i can be constructed as

1 Cm(r) = Θ{r − d[X(i), X(j)]} (31) i − ( − ) ∑ N m 1 j6=1 the Heaviside step function, Θ(x) is similar to the symbol in correlation dimension where Θ(x) = 0 if X ≤ 0 and Θ(x) = 1 if X > 0. The symbol r in (31) denotes a predetermined tolerance value, deﬁned as

r = k.std(Y) (32) where std(Y) refers to the standard deviation of vibration signal, Y and k is a constant (k > 0). By deﬁning 1 ϕm(r) = ln[Cm(r)], i = 1, 2, . . . , N − m + 1 (33) − + ∑ i N m 1 i the approximate entropy value of time series can be computed as

ApEn (m, r) = lim [ϕm(r) − ϕm+1(r)] (34) N→∞

3.4.4. Largest Lyapunov Exponent The largest Lyapunov exponent (LLE) algorithm is an established method and has been used in some areas such as biomedical signal processing analysis especially for electroencephalography (EEG) signal [19]. The LLE algorithm calculates the exponential divergence (either positive or negative exponential) of two initial neighboring trajectories in a phase-space matrix. In other words, the LLE algorithm quantiﬁes the degree of chaos of vibration signal in certain time. The degree of chaos corresponds to any local instability vibration signal due to the dynamic contact between rolling elements and defect spots during the data acquisition. The detailed discussion and the application of LLE in slew bearing vibration data is presented in [64]. The features extraction result for correlation dimension, fractal dimension and approximate entropy from slew bearing vibration data is presented in Figure 18. It can be seen that the approximate entropy shows better result in presenting the degradation condition of the slew bearing than the correlation dimension and fractal dimension. A ﬂuctuation in the approximate entropy in the last measurement (after the 90th day) indicates that the slew bearing condition has changed. Machines 2017,, 5,, 2121 22 of 28

1.14

1.12

1.1

1.08 Fractal dimension 1.06 0 20 40 60 80 100 120 140 t, days

-2.37

-2.375

-2.38

-2.385 Correlation dimension -2.39 0 20 40 60 80 100 120 140 t, days

0.8

0.6

0.4

0.2 Approximate entropy 0 0 20 40 60 80 100 120 140 t, days

Figure 18.18. FractalFractal dimension, dimension, correlation correlation dimension dimension and and approximate approximate entropy entropy features features calculated calculated from thefrom vibration the vibration slew bearingslew bearing signal. signal.

3.5. Category 5: Complexity Measurement

3.5.1. Kolmogorov-Smirnov Test The Kolmogorov-Smirnov (KS) test is a nonparametricnonparametric test that can be used to compare or to measure the similarity of two cumulative distribution functions (CDFs),(CDFs), TT((xx) )and andR (xR() x[65 ) ].[65].T(x )T(isx a) targetis a target CDF CDF and Rand(x) isR( ax reference ) is a reference CDF. Because CDF. Because the signature the signature of bearing of condition bearing condition can be identified can be by its CDF, therefore the KS test can be applied in bearing condition monitoring to distinguish bearing identified by its CDF, therefore the KS test can be applied in bearing condition monitoring to health signal from faulty signal. In order to distinguish the two CDFs, T(x) and R(x), a statistical distinguish bearing health signal from faulty signal. In order to distinguish the two CDFs, T(x ) and distance D is calculated. The distance D can be defined as the maximum absolute distance between R(x ) , a statistical distance D is calculated. The distance D can be defined as the maximum T(x) and R(x). This is defined as follows absolute distance between T(x ) and R(x ). This is defined as follows D = max |T(x) − R(x)| (35) DmaxT()R()=−−∞

The KS feature of slew bearing vibration data is presented in Figure 1919.. The ‘0’ represents normal condition andand ‘1’ ‘1’ represent represent abnormal abnormal condition. condition. It can beIt seencan thatbe seen at the that beginning at the of beginning the measurement, of the themeasurement, condition of the bearing condition was of still bearing normal. was Then still norm it changedal. Then to it abnormal changed conditionto abnormal when condition bearing when had runbearing for ahad couple run for of days. a couple This of result days. needsThis result further needs investigation further investigation as the abnormal as the detectionabnormal happened detection toohappened early. too early.

MachinesMachines 20172017,, 55,, 21 21 2323 ofof 2828

0.80.8

0.60.6

0.40.4 KS feature KS feature

0.20.2

00 2020 4040 6060 8080 100100 120120 140140 tt,, daysdays

FigureFigure 19.19. KSKS testtest featurefeature calculatedcalculated extractedextracted fromfrom thethe vibrationvibration slew slew bearing bearing signal. signal.

3.5.2.3.5.2. SampleSample EntropyEntropy TheThe samplesample entropyentropy hashas beenbeen proposedproposed byby Richman Richman andand MoormanMoorman [66][66] [66] toto improve improve the the approximateapproximate entropy.entropy.entropy. Chen ChenChen et et al.et [al.al.67 ][67][67] presents presentspresents an application anan applicationapplication of sample ofof sample entropysample entropy forentropy features forfor extraction featuresfeatures extractionextractionof gearbox ofof vibration gearboxgearbox signals.vibrationvibration However, signals.signals. However,However, the application thethe applicationapplication of the sample ofof thethe entropy samplesample for entropyentropy bearing forfor vibration bearingbearing vibrationvibrationsignal is still signalsignal limited isis stillstill [68 limitedlimited]. The sample [68].[68]. TheThe entropy samplesample feature entropyentropy for slewfeaturefeature bearing forfor slewslew vibration bearingbearing data vibrationvibration is presented datadata is inis presentedpresentedFigure 20. in Itin canFigureFigure be observed 20.20. ItIt cancan from bebe observedobserved Figure 20 from fromthat theFigureFigure sample 2020 thatthat entropy thethe sample featuresample isentropyentropy not effective featurefeature for isis slew notnot effectiveeffectivebearing damage forfor slewslew detection. bearingbearing damagedamage detection.detection.

2.52.5

1.51.5 Sample entropy Sample entropy 11

0.50.5 00 2020 4040 6060 8080 100100 120120 140140 tt,, daysdays

FigureFigure 20.20. SampleSampleSample entropyentropy featurefeature extractedextracted fromfrom thethe vibration vibration slew slew bearing bearing signal. signal.

Machines 2017, 5, 21 24 of 28

3.6. Category 6: Other Features

3.6.1. Singular Value Decomposition (SVD) The singular value decomposition (SVD) is usually applied to quantify the periodicity of the time series data [69]. A comparison study [70] has been shown that SVD performs the Fourier decomposition in terms of information content and robustness. Formula of SVD can be found in [71].

3.6.2. Piecewise Aggregate Approximation (PAA) and Adaptive Piecewise Constant Approximation (APCA) In online bearing condition monitoring cases, the changes of bearing condition can be effectively identiﬁed by measuring the similarity between two vibration signals (i.e., normal or reference and monitored signal) using Euclidean distance. However, the size of the measured data is a problem in Euclidean distance method. The similarity search methods are techniques that perform dimensionality data reduction methods such as piecewise aggregate approximation (PAA) [72,73] and adaptive piecewise constant approximation (APCA) [72]. The APCA [74], is another approach of PAA that allows arbitrary length segments. Either PAA or APCA have particular advantages; for example, PAA has twice as many approximating segments, and APCA is able to separate a single segment in an area of low activity and many segments in areas of high activity [74]. The PAA has been applied in previous study [39] for data pre-processing stage of circular domain features extraction. The detailed discussion and the application of PAA in slew bearing vibration data is presented in [39].

4. Conclusions Features extraction methods in wide range application such as mechanical engineering (i.e., for bearing vibration signals), biomedical engineering area (i.e., for EEG and ECG signals) and ﬁnance area (i.e., for time series signals) have been presented. According to the certain characteristic of existing feature extraction methods, the methods in this paper were classiﬁed into six categories: (1) time-domain features extraction; (2) frequency-domain features extraction; (3) time frequency representation; (4) phase-space dissimilarity measure; (5) complexity measure; and (6) other features. Some of features extraction methods in each category have been empirically tested using actual normal to failure slew bearing vibration signal. It was shown that some features do not show bearing degradation trend clearly and the others are shown to be potential slew bearing condition parameters. The potential features are: impulse factor, margin factor, mathematical morphology operators, wavelet decomposition-kurtosis, approximate entropy and largest Lyapunov exponent. Since slew bearing operating at very low speed with highly applied load, the vibration signals exhibit non-linear, non-stationary and chaotic behaviour. The methods such as impulse factor and margin factor measured the statistical properties of vibration signal and therefore could revealed the information related to the impulse level. Other potential methods based on the phase-space dissimilarity measurement such as approximate entropy and LLE algorithm can be used to extract pertinent bearing signal corresponding to the bearing damage progression. This is because the methods analyzed the vibration signal in certain time-frame to extract the useful information from the non-linear and non-stationary behaviour.

Acknowledgments: Wahyu Caesarendra thanks the University of Wollongong, Australia for the financial support through International Postgraduate Research Scholarship (IPRS) during this study. Author Contributions: Wahyu Caesarendra conducted Conceptualization, resources, experimental setup, data collection, data processing, data analysis and original draft preparation. Tegoeh Tjahjowidodo provided Funding acquisition, review and editing. Conflicts of Interest: The authors declare no conflict of interest. Machines 2017, 5, 21 25 of 28

Appendix A. The Formula for Calculating Bearing Fault Frequencies [75] • Fault frequency of outer ring:

IR − OR ( ( )) · rpm rpm cos α dr FOR = · 1 − · z (A1) 2 dm

• Fault frequency of inner ring:

IR − OR ( ( )) · rpm rpm cos α dr FIR = · 1 + · z (A2) 2 dm

• Fault frequency of rolling element:

" # IR − OR ( ( ))2 · rpm rpm dm cos α dr FR = · − (A3) 2 dr dm where IRrpm and ORrpm are the rotational speeds of the inner ring and outer ring. For 1 r/min the value of IRrpm is 1 and the value of ORrpm is 0. dm denotes the mean bearing diameter, dr is diameter of the rolling element and z is number of rolling elements.

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