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Gear Fault Diagnosis Using Bispectrum Analysisof Active Noise Cancellation-Based Filtered Soundand Vibration SignalsDibya Prakash Jena and S. N. PanigrahiSense and Process (SnP) Laboratory, School of Mechanical Sciences Indian Institute of Technology, Bhubaneswar- 751013, India

(Received 25 October 2011; Revised 8 January 2013; Accepted 6 February 2013)

Fault diagnosis using acoustical and vibration signal processing has received strong attention from many re-searchers over the last two decades. In the present work, the experiment has been carried out with a customizedgear mesh test setup in which the defects have been introduced in the driver gear. Classical statistical analysis in-cluding higher-order statistics, namely bispectrum analysis, has been incorporated to detect the defects. However,in order to improve the signal-to-noise ratio of the captured signals for accurate defect detection, an adaptive filter-ing has been proposed. Active noise cancellation (ANC) has been applied on the acoustical and vibration signalsas a denoising filter. The least mean square based ANC technique has been implemented considering the signalsfrom healthy gear meshing as the background noise. The focus of this experimental research is to evaluate theappropriateness of the ANC technique as a denoising tool and the subsequent bispectrum analysis for identifyingthe defects. The performance of the ANC filtering was evaluated with most widely accepted standard filters. Asynthetic signal, close in nature to the actual signal, has been investigated to ascertain the adequacy.

1. INTRODUCTION

Safety, reliability, efficiency, and performance of rotatingmachinery are major concerns in the industry. In this situa-tion, the task of condition monitoring and fault diagnosis ofrotating machinery has significant importance. Many meth-ods have already been widely used in a variety of industriesfor predictive maintenance. It has been widely accepted thatthe structural defects in rotating machinery components can bedetected through monitoring acoustical and/or vibration sig-nals. The machine condition monitoring process consists ofthree stages of signal processing: (1) acquisition of acoustic orvibration signals, (2) signal pre-processing and extraction ofthe fault feature, and (3) diagnosis of the defect.

Various signal processing techniques have been identifiedand proposed by many researchers in this context. The mostcommon method is the fast Fourier transform (FFT) to obtainthe power spectrum to investigate the frequency componentsof the entire signal. However, it is a well-known fact is that theacoustical or vibration signals from a rotating system is com-posed of a large number of non-stationary signals, particularlyin the presence of a localized, defect such as bearing pitting orgear tooth fracture. Short-time Fourier transform and relatedtime-frequency and time-scale techniques have often been usedto detect such non-stationary defect signatures. Another fact isthat typical defects in machinery components have been char-acterized by particular vibration patterns.1 Loutas et al. havereported that the acoustical emission (AE) technique is moreeffective in the early stages of defect identification comparedto vibration monitoring, particularly in the case of a crack inthe gear. Regionally linear behaviour of AE parameters hasbeen observed by them where the associated gradients change

proportionally with the crack propagation rate.2 Some poten-tial defects, namely spalling of the gears and bearings, clear-ances, etc., induce periodic impulses in acoustical and vibra-tion signals of rotating machines. Such impulses may excitethe eigenmodes of the structure and the sensor. The statisticalparameters such as the root mean square (RMS) value, kur-tosis, crest factor, skewness, peak value, and signal-to-noiseratio (SNR) are most widely used to detect the defect. Theseindicators are easy to implement; however, the complexity ofthe mechanisms involved may give rise to serious errors in in-terpretation. A detailed study was conducted by Dron et al.on the influence of certain parameters on the value of the crestfactor, kurtosis, and RMS value.3 In order to carry out con-dition monitoring of the gear meshing using acoustical andvibration signals analysis, the selection of such statistical in-dicators needs to be well suited to the impulsive nature of theexcitatory forces generated by the defects. Thus, reliability infault diagnosis can be achieved if, and only if, the acquired sig-nals are free from any background noise. Therefore, the majorchallenge is to remove background noise from acoustical andvibration signals captured at the faulty condition using appro-priate signal processing. The intention of signal denoising is tominimize the influence of the unwanted noise without affectingthe defect signatures.

Over the last decade, the performance of active noise can-cellation (ANC) systems has been improved primarily due tothe integration of digital signal processing. The merging of ac-tive noise cancellation techniques and digital signal processinghas enabled the control of noise dynamically and adaptively. Inpractice, active noise control is mainly used for duct-like sys-tems such as blowers, ventilation systems, or enclosures likeaircraft and vehicle cabins, headphones, and control rooms.

58 (pp. 58–70) International Journal of Acoustics and Vibration, Vol. 18, No. 2, 2013

D. P. Jena, et al.: GEAR FAULT DIAGNOSIS USING BISPECTRUM ANALYSIS OF ACTIVE NOISE CANCELLATION-BASED FILTERED SOUND. . .

The adaptive filtering deals with a least mean square (LMS)algorithm and an approach with reference signals with fixedlength of filter coefficients. The LMS algorithm is widely usedin adaptive filtering due to its computational simplicity and un-biased converging nature in a stationary environment.4 Adap-tive digital filters consist of two distinctive parts: (1) a digitalfilter to perform the desired signal processing and (2) an adap-tive algorithm using a reference signal and a residual error toadjust the coefficients (weights) of that filter. Based on a finiteimpulse response (FIR) structure, an adaptive digital filter witha filtered-x least mean square algorithm has been widely im-plemented in ANC applications due to its relative simplicity indesign and implementation.5 Gonzalez et al. have investigatedthe attenuation of engine noise using active noise cancellation.They have demonstrated that ANC can be considered a usefultool to reduce the sound pressure level of low-frequency noisesfor improving the acoustical comfort.6 On the other hand, theuse of higher-order spectral (HOS) analysis is an emerging areaof interest, particularly when the nonlinear system analysis isof interest.7–9 HOS analysis demonstrates the groups of fre-quencies and their interrelationships. This technique is alsocapable of explaining the origins of spectral peaks at certainvalues in the frequency spectrum. An HOS, especially bispec-trum analysis, has been used by many investigators for machin-ery condition monitoring.7 HOS has the potential to quantifynonlinearities using the time series signal. As a result, HOSmethods have been applied with partial success to rotating ma-chinery condition monitoring and fault diagnosis.8 In sum-mary, bispectrum analysis can identify the non-Gaussian pro-cesses.9 Collis, White, and Hammond have reported the basicprinciples of HOS and also have demonstrated the capabilityof HOS in yielding information which is unavailable throughinspection of second-order statistics such as the spectrum orcorrelation function.10

However, analysis using a bispectrum technique has beenused predominantly for condition monitoring purposes. Thebispectrum of a signal is the decomposition of the third mo-ment skewness of the signal over frequency, which supports theanalysis of systems with asymmetric nonlinearities. Similarly,another powerful technique known as trispectrum represents adecomposition of kurtosis over frequency. The bispectrum ofacoustical, vibration, or current signals has been extensivelyinvestigated by many researchers and reported suitable in faultdiagnosis of rotating system components, such as a damagedgear or bearing.9 Montero and Medina have demonstrated thetheoretical approach of implementing a bispectrum techniqueto identify a rolling bearing element defect.11 Li et al. havepresented fault detection and diagnosis of a gearbox in ma-rine propulsion systems using bispectrum analysis and artifi-cial neural networks. Incipient gear fault vibration signals of-ten have non-stationary features, are usually heavily corruptedby noise, and are often strongly coupled with the faults of othercomponents. Li et al. have demonstrated that the featuresof the gear fault vibration signals can be extracted effectivelyusing bispectrum analysis. Both the amplitude and phase in-formation can be preserved, and distinguished features can beextracted along the parallel lines of the bispectrum diagonal.12

Time-frequency analysis has been used by many investiga-

tors to analyse the non-stationarity of the signal.13 Sung, Tai,and Chen have employed wavelet transform to detect the lo-cation of tooth defects in a gear system precisely. They haveimplemented wavelet analysis on vibration signals for locat-ing gear defects and advantages of multi-resolution propertyof the wavelet.14 Jena, Panigrahi, and Kumar recently investi-gated adaptive wavelet transform for analysing non-stationaryvibration response from a faulty gear mesh system.15 Anotherimportant aspect of condition monitoring is establishing a re-liable online system to offer accurate fault diagnostic informa-tion for mechanical systems to prevent machinery performancedegradation, malfunction, or even catastrophic failures. More-over, machinery fault diagnosis information can also enablethe establishment of a maintenance program based on an earlywarning of incipient defects.

In the present work, an experiment has been carried out witha customized gear-meshing test setup in which defects havebeen introduced in the driver gear teeth. Two different defectconditions have been analysed: (1) a defect in one tooth and (2)a defect in two teeth. The acoustical and vibration signals arecaptured both in healthy conditions and with faulty gears. TheANC has been proposed for denoising purposea. An LMS-based adaptive filtering technique has been used on acousticaland vibration signals to improve the SNR of the signals in de-fect scenarios. The acoustical and vibration signals at healthyconditions have been used as reference signals for the adaptiveLMS filter. The statistical and bispectrum analysis have beenincorporated to evaluate the performance of the filtering tech-nique. A synthetic signal simulation analysis has been carriedout to understand the capability of the proposed method fol-lowed by experimental validations, which are explained andpresented in subsequent sections.

2. UNDERLYING THEORY

2.1. Hunting Tooth FrequencyIt is a well-known fact that gear-mesh frequency can be com-

puted by multiplying the number of teeth (T ) of a gear by thespeed of the gear (Gs). The number of teeth on the drive gearmultiplied by the speed of the drive gear must equal the num-ber of teeth on the driven gear multiplied by the speed of thedriven gear. The fractional gear-mesh frequencies generatedmay be caused by the common factor and the eccentric gear.Hunting tooth frequency (HTF) occurs when the same toothon each gear comes into mesh again.16 The HTF can be deter-mined by dividing the least common multiple of the teeth onthe two gears by the uncommon factor of the gear of interest.The product equals the number of revolutions the gear mustmake before the HTF occurs and can be expressed as

HTF = [1/(L/U)]× (1/Gs); (1)

where L is the least common multiple and U is the uncommonfactor of the gear. The HTF is not normally measurable be-cause it occurs infrequently; however, one can notice the samesituation occurring in the time domain signal. The broken toothon each gear generates a pulse each time when it goes intomesh. In the case of two broken teeth, consecutive pulses are

International Journal of Acoustics and Vibration, Vol. 18, No. 2, 2013 59


generated. The frequency of such impulse events observed ina time-domain signal is called HTF.16 Amplitude modulationof gear-mesh frequency and harmonics reveals useful informa-tion about the mesh (e.g., misaligned gears, improper backlash,loading, eccentricity, etc.). Gear-mesh frequency does not usu-ally reveal a broken, cracked, or chipped tooth except in rarecases when natural frequencies are not measurable. Gear-meshfrequency and amplitude can be modulated by the speed of theproblem gear. Some ghost frequencies may also be present dueto errors during gear manufacturing.17

2.2. Active Noise Cancellation Using an LMSAlgorithm

Acoustical and vibration signals acquired from the machinesfor diagnostic purposes may be either deterministic or ran-dom. Deterministic signals can be further classified as peri-odic and non-periodic, whereas random signals can be clas-sified as stationary and non-stationary.18 Useful informationcan be extracted from these signals by appropriate signal pro-cessing techniques. However, these acoustical and vibrationsignals often contain a lot of noise, which may also lead toincorrect conclusions. In such cases, techniques that enhancethe SNR are highly desired. Adaptive noise cancellation is onesuch technique that enhances SNR.17 An adaptive digital fil-ter consists of two stages of signal processing. The first stageis a digital filter, which processes the expected output signal,and second one is an algorithm to adjust weighting coefficientsof the digital filter. Two kinds of digital filters can be usedin ANC, namely infinite impulse response or FIR filters. Inthe present study, an FIR filter is used as the controller of thesystem.

Mathematically, if a linear FIR filter (discrete-time) has aseries of coefficients wl(n); (l = 0, 1 . . . , L − 1) and a seriesof continuous inputs {x(n) x(n− 1) . . . x(n−L+ 1)}, thenan expected output, d(n), occurs, and the output signal is usedto reduce noise from filter.19

Briefly, the procedures for the ANC system can be describedas follows:

1. Optimal selection of the order of filter coefficient L, stepsize µ, and the initial filter coefficients w0(n)

2. Evaluation of the output signal from the adaptive filter:

y(n) =

L−1∑l=0

wl(n)x(n− 1) (2)

3. Measurement of the error signal as:

e(n) = d(n)− y(n) (3)

4. Updating of the adaptive filter coefficient using LMS al-gorithm

wl(n+ 1) = wl(n) + µx(n− l)e(n); (4)

where l = 0, 1, . . . , L− 1.

The step size and filter length are major parameters in adaptivefiltering. The step size value affects the convergence speed,steady-state error, and stability of the adaptive filter. A smallstep size ensures low steady-state error and decreased conver-gence speed. Large step size improves the convergence speedbut might cause instability.

The filter length affects the computational resource require-ments, convergence speed, and steady-state error. The optimalfilter length is decided by a trial-and-error process. However,simulation of adaptive filter is advised to be carried out to de-termine the most appropriate filter length. The filter lengthmust be greater than the number of significant taps in the im-pulse response of the unknown system.20 A long filter lengthcan reduce the steady-state error. Optimization of the filterlength that satisfies the application is highly desired.

2.3. Statistical ParametersTo obtain useful information from the time-domain acoustic

and vibration signals various statistical techniques have beendeveloped over the years. One of the parameters, namely, thecrest factor, which is defined as the ratio of maximum absolutevalue to the RMS value of the vibration signal, gives an ideaabout the occurrence of impulse in the time-domain signal. Inreal-time condition monitoring, an increased value of the crestfactor over a period of time indicates the presence of wear orpitting. Another powerful parameter called kurtosis measuresthe degree of peakiness of a distribution compared to a normaldistribution. In general, even statistical moments give infor-mation about spread. Mathematically, crest factor and kurtosisfor signal x(n) with N number of samples in the time domaincan be expressed as:

CrestFactor =CrestValueRMS value

=sup |x(n)|√

(1/N)∑N

n=1[x(n)]2(5)

and

Kurtosis =M4

M22

=(1/N)

∑Nn=1 (x(n)− x̄)

4[(1/N)

∑Nn=1 (x(n)− x̄)

2]2 ; (6)

where M4 and M2 are the fourth-order and second-order sta-tistical moment, respectively, and the x̄ is the mean of the sig-nal.3, 21 The kurtosis and the crest factor parameters are verysensitive to the shape of the signal. The fourth-order momentof the signal gives a substantial weight to high amplitudes ofthe kurtosis. The crest factor attenuates the impact of an iso-lated event with high crest amplitude that only takes into ac-count the crest amplitude of this event. The kurtosis thereforeappears as a better indicator than the crest factor.21 But, inthe case of severe defects or a multiple defect scenario, thesestatistical parameters only provide the indication of the pres-ence of the defect but do not provide any information aboutthe severity of the defect. One of the limitations of the kurtosismethod is that the kurtosis value falls to 3 when the damage iswell advanced. This may be due to the higher relaxation timeof the impulsive response than the impulsive repetition period.Monitoring the overall RMS values can be more useful in suchcases. Another limitation is that the kurtosis method can give

60 International Journal of Acoustics and Vibration, Vol. 18, No. 2, 2013


variable and misleading results if measurements are taken onmachines in an unloaded condition.17

SNR is a measure used to compare the level of a desiredsignal to the level of the background noise. SNR can be definedas the inverse of coefficient of variation (Cv), (i.e., SNR =

1/Cv and Cv = σ/|µ|), where σ is the standard deviation andµ is the mean of a discrete signal. The absolute value is takenfor the mean to ensure Cv will be always positive.

The mean-square error (MSE) is widely used for filter-ing performance analysis. MSE measures the average of thesquares of the errors. The error is the difference between theobserved and estimated values. MSE is the second moment(about the origin) of the error and thus incorporates both thevariance of the estimator and its bias. For an unbiased estima-tor, the MSE is the variance.22 Mathematically, MSE can beexpressed as

MSE = (1/n)

n−1∑i=0

(xi − yi)2; (7)

where n is the number of data points, xi is the i-th element ofx and yi is the i-th element of y.

2.4. Bispectrum EstimationHigher-order statistics is effective in studying feature extrac-

tion of the non-stationary signal. Higher-order statistics usu-ally refers to four major forms: (1) high-order moments, (2)high-order cumulants, (3) high-order moment spectrum, and(4) the high-order cumulant spectrum. The high-order cumu-lant of the random signal can be generated by the derivative ofthe second characteristic function. In practice, the higher-orderspectra of a signal must be estimated from a finite set of mea-surements. Essentially, there are two broad non-parametric ap-proaches: (1) the indirect method based on estimating the cu-mulant functions and then taking the Fourier transform and (2)the direct method based on a segment averaging approach.23

In the present analysis, indirect method is used to estimate thebispectrum.

Mathematically, for a random signal x(t), the second-orderspectral density (power spectrum) is given by

P (f) = E [X(f)X∗(f)]; (8)

where X(f) is the Fourier transform of x(t), E[. . .] indicatesthe expectation value (or equivalently the average over a statis-tical ensemble) and ∗ denotes the complex conjugate. Mathe-matically, the bispectrum can be expressed as

B(f1,f2) = E [X(f1)X(f2)X∗(f1 + f2)]. (9)

Due to the symmetries in the bispectrum, the region boundedby the lines f1 = 0 and f1 = f2 contains all the available in-formation. It is worth noting that if X(f1) = 0; X(f2) = 0; orX(f1 + f2) = 0, the bispectrum at f1, f2 is also zero, whichis not obvious. In short, signals resulting from the nonlin-ear interaction of some excitation components have a specificphase relationship with the excitations that caused them. Inthe power spectrum, the phase information is lost, and hence,this phase relationship between different frequencies cannot beexploited.12, 24

3. SIMULATED ANALYSIS

This section is intended to evaluate the performance of theproposed method of analysis using synthetic test signals. Asignal containing a number of sinusoidal bursts of progres-sively increasing frequency with respect to time was chosenfor the analysis in order to observe the frequency resolutiondependencies. The signal used can be mathematically writtenas

X[n] = x[n] +Gn; (10)

where

x[n] =

m=10∑m=1

cos

[2πn

fmfs

(t− τm)

][u(t− τm)− u(t− δm)] ;

where fm = 15 mHz; τm = 0.0006 + 0.002(m − 1); δm =

τm + 0.0006; Gn ∼ N(0, 0.25); and fs = 50 kHz is the sam-pling frequency that is used to convert the continuous signalto a discrete one. The corresponding signals x[n] and X[n]

are shown in Figs. (1a) and (1b), respectively. To understandthe effect of Gaussian noise Gn on the synthetic signal x[n],the power spectrum density (PSD) has been extracted for bothx[n] and X[n], as shown in Figs. (1c) and (1d), respectively. Itis worth noting that from the PSD spectra, the impact of Gn isnot observed prominently.

Traditional correlation and power spectral analysis based onFourier transform could not extract useful information from thenonstable and nonlinear signals because the Fourier transformis based on the assumption that the signal is stationary. Thebispectrum has been proven to be effective in this situation. Itcan capture characteristic frequency, identify the phase infor-mation, and extract nonlinearity. To demonstrate the effect ofGn on synthetic signals, the bispectrum has been investigatedfor x[n] andX[n], as shown in Figs. (2a) and (2b), respectively.The bispectrum has been evaluated with 2048 numbers of fre-quency bins, 256 points of window length, and linear peak holdaveraging with 50% overlapping. A large window generates aPSD with small bias but results in a coarse PSD plot.

A small window generates a smooth PSD plot but leads tolarge bias. Overlap specifies the overlap in percentage, of themoving window that applies to the time series. This parameterdetermines how much data of the signal will be used for spacematrix. A large overlap reduces the variance of the resultingpower spectrum but increases computation time. The resultingbispectrum can detect the asymmetric nonlinearities in the in-put time series. From Figs. (2a) and (2b), one can note thatadditional spectrum spikes are visible after introducing Gn.The bispectrum of x[n] (see Fig. (2a)) explains the presenceof periodic impulses at different frequencies, which creates theprominent side band spikes. The bispectrum of X[n] has addi-tional spikes due to well-known quadratic phase coupling withGaussian noise Gn and synthetic signal x[n]. It is well knownthat the nonlinear couplings, including quadratic phase cou-pling, can be identified using bispectrum analysis.

In line with the proposed technique, adaptive noise cancel-lation was implemented to improve the SNR of the signal. Theperiodic burst signal x[n] is the desired signal, where Gaussiansignal Gn acts as the background noise. The resultant signal



Figure 1. Synthetic signal used for the performance analysis of the proposedmethod: (a) synthetic signal x[n], (b) synthetic signal X[n], (c) PSD of syn-thetic signal x[n], and (d) PSD of synthetic signal X[n].

X[n] can be expressed as

X[n] = x[n] +Gn. (11)

The schematic of active noise cancellation process is shownin Fig. (3). Analogous to a real-time scenario, the Gaussiannoise Gn is treated as an acoustical signal or vibration signalat a healthy condition, and X[n] is the impulsive noisy signalat the faulty condition.

In the present simulation study, the objective was to achievex[n] from X[n]. An ANC technique has been implemented asan adaptive denoising filter. The adaptive filter creates an FIRfilter based on an LMS algorithm, as shown in Fig. (3). Theadaptive filter considers the Gn as back ground noise and fil-ters out the noisy signal X[n] using a filter size of 10 and stepsize of 0.1. When the output signal y(n) becomes close to Gn,then the adaptive system can remove the background noise. InFig. (3), the error signal e(n) from the adaptive filter denotesthe resultant signal, which needs to be similar to x[n]. The er-ror signal from the adaptive filter and the denoised signal havebeen shown in Figs. (4a) and (4b), respectively. The overlap-

Figure 2. Bispectrum of synthetic signal: (a) bispectrum of x[n] and (b) bis-pectrum of X[n].

Figure 3. ANC schematic: FIR adaptive filter with the standard LMSalgorithm.

ping of raw noisy signal on denoised signal has been shownin Fig. (4c). Now, one can note the suitability of the ANCtechnique as a denoising tool. Next, the bispectrum of the de-noised signal has been extracted and is shown in Fig. (5). Itis worth noting that the reduction of spectrum spikes. Fromthe synthetic signal analysis, it is very clear that ANC can beused to remove Gaussian noise and that bispectrum analysis isan appropriate tool to understand the presence of different fre-quencies and nonlinearity of the signal. The adaptive filteringparameters for the synthetic signal are not optimal but explainthe denoising capability of adaptive filter. The optimal filtersize and step size parameters have been extracted for real-time



Figure 4. Denoising by ANC: (a) adaptive filter output y(n), (b) error signale(n), i.e., denoised synthetic signal, and (c) denoised signal overlapped withoriginal noisy synthetic signal X[n].

Figure 5. Bispectrum of denoised signal (i.e., error signal from adaptive filtere(n)).

acoustical and vibration signals, captured from an experimentwhich is explained in the next section.

4. EXPERIMENT

In the present work, the experiments were conducted on agear mesh assembly fabricated for the purpose as shown inFig. (6). The pinion with 15 teeth was mounted on a drivershaft coupled with a Crompton© single phase 50 Hz AC induc-tion motor of power rating 0.5 hp with the help of a belt drive.The driver shaft was supported on two P204 bearing blocks.

Figure 6. Test setup: (a) gear meshing test setup mounted with accelerometerand microphone, (b) driver gear of one defective tooth, and (c) driver gear oftwo defective teeth.

The gear on the driven shaft had 30 teeth. The other end of thedriven shaft had means to apply the load on the shaft. The ex-periment was conducted by loading the driven shaft by differ-ent weights. From the experiment, it has been found that 2 kgload is suitable to carry out the experiment with adequate dy-namic stability. The operating frequency observed was 7.62 Hz(457 rpm).

In order to acquire the acoustical signal, one microphone(Behringer ECM-8000©) was mounted near the gear meshing.A PCB© ICP-type accelerometer was mounted on the drivenshaft bearing block, as shown in Fig. (6a), to capture the vi-bration signal. The acoustical signals and the vibration signalswere acquired with the help of a National Instruments© SCXI-1530© data acquisition system and a customized LabVIEW-based application software. The signals were captured at sam-pling rate of 50 K samples per second. The data samples havebeen taken after running the system for 15 minutes to avoidinitial dynamic instability and were processed offline.

The experiment was carried out in three phases. In the firstphase, the healthy gears were mounted, and the correspondingacoustical and vibration signals were captured. Sample data ofone second duration are shown in Figs. (7a) and (8a), and thecorresponding PSD spectra are shown in Figs. (7b) and (8b).In the second phase, the defective driver gear (gear defect-1)was mounted on the driver shaft. One tooth of the driver gearwas artificially damaged (defect-1) as shown in Fig. (6b). Thecorresponding acoustical and vibration signals were captured,and sample data of one second duration are shown in Figs. (9a)and (10a), respectively. From Figure, it is worth noting that thesignal is modulated with the impulse generated by the defect.In the last phase of the experiment, the defective driver gear(gear defect-2) was mounted on the driver shaft. Two teethof driver gear were artificially damaged (defect-2), as shownin Fig. (6c). The corresponding acoustical and vibration sig-nals were captured, and sample data of one second durationare shown in Figs. (11a) and (12a), respectively.

5. RESULTS AND DISCUSSION

The gear meshing frequency (GMF) at 114.25 Hz was ob-served for an operating frequency of 7.62 Hz (457 rpm). Onecan note the HTF at 7.68 Hz from the raw acoustical and vi-bration time-domain signals at faulty conditions. From thePSD spectrum of the acoustical signal for healthy gear mesh-ing, the GMF at 113 Hz was prominently visible along with



Figure 7. Raw acoustical signal of healthy gear meshing: (a) acoustical signaland (b) PSD of acoustical signal.

Figure 8. Raw vibration signal of healthy gear meshing: (a) vibration signaland (b) PSD of vibration signal.

higher harmonics (6th, 10th, etc.) of high amplitudes. Simi-larly, the PSD of the vibration signal for healthy gear meshingalso carried 6th and 16th harmonics of the GMF prominently,as shown in Figs. (7b) and (8b), respectively. The PSD ofacoustical signals for damaged driver gear meshing, namelyone tooth defect (broken) and two teeth defects (broken), areshown in Figs. (9b) and (10b), respectively. The PSD spectrumof defect-1 carried the peak at GMFs predominately. How-ever, in defect-2 condition, the 5th, 9th, and 10th harmonics ofthe GMF were also visible prominently. Similarly, the PSDof vibration signals from the defective (defect-1 and defect-2)driver gear meshing was extracted and is shown in Figs. (11b)and (12b), respectively. In such situations, one can observe thatthe PSD spectra carried the spike at GMFs with very lower am-plitude. The higher harmonics of GMFs (5th, 16th, 19th, 25th,and 26th) were noticeable with higher amplitudes and with highside bands.

Figure 9. Acoustical signal from gear mesh with defect-1 driver gear: (a) rawacoustical signal and (b) PSD of the acoustical signal.

Figure 10. Acoustical signal from gear mesh with defect-2 driver gear: (a) rawacoustical signal (b) PSD of the acoustical signal.

Table 1. Statistical parameters of the acoustical signal.

RMS SD Crest SNR KurtosisFactorHealthy gear meshing 0.20361 0.20105 4.91126 0.79445 3.60215Defective driver gear-1 0.20130 0.2012 4.9678 0.74176 5.46935Defective driver gear-2 0.28422 0.28414 4.06056 0.76664 3.89425

From the PSD spectra, it is worth noting that the acousti-cal signal carried high-frequency noise at smaller amplitudesand that the vibration signal carried high-frequency noise at alarger amplitude. Next, the statistical quantities such as RMSvalue, standard deviation, crest factor, SNR, and kurtosis forthe acoustical and vibration signals were evaluated and havebeen tabulated in Tables 1 and 2, respectively. From Tables 1and 2, it can be observed that the statistical parameters’ varia-tions with respect to defect severity are inconsistent; only thekurtosis of acoustical and vibration signals varied significantlywith respect to defect, but the variation shows a decreasing pat-



Figure 11. Vibration response from gear mesh with defect-1 driver gear:(a) raw vibration signal and (b) PSD of the vibration signal.

Figure 12. Vibration response from gear mesh with defect-2 driver gear:(a) raw vibration signal and (b) PSD of the vibration signal.

Table 2. Statistical parameters of vibration signal.

RMS SD Crest SNR KurtosisFactorHealthy gear meshing 0.23287 0.23287 4.29431 0.76149 4.25658Defective driver gear-1 0.14548 0.08068 5.15058 1.59379 31.71135Defective driver gear-2 0.12390 0.12080 5.74989 0.67445 13.76904

tern with severe defect condition. This is possible due to thesharpness of the burst visible for defect-1 condition.17

For effective feature extraction of the nonstationary acous-tical and vibration signals, we implemented bispectrum analy-sis, which computes the single-sided bispectrum of a univari-ate time series using the FFT method. The bispectra were ex-tracted with 2048 frequency bins using 256 points of windowlength and linear peak hold, with parametersaveraging at 50%overlap. The resultant bispectrum had 24.42 Hz frequency res-olution.

The bispectrum for acoustical signals (healthy condition,

Figure 13. Bispectrum analysis of gear meshing acoustical signals: (a) healthygear meshing, (b) defective (one failure tooth) gear meshing, and (c) defective(two failure teeth) gear meshing.

defect-1, and defect-2) was extracted and are shown inFigs. (13a), (13b), and (13c), respectively. The bispectrum ofthe acoustical signal from healthy gear meshing had a majorspike at the GMF frequency (at ∼110 Hz). From Fig. (13),it can be noted that the bispectrum of faulty condition carriesa prominent peak at about 1500 Hz with wide side band. Insuch cases, it can also be observed that the high-frequency sig-nal components are introduced with a linear phase. So, thelow-frequency peaks for HTF and GMF are not discernible.Similarly, the bispectra for vibration signals (healthy condi-tion, defect-1, and defect-2) have been extracted and are shown



Figure 14. Bispectrum analysis of gear meshing vibration signals: (a) healthygear meshing, (b) defective (one failure tooth) gear meshing, and (c) defective(two failure teeth) gear meshing.

in Figs. (14a), (14b), and (14c), respectively. The bispectraof the vibration signals from healthy gear meshing shows ma-jor spikes at around 670 Hz, 1800 Hz, and 2100 Hz with wideside band. From Fig. (14), it can be noted that the bispectrumcarries major peaks at around 2100 Hz, 2500 Hz, and 2800 Hzwith wide side bands for faulty conditions. In such cases, an-other observation is that the high-frequency components areintroduced with the linear phase. The low-frequency peak ofHTF is visible for defect-1 condition but not prominently vis-ible for defect-2. It is worth noting from the bispectrum thatthe three major spikes are visible, which dominates other low-

frequency spikes in the faulty condition.From statistical analysis and bispectrum analysis, it is under-

stood that an adequate filtering is required to remove noise andmake signals more informative. Next, in line with the proposedsignal processing scheme, an ANC-based filtering techniquewas implemented. The filtering performance was comparedwith standard filters such as (a) FIR-based high-pass filter witha high cut-off frequency of 100 Hz with 51 taps; (b) FIR-basedlow-pass filter with a low cut-off frequency 5 kHz with 51 taps;(c) FIR-based band-pass filter with a high cut-off frequencyof 100 Hz and low cut-off frequency of 5 kHz with 51 taps;(d) rectangular half the width of the moving average filter; and(e) wavelet denoising using undecimated wavelet transform upto decomposition level 3, using db-2 as the mother waveletand soft thresholding at multiple levels using the SURE tech-nique.25

5.1. Acoustical Signal Analysis

Active noise cancellation has been implemented with anLMS-based adaptive filtering for acoustical signals from thedefect conditions (defect-1 and defect-2). The acoustical signalfrom the healthy gear meshing was used as the reference back-ground signal in the adaptive filter. The filtering was carriedout with manual tuning of FIR filter length (200) and step size(0.093). The filtered acoustical signal of defect-1 and the cor-responding PSD spectrum are shown in Figs. (15a) and (15b),respectively. Similarly, the filtered signal of defect-2 conditionand the corresponding PSD spectrum are shown in Figs. (15c)and (15d), respectively.

From the PSD spectra of the denoised signals, it can benoted that the major low-frequency component is visible at99.9 Hz (GMF –[Operating Frequency + HTF]). The high-frequency components are also visible, but their magnitudeshave been reduced significantly. The denoised acoustical sig-nal for defect-1 and defect-2 were overlapped with the corre-sponding raw acoustical signal and are shown in Figs. (16a)and (16b), respectively.

The statistical parameters, as discussed previously, wereevaluated and have been tabulated in Table 3. In order to eval-uate the performance of ANC-based filtering, well-establishedfilters have been investigated. The statistical parameters of cor-responding denoised acoustical signals were evaluated and arepresented in Tables 4 and 5 for defect-1 and defect-2, respec-tively. It is worth noting from Tables 4 and 5 that the RMSand standard deviation decreased for the ANC-based denoisedsignal. However, the crest factor, SNR, and kurtosis values in-creased significantly compared to the raw acoustical signal anddenoised signals from other standard filters. The MSE param-eter was the lowest compared to other standard filters, whichexplains the retaining of the signal signature with minimal dis-tortion after ANC-based filtering.

Next, the bispectrum of denoised acoustical signals fordefect-1 and defect-2 conditions were extracted with the pa-rameters in Tables 4 and 5. The corresponding spectra areshown in Figs. (17a) and (17b), respectively. Based on bispec-trum analysis, it can be observed that the low-frequency com-ponent of HTF is clearly visible with high amplitudes. Other



Table 3. Statistical parameters of ANC-based denoised acoustical signals.

RMS SD Crest Factor SNR Kurtosis MSEDefective driver gear-1 0.13063 0.13044 6.0714 0.75175 5.63802 0.0367Defective driver gear-2 0.18015 0.18001 4.46279 0.76608 4.20451 0.07234

Table 4. Statistical parameters of denoised defective acoustical signals (failure in one tooth) with standard filters.

RMS SD Crest Factor SNR Kurtosis MSEHigh pass 0.19728 0.1972 5.04753 0.73888 5.61372 0.08663Low pass 0.19314 0.19303 5.55174 0.74679 5.21144 0.08532Band pass 0.18941 0.18932 5.60676 0.74436 5.34275 0.0836Smoothing 0.19386 0.19376 5.61519 0.74545 5.25884 0.00429Wavelet denoised 0.17696 0.17685 5.69978 0.73295 6.01631 0.00201ANC denoised signal 0.13063 0.13044 6.0714 0.75175 5.63802 0.0367

Figure 15. Denoised acoustical signal of defective gear meshing: (a) defect inone tooth, (b) PSD in one tooth, (c) two teeth, and (d) PSD in two teeth.

high-frequency components were also observed but with sig-nificantly lower magnitude compared to the bispectrum of theraw acoustical signal (significant peak at 1015 Hz ∼ 9th har-monic of GMF; see Fig. (13)).

5.2. Vibration Signal Analysis

In last phase of signal processing, the ANC was imple-mented on vibration signals from the defect conditions (defect-1 and defect-2) with LMS-based adaptive filtering. Similar to

Figure 16. Denoised acoustical signal of defective gear meshing overlappedwith raw signal: (a) denoised acoustical signal overlapped with the raw acous-tical signal for a defect in one tooth and (b) denoised acoustical signal over-lapped with the raw acoustical signal for a defect in two teeth.

the acoustical signal processing, the vibration signal at healthygear meshing was used as the reference background signalin adaptive filtering. The filtering was carried out with man-ual tuning of filter length200 and step size 0.02. The filteredvibration signal of defect-1 condition and the correspondingPSD spectrum are shown in Figs. (18a) and (18b), respectively.Similarly, the filtered vibration signal of defect-2 condition andthe corresponding PSD spectrum are shown in Figs. (18c) and(18d), respectively.

From the PSD spectra of denoised signals, it can be observedthat the low-frequency component is visible at 98 Hz (GMF –[Operating Frequency + HTF]). The PSD of the ANC-basedfiltered vibration signal carried additional prominent frequencypeaks (at 224 Hz, 687 Hz, and 1689 Hz, which is equivalentto 2nd, 6th, and 15th harmonics of GMF). The major spike at2854 Hz with a wide side band was observed in both the de-fect conditions, which was introduced due to the defect. Thedenoised vibration signal for defect-1 and defect-2 were over-lapped with the corresponding raw vibration signal and areshown in Figs. (19a) and (19b), respectively.

The statistical parameters, as discussed previously, wereevaluated and have been tabulated in Table 6. In order to eval-uate the performance of ANC-based filtering, well-establishedfilters were investigated. The statistical parameters of corre-sponding denoised vibration signals were evaluated and have



Table 5. Statistical parameters of denoised defective acoustical signals (failure in two teeth) with standard filters.

RMS SD Crest Factor SNR Kurtosis MSEHigh Pass (100 Hz) 0.27711 0.27705 3.58247 0.7645 3.96218 0.18548Low Pass (5000 Hz) 0.27191 0.27183 4.17771 0.77238 3.78861 0.18355Band Pass (100–5000 Hz) 0.26517 0.2651 4.22489 0.77082 3.84271 0.1785Smoothing (rectangular) 0.27458 0.27451 4.22522 0.77017 3.79728 0.00785Wavelet denoised (UWT) 0.25239 0.25231 4.28111 0.76607 3.9837 0.00408ANC denoised signal 0.18015 0.18001 4.46279 0.76608 4.20451 0.07234

Table 6. Statistical parameters of the ANC-based denoised vibration signal.

RMS SD Crest Factor SNR Kurtosis MSEDefective driver gear-1 0.14762 0.06516 7.37072 2.09151 20.75317 0.00453Defective driver gear-2 0.09081 0.08566 8.59815 0.74164 11.54897 0.00869

Figure 17. Bispectrum of denoised acoustical signals by ANC: (a) bispectrumspectrum of the denoised acoustical signal at defect in one tooth and (b) bis-pectrum spectrum of the denoised acoustical signal at defect in two teeth.

been presented in Tables 7 and 8 for defect-1 and defect-2, re-spectively. Similar to the statistics of the denoised acousticalsignal, it is worth noting from Tables 7 and 8 that the RMS andstandard deviation are decreased for the ANC-based denoisedsignal. However, the crest factor, SNR, and kurtosis values in-creased significantly compared to the raw vibration signal anddenoised signals from other standard filters. The MSE param-eter is was the lowest compared to other standard filters, whichexplains the retaining of signal signature with minimal distor-tion after ANC-based filtering.

Finally, the bispectrum of denoised vibration signals fordefect-1 and defect-2 conditions was extracted, similar to theacoustical signal processing. The corresponding spectra areshown in Figs. (20a) and (20b), respectively. From the bis-

Figure 18. Denoised vibration signal of defective gear meshing: (a) one tooth,(b) PSD in one tooth, (c) two teeth, and (d) PSD in two teeth.

pectrum, it can be observed that the low-frequency componentof HTF is clearly visible with high amplitudes. Other high-frequency components were also observed but with signifi-cantly lower magnitude compared to the bispectrum of the rawacoustical signal (significant peak at ∼2854 Hz, see Fig. (14)).With an increase in defect severity, other high-frequency com-ponents are introduced which dominate the HTF spike magni-tude. The proposed method was tested with 20 test samples,and the observed results are repeatable. in summary, the pro-



Table 7. Statistical parameters of the denoised defective vibration signal (failure in one tooth) with standard filters.

RMS SD Crest Factor SNR Kurtosis MSEHigh pass (100 Hz) 0.13635 0.07964 7.1434 1.48993 32.72079 0.01597Low pass (5000 Hz) 0.14491 0.07976 7.38803 1.60825 30.62829 0.01584Band pass (100–5000 Hz) 0.13611 0.07886 7.46802 1.50524 31.67959 0.01578Smoothing (rectangular) 0.14399 0.07796 7.43385 1.64244 30.6741 0.00064Wavelet denoised (UWT) 0.1436 0.07723 7.41479 1.65786 34.98493 0.00005ANC denoised signal 0.14762 0.06516 7.37072 2.09151 20.75317 0.00453

Table 8. Statistical parameters of the denoised defective acoustical signal (failure in two teeth) with standard filters.

RMS SD Crest Factor SNR Kurtosis MSEHigh pass (100 Hz) 0.12066 0.11801 8.19805 0.66358 14.50472 0.02853Low pass (5000 Hz) 0.12315 0.12004 8.23485 0.67596 13.41093 0.0291Band pass (100–5000 Hz) 0.12019 0.11752 8.27487 0.66537 14.13218 0.02837Smoothing (rectangular) 0.12107 0.1179 8.28729 0.67947 13.30746 0.00103Wavelet denoised (UWT) 0.11854 0.1153 8.32403 0.66858 14.78804 0.00014ANC denoised signal 0.09081 0.08566 8.59815 0.74164 11.54897 0.00869

Figure 19. Denoised vibration signal of defective gear meshing overlappedwith raw signal: (a) one tooth and (b) two teeth.

posed method can be deemed suitable and reliable in gear teethfault diagnosis using emitted acoustical signals or induced vi-bration response.

6. CONCLUSIONS

Statistical parameters provide information about the pres-ence of the defects. However, they cannot explain the severityof the defect. The HTF is not observed from the PSD spec-tra of the acoustical and vibration signals from faulty condi-tions. The experimental analysis reveals that the ANC is anappropriate denoising tool for these faulty conditions. Acous-tical and vibration signals from the healthy gear meshing sce-nario can be used as the background noise for the LMS-basedadaptive filter. The performance of the proposed denoisingtechnique is superior to most of the well-established standardfilters. The improvement in SNR, crest factor, and kurtosisare significantly noteworthy after the denoising of acousticaland vibration signals from the faulty condition (defect-1 anddefect-2) as compared to the raw signal. The bispectrum of de-noised acoustical and vibration signal carries prominent spikesat lower frequencies, which explains the presence of HTF inthe faulty condition. From identification of HTF, one can en-sure the presence of faulty teeth in gear meshing. In summary,

Figure 20. Bispectrum of denoised vibration signals by ANC: (a) bispectrumspectrum in one tooth and (b) bispectrum spectrum in two teeth.

the bispectrum of denoised (ANC-based) acoustical and vibra-tion signals provides a clearer view about the presence of faultin gear teeth.

ACKNOWLEDGEMENTS

The authors want to acknowledge support from Dr. RajeshKumar, Mechanical Engineering Department, SLIET Lon-gowal for execution of the experiment. The first author ac-knowledges generous funding received from the Ministry ofHuman Resource Development, Government of India that al-lowed this work to be carried out.



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