on measured vibration signals

Liu Hong, Jaspreet Singh DhupiSchool of Mechanical and Aerospace Engineering, Nan

a r t i c l e i n f o

Article history:Received 6 June 2013Received in revised form28 October 2013Accepted 21 November 2013Handling Editor: I. TrendafilovaAvailable online 3 January 2014

st DTW algorithm.All rights reserved.

Early detection of local gear faults in practical industrial environments is crucial to optimize the maintenance schedule

operation of the system [4,5]. If there is any local gear fault, e.g., such as a scuffing, cracking or a fretting corrosive tooth as

Contents lists available at ScienceDirect

Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 216421800022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jsv.2013.11.033n Corresponding author at. Division of Mechatronics and Design, School of Mechanical and Aerospace Engineering, Nanyang Technological University,N3-02c-78, 50 Nanyang Avenue, Singapore 639798, Singapore. Tel.: 65 6790 5506; fax: 65 6792-4062.

E-mail address: djaspreet@ntu.edu.sg (J.S. Dhupia).and reduce the financial cost of gearbox damage [13]. Vibration based diagnosis using a sensor fixed to a gearbox housingis the most preferred monitoring technique because of the ease of measurement and no interference with the normalthe planet motion with respect to the fixed sensor, which is experimand is later employed for the estimation of reference signal used in Fa

& 2013 Elsevier Ltd.

1. Introductiondetection of gears is validated using experimental signals from a planetary gearbox test rig.For fault detection in planetary gear-sets, a window function is introduced to account for

entally determinedtechnique is beneficial in practical analysis to highlight sideband patterns in situationswhere data is often contaminated by process/measurement noises and small fluctuationsa n

yang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

a b s t r a c t

Spectral analysis techniques to process vibration measurements have been widely studiedto characterize the state of gearboxes. However, in practice, the modulated sidebandsresulting from the local gear fault are often difficult to extract accurately from anambiguous/blurred measured vibration spectrum due to the limited frequency resolutionand small fluctuations in the operating speed of the machine that often occurs in anindustrial environment. To address this issue, a new time-domain diagnostic algorithm isdeveloped and presented herein for monitoring of gear faults, which shows an improvedfault extraction capability from such measured vibration signals. This new time-domainfault detection method combines the fast dynamic time warping (Fast DTW) as well as thecorrelated kurtosis (CK) techniques to characterize the local gear fault, and identify thecorresponding faulty gear and its position. Fast DTW is employed to extract the periodicimpulse excitations caused from the faulty gear tooth using an estimated reference signalthat has the same frequency as the nominal gear mesh harmonic and is built usingvibration characteristics of the gearbox operation under presumed healthy conditions. This

in operating speeds that occur even at otherwise presumed steady-state conditions. Theextracted signal is then resampled for subsequent diagnostic analysis using CK technique.CK takes advantages of the periodicity of the geared faults; it is used to identify the positionof the local gear fault in the gearbox. Based on simulated gear vibration signals, the FastDTW and CK based approach is shown to be useful for condition monitoring in both fixedaxis as well as epicyclic gearboxes. Finally the effectiveness of the proposed method in faultA time domain approach to diagnose gearbox fault based

journal homepage: www.elsevier.com/locate/jsvi

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2165shown in Fig. 1, it results in amplitude and frequency modulations in the original vibration measurements whose sidebandsare periodic at the shaft rotation frequency and its harmonics [6,7]. However, gearboxes often operate under some smallfluctuation around nominal load/speed conditions during their normal service [8,9]. These fluctuations result in a variation ofboth the modulations and their carrier frequencies (gear mesh harmonics) that blurs the sideband components in the spectraof the vibration measurement, often making it difficult to be recognized [10]. Such smearing effect can be abated by the ordertracking technique or the time synchronous averaging (TSA) that acquires the measurements synchronized at identical angleincrement instead of the identical sampling period [11,12]. Although TSA is a well-established technique for analyzing gearboxvibration signal [1315], its commercial implementation is limited because of the requirement for additional shaft mountedencoders to provide a measure of shaft angular position and sophisticated interpolation algorithms to resample the vibrationdata. Since such equipment and resources lead to increased cost to applications, they are usually absent in most industrialapplications [16]. In such cases, the conventional method is to extract the measurement over a shorter time duration using asliding window during which the gearbox is presumed to operate under stationary condition. However, these shorter lengthvibration signals are usually analyzed using Fourier transforms that has limitations such as the limited frequency resolutionand spectral leakage, while the small operational speed oscillations continue to exist.

To avoid the extra cost incurred in implementation of TSA and shortcomings of the Fourier transform based analysis,a time domain method that uses dynamic time warping (DTW) was recently employed to detect common faults in areciprocating compressor through current signals of the driving motor in [17], though no identification or diagnosis of faultsignals was demonstrated. In most industrial gearboxes, which contain several gear pairs, the fault identification anddiagnosis is of interest as replacement gears can be ordered before the actual disassembly and physical inspection of gears,which in turn reduces the machine downtime [10]. Further, even though DTW algorithm was indicated to be effective inrecognition, data mining and signal processing [18], it has an O(N2) time and space complexity that limits its usefulness onlyto small time series containing at most a few thousand data points [19]. This makes the DTW algorithm hard to be applied tovibration signal monitoring for applications such as gearboxes, where data is usually measured for several seconds at asampling rate which is of the order of thousands of Hertz to capture the characteristic vibration phenomena occurringaround the meshing frequency and its harmonics.

To address the limitations of DTW time and space complexity, as well as, characterize the local gear fault by identifyingthe corresponding faulted gear and its position, a new time-domain diagnostic algorithm combining the fast dynamic timewarping (Fast DTW) as well as the correlated kurtosis (CK) techniques is proposed herein. Fast DTW runs in linear O(N) timeand space complexity [19], has been applied to fault diagnostic application in geared transmissions for the first time in thiswork to highlight the sideband patterns resulting from the local gear fault. The fault diagnosis algorithm introduces anestimated reference signal that has the same frequency as the nominal gear mesh frequency and is built using vibration

Fig. 1. Examples of the local gear faulty problems in a wind turbine gearbox: (a) scuffing on the ring gear (credit: GEARTECH, NREL #19855) and (b) frettingcorrosion occurred along a line of action on the sun pinion (credit: GEARTECH, NREL #19750).characteristics of the gearbox when operating under presumed healthy conditions. The effect of shifting and distortion thatusually occurs between a measured signal and an estimated reference signal due to errors in estimation as well as smallspeed fluctuations is minimized by Fast DTW as it eliminates these distortion effects by allowing an elastic stretching orcompressing within the two time series to determine the best fit between the estimated reference and measured signal.Thus, it enables an improved time-domain fault diagnosis algorithm wherein the residual signal is more sensitive to faultsthrough filtering out of normal process disturbances. After processing the vibration signal using FAST DTW, the resultingresidual signal is resampled for subsequent identification of damaged gear and its position using correlated kurtosis. Thisstep is necessary as the correlated kurtosis analysis takes advantages of the rotating periodicity of the local gear faults toidentify the position of the damaged gear tooth, which is restored during the resampling. The applicability of the techniquehas been demonstrated through both simulations and experiments. Simulations for monitoring of fixed-axis and epicyclicgear-sets show that the vibration signals processed using the proposed technique have residual signal containing rich faultinformation which is more sensitive to the gear faults. The effectiveness of this approach is also demonstrated experi-mentally through measured vibration signals from a 4 kW planetary gearbox test rig.

The rest of the paper is organized as follows. In Section 2, the proposed time domain approach for gear fault diagnosisbased on the Fast DTW and correlated kurtosis using vibration signal is described. Section 3 investigates the effectiveness of

the proposed method using MATLAB/Simulink simulation studies. Experimental validation using a controlled planetarygearbox test rig is described in Section 4. Finally, Section 5 concludes the paper.

2. Proposed algorithm for gearbox fault diagnosis in time-domain

Dynamic time warping has been applied to process data from motor currents to detect mechanical faults in a recipro-cating compressor [17]. But the quadratic time and space complexity of this algorithm limits DTW's application. To addressthis limitation, an improved time domain fault diagnosis method, which combines Fast DTW and correlated kurtosis, basedon vibration measurement from a rotary machine is proposed in this work. This approach demonstrates good performancefor extracting fault signature from measured vibration data, and is able to identify the local gear fault, i.e., the position of thedamaged gear in the gearbox. The background of dynamic time warping, fast dynamic time warping and correlated kurtosistechniques are summarized in the first part of this section for the ease of the readers. Afterwards, a brief description of theproposed algorithm is presented.

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 216421802166Y y1; y2;; yj;; yM (2)Construct a warp path W:

W w1;w2;;wk;;wKwhere K is the length of the warp path and the kth element of the warp path is

wk i; j (3)such that i is an index from time series X, and j is an index from time series Y. The warp path must satisfy the followingconditions:

a) w1(1,1) which implies that the warp path must start at the beginning of each time series,b) wK(N,M) which implies that the wrap path must finish at the end of both time series, andc) if wk(i, j) and wk1(i, j); then iA(i, i1), jA(j, j1), which implies that between the start and end of the wrap path

every index in both of the given time series must be utilized.

Dynamic time warping finds the optimal warp path that minimizes the accumulative distance (usually Euclideandistance) between the two time series by typically using a dynamic programming approach. For this approach, a twodimensional cost matrix D (also referred as the accumulative distance matrix) of dimension NM is constructed. Fig. 3shows an example to determine optimal warp path based on the cost matrix for the alignment of two time series signalsshown in Fig. 2. The detailed dynamic programming approach is described in [18]. This algorithm is quadratic in both timeand space complexity, as each cell in cost matrix D is required to be filled exactly once.

Fig. 2. Dynamic time warping of two time series: (a) before and (b) after processing. (For interpretation of the references to color in this figure, the readeris referred to the web version of this article.)X x1; x2;; xi;; xN (1)

respectively.2.1. Dynamic time warping

Dynamic time warping technique finds the optimal alignment between two time series by allowing the given time seriesto be warped nonlinearly by stretching or shrinking along its time axis. Thus, DTW can be used to determine the similaritybetween the two time series. Fig. 2 illustrates the alignment of two time series processed by DTW. Note that, DTW algorithmis able to achieve alignment by non-uniform warping, i.e., while the time series Y is shrunk in the start, it is stretched laterto align with the time series X.

The dynamic time warping problem can be stated formally as follows: Given two time series, X and Y, of length N and M,

2.2. Fast dynamic time warping

As a result of quadratic time and space complexity of DTW algorithm, its implementation to monitoring data containing onlyaround 10K measurement points may require a gigabyte range of memory space. Therefore, speeding up computational time andreducing memory requirement is crucial for successful implementation of DTW algorithm to most vibration based fault diagnosisproblems. A multi-level approach, called Fast DTW algorithm, has been developed that runs in linear O(N) time and spacecomplexity [19]. Fast DTWalgorithm can be implemented using a three step recursive approach. First, a lower level/resolution timeseries is created that has half as many points as the input time series. Next, an optimal path in this low level/resolution is found,which is projected to the original higher level/resolution input time series. Finally, the projected path is expanded by a pre-definedradius to form a search window that is passed to the DTW algorithm. Thus, the fast implementation the DTW algorithm onlyevaluates the cells in the search window rather than the complete cost matrix of O(N2) dimension. Fig. 4 shows this three stepapproach of Fast DTWalgorithm pictorially using the same time series as given in Fig. 2. Shaded cells represent the search window,in which red shaded cells represent the projected path and green shaded cells represent the pre-defined radius.

The performance of the programmed Fast DTW code to remove small fluctuations that can be found in the measuredsignals can be further evaluated using a simulated case. Consider a test signal x(t) and reference signal y(t):

xt A cos 2f t0:02t2 0rtr1;

A cos 2f t0:01t2 1otr2; 2f 1:02

((4)

yt A cos 2f t (5)where tA 0;2, amplitude A1 and rotational frequency f10 Hz. For this test, a simulation time step dt of 1104 s andpre-defined radius of 20 cells is chosen. Fig. 5 shows the test signal and reference signal before and after application of Fast

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2167Fig. 3. Optimal warping path searched by dynamic programming.

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 216421802168DTW algorithm. From Figs. 2 and 5, it can be concluded that DTW/Fast DTW can align two similar signals that contain smallphase difference or small speed oscillation that is often the case in practical applications. This is an important propertywhich is responsible for the improved sensitivity of residual signal obtained from the fault detection algorithm, which isdescribed in detail in Section 2.4.

Fig. 5. Fast dynamic time warping of the two exampled time series: (a) before and (b) after processing.

Fig. 4. Window and optimal path evaluated during different level/resolutions when running recursive Fast DTW algorithm with pre-defined radiusof 4 cells.2.3. Correlated kurtosis

A fault in rotating machine such as a local gear fault introduces periodic impacts that appear as impulse-like peaks in themeasurements [20]. Kurtosis has been recommended for detection of such peaks in measurement signal [21]. Although,a high kurtosis value for a given data set indicates a presence of a distinct peak; however, kurtosis value decreases when themeasured data set contains periodic impulses repeating at the period of a fault. Correlated kurtosis takes advantage of theperiodicity of the faults, and therefore, is used in this work to detect periodic impulses introduced by a faulted gear tooth.Correlated kurtosis of M-shift for measured data set X is defined as [20]

CKMT Nn 1Mm 0xnmT 2

Nn 1x2nM1(6)

where T is the period of interest (the period for fault signature that needs to be detected). Correlated kurtosis of first and second-shift,M1 or 2, is studied in this paper. From (6), it can be seen that the CK value approaches a maximum only when the periodof interest Tmatches with the period of the impulses. Fig. 6 illustrates the CKM versus Kurtosis for several simple signals. Fig. 6(a)

shows a signal with distinct peak, while Fig. 6(b) and (c) are defined as 0:3floorn=100k 0 nk100 and 0:1floorn=100k 0 nk100,

respectively. It can be seen that while Kurtosis value decreases as the peaks in data set become periodic, the CK value increasesabout the specified period. However, comparing Fig. 6(b) and (c), both Kurtosis and CK values are not sensitive to the amplitudeof the peaks in the data set. Therefore, to investigate the variations in amplitude of data sets, which indicates the severity of thefault, the rms value is also calculated for the presented simulations and experimental results in the later sections.

2.4. Proposed fault detection algorithm

The key steps of the proposed diagnosis approach in this study are band-pass filtering, reference signal estimation, FastDTW implementation and residual signal analysis of processed vibration measurement to detect gear faults in gear-sets,which are presented as a flowchart in Fig. 7. The details of this approach are given as follows:

Step 1: Band-pass filtering. The measured vibration signal is pre-processed by band-pass filtering around the dominantgear mesh frequency harmonic fm to remove other gear mesh harmonics as well as remaining non-fault related frequency

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2169comdriv

Sundbanbotsigninvo

(1)ponents resulting from high frequency noise and other process disturbances that are commonly injected in an industriales. The pre-processed vibration signal is denoted as x(t).tep 2: Reference signal estimation. An estimated signal is built using vibration characteristics of the gear-set operationer presumed healthy conditions, which has the same frequency as the nominal gear mesh harmonic fm after thed-pass filtering. This reference signal is used as an input in Step 3. The procedure for estimating the reference signal forh fixed axis as well as the planetary gear-set is described separately. This is because the planetary gear-set vibrationals have more spectral components than a fixed axis gear systems and therefore, the estimation technique is also morelved.

Fixed axis gear-set: Considering an error-free fixed axis gear pair meshing under a constant load and speed, its meshvibration y(t) around a certain gear mesh frequency harmonic fm can be expressed as a sinusoidal signal [6]:

yt A cos 2f mt (7)

Kurtosis = 98.01, CK1(100) = 0.09, CK2(100)=0.008, CKM(T

Kurtosis = 998, CK1(100) = 0, CK2(100) = 0, CKM(T 100) = 0

Kurtosis = 98.01, CK1(100) = 0.09, CK2(100) = 0.008,CKM(T

Fig. 6. CK versus Kurtosis for different signals containing distinct peaks.

which can be used as the reference signal y(t) in the subsequent Fast DTW algorithm. The amplitude of reference signalis estimated by finding the root mean square (rms) value for the raw measured vibration signal after band-pass filteringstage, x(t) that has a length of N as

A2

pxrms

2NNx2n

s(8)

The nominal gear mesh frequency fm can be calculated from the nominal operational speed of the system. Its time axis tis chosen to have the same length N and time step as x(t) in this study. The initial phase of the reference signal isestimated by sweeping the phase angle of the reference signal from 0 to 2 in small increments and finding the phaseangle that yields the least mean square error between x(t) and y(t).

(2) Planetary gear-set: When a planet pinion moves with the carrier toward the vibration sensor mounted on the stationarygearbox housing, the level of the measured vibration signal increases, reaches a peak (r1) when the planet is closest tothe sensor, and then decreases as the planet gear recedes (Z0). The window function wtA 0;1, models the effect ofthis amplitude modulation (AM) phenomenon that is periodic with the frequency of carrier rotation [22]. Therefore, themesh vibration signal y(t) around a certain gear mesh frequency harmonic fm of a healthy planetary gear-set can be

Fig. 8. Scheme of determining the pre-defined window function.

Fig. 7. Flowchart of the proposed time domain fault detection method.

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 216421802170Fig. 9. Reference signal estimation for planetary gear-set.

expressed as sinusoidal signal with AM effect [23]:

yt wtA cos 2f mt (9)Further, w(t) can be expanded as a Fourier series consisting of a sum of sinusoidal functions containing the carrierrotational frequency and its harmonics as

wt J

j 0Wj cos 2jf ct j (10)

where fc is the nominal speed of the carrier that can be calculated from the nominal operational speed of the gear-set.Since the actual pattern of the window function w(t) relates to the structure of the gearbox and the position of thesensor only, the value of amplitudeWj and initial phase j of jth harmonic in Eq. (10) should be independent from carrierrotational frequency fc. The value of amplitudeWj and initial phase j of jth harmonic in Eq. (10) can be found and storedas pre-defined parameters to describe the window function for a given planetary gearbox. These values, Wj and j, aredetermined using the steps described in Fig. 8. First, the measured vibration signal under healthy condition is firstpassed through a band pass filter around the mesh harmonic of interest. Then, the envelope of the measured vibrationsignal can be extracted by hardware/software envelope detector/demodulation algorithm. Afterwards, this envelopesignal is normalized within the [0, 1] range to form the window function w(t). Further, the order tracking and timesynchronous averaging (TSA) techniques can be applied to synchronize the window signal based on the carrierrotational angle to attenuate aperiodic noise. Finally, the discrete Fourier series transform is employed to determine thediscrete magnitude spectrum ofWj and discrete phase spectrum of j. The overall scheme of reference signal estimationfor planetary gear-set using these evaluated parameters for window function,Wj and j, is illustrated in Fig. 9. The initialphase of the window function can be found by sweeping from 0 to 2 in small increments to find the value of thatyields the least mean square error between the envelop of x(t) and w(t). Further, the amplitude A and the initialphase of Eq. (9) can be also estimated by the same methods presented in fixed-axis case. Afterwards, Eqs. (9) and (10)are used to generate the reference signal y(t) as shown in Fig. 9 for the planetary gear-set, which is employed in thesubsequent Fast DTW algorithm.

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2171Step 3: Fast DTW implementation and residual signal analysis. The two signals, pre-processed measured signal x(t) andestimated reference signal y(t), are matched in time domain using the Fast DTW algorithm described in Section 2.2. Theaim of applying Fast DTW is to reveal the difference between the two signals x(t) and y(t), which is highlighted byevaluating the residual signal. The raw residual signal is defined as |xwarpedywarped|, where xwarped and ywarped areobtained from signals x and y respectively, after transforming with the wrap path obtained from Fast DTW. If thegearbox is operating under ideal healthy condition, the measured vibration signal after band-pass filtering should besimilar to the estimated reference signal. Small phase/rotational frequency differences that may occur between x and ybecause of the typical machine operation characteristics can be removed by warping them along the time axis (similar

Fig. 10. The time resampling algorithm.

3.

Tpermo

3.1.

Ateet

rota

Fig.andto the illustrations in Figs. 2 and 5) using Fast DTW algorithm. Thus, the residual signal under healthy condition issmoothed out and has lower rms value when processed by Fast DTW. If a damaged gear exists, it introduces periodicimpact/impulse-like response in the measured signal at its characteristic fault period T. Thus, the residual signal underfaulty condition contains periodic peak values with high amplitude. The rms value of the residual signal is employed todetect variation in the amplitude of the residual. A higher rms value indicates a larger difference between the measuredsignal and the reference signal and hence indicates the severity of the fault. Moreover, damaged gear teeth on differentgears have different characteristic fault period T related to rotational frequency of the shaft carrying the gear. Thus, by

11. Performance of time resampling algorithm: (a) test and reference signal, (b) test and reference signal after Fast DTW, (c) the raw residual signal,(d) the residual signal after the proposed time resample algorithm.L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 216421802172evaluating the CKM(T) for all possible characteristic fault frequencies arising from possible tooth damage at differentgears from the residual signal can identify the position of the fault. The challenge to evaluate the CKM(T) on the rawresidual signal obtained after the application of Fast DTW algorithm is that the length of residual signal K is usuallydifferent from the length N of the original signals. The reason for this difference in length of data can be deduced fromFig. 3 and constrain (c) on the wrap path wherein it can be observed that the index jmay equal to j in the warping pathfunction wk(i, j), wk1(i, j). Therefore, a resampling algorithm is required to restore the length of the raw residualsignal back to the original measured signal before employing CKM(T) to identify the fault position. This time resamplingalgorithm is presented in Fig. 10. The capability of the time resample algorithm is illustrated in Fig. 11. A periodicreference signal with period of 275 data points and a quasi-periodic test signal with a period fluctuating around thenominal period of 275 data points is shown before and after application of Fast DTW in Fig. 11(a) and (b) respectively.The peaks of the raw residual in Fig. 11(c) are also a quasi-periodic signal due to the quasi-periodic characteristics of thetest signal and the warp path generated by the Fast DTW algorithm. Fig. 11(d) gives the residual signal that is resampledusing the index j of the periodic reference signal along the original time axis of the reference. The period of this residualsignal is restored back to 275 data points, which is the same as the reference signal.

Simulations based investigation of proposed algorithm's performance

he performance of the proposed approach is investigated using both simulations and experiments. In this section, theformance of algorithm is tested by developing an analytical model for a single stage of fixed axis gearbox and a dynamicdel for an equally-spaced planetary gearbox.

Detection and location of a gear defect in single stage fixed axis gearbox

simulated case is presented herein: a single stage fixed axis gear-set with a pinion having 10 teeth and a gear having 13h. Let mesh harmonic order m1, its amplitude A11, number of pinion teeth Np10, number of gear teeth Ng13 and

tional frequency of pinion is defined as f s 0:08t10; tA 0;10:04t10; tA 1;2

(to simulate small fluctuations around the nominal

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2173rotational speed fns10 Hz. The amplitude and phase modulation (AM & FM) functions due to a local fault on the pinion aredefined as at 0:3 cos 2f st

and bt 0:1 cos 2f st. The simulated vibration signals of the gearbox under healthy and

pinion faulty condition are generated as xh(t) and xpf(t), respectively:

xht A1 cos 2Npf stnt (11)

Fig. 12. (a) Simulated signals of the gearbox under healthy and pinion faulty condition xh(t) and xpf(t), (b) spectra of xh(t) and xpf(t) after band-pass filter, (c)xh(t) and reference signal y(t) after initial phase estimation but before Fast DTW, and (d) xh(t) and reference signal y(t) after Fast DTW.xpf t A11at cos 2Npf stbtnt (12)where n(t) is the Gaussian white noise whose SNR0 dB. This mathematical signal characteristics (Eqs. (11) and (12)) forhealthy and faulty cases respectively has been widely used for simulation studies of gear diagnosis by researchers [4,6,10].Fig. 12(a) plots the time domain waveform of simulated healthy signal xh(t) and faulty signal xpf(t), which indicates that theAMFM features are mostly concealed by the noise. Afterwards, the proposed time domain fault diagnosis approachdescribed in last section is applied to both xh(t) and xpf(t). Due to the fluctuations in the operational speed, the fault featurecannot be clearly identified even from the spectra obtained after the band-pass filtering of raw signals to remove the highfrequency noise (as seen from Fig. 12(b)). Fig. 12(c) and (d) presents the simulated vibration signal xh(t) and the referencesignal y(t) before and after application of the proposed Fast DTW process. It can be seen that the simulated and referencesignals are aligned together as a result of Fast DTW application. For calculation of CKM(T), T corresponding to local faultperiod of pinion is Tp(1/fns)/dt1000 while T corresponding to the local fault period for the gear is Tg(1/(Npfns/Ng))/dt1300. Fig. 13(a) and (b) gives the raw residual signals rz (just after the application of Fast DTW) and the residual signals z(after subsequent application of the proposed resampling algorithm). CKM(T) is calculated for both residual signals forhealthy as well as faulty case. It can be observed from Fig. 13 that the proposed resampling algorithm for the raw residualsignal after Fast DTW is necessary to employ CKM(T) to identify the fault position. For example, the rz based CK1(1000) underhealthy and local pinion faulty condition remains at small values 0.6105 and 1.6105, respectively. This variation inCK1 value is insensitive to the fault at the pinion location when compared to the z based CK1(1000) under healthy and faultyconditions with values of 1.3105 and 23.3105 respectively. Further, the z based CK1(1300) is insensitive to pinionfault and remains at smaller value of 2.7105, which indicates that the CK1(T) value is sensitive only for values of Tcorresponding to the time period of the characteristic fault frequency and its multiples. Similar trend can be also observed inthe CK2 values in Fig. 13.

3.2. Dynamic model of planetary gear transmission

The dynamic model of planetary gear system used in this study is given in Fig. 14 and described in detail in [24]. Thegears are assumed as rigid bodies connected to each other along the line of action through the corresponding gear meshstiffness and viscous damping [25,26]. These gears are held by bearings, which allow them to translate in x and y directions

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 216421802174and freely rotate about their centers in the xy transverse plane of gear. Such model has been widely applied to study thedynamics of industrial planetary gearbox [27,28]. The motion of the sun gear is defined with the translational displacementxs and ys, and the angular coordinate s. Similarly, the motion of the carrier is defined by xc, yc, and c. is the pressure angleand i is the initial angle location for planet i. An error function espi and erpi with 5 m amplitude error and a profile similarto saw-tooth [9,29,30] are used to describe gear imperfections such as deviations in the dimensions and shape of the gearsdue to the manufacturing error as shown in Fig. 15(a). Thus, the gear mesh deformation along the line of action between thesun gear and the ith planet gear can be defined as (R is the radius of the base circle):

spi xsxiRc cos i sin iysyiRc sin i cos iscRspcRpespi (13)

Similarly, the gear mesh deformation between the ring gear and the ith planet gear can be written as

api xiRc cos i sin iyiRc sin i cos iicRp0cRrerpi (14)

Healthy condition: RMS=0.06, CK1(Tp)=0.610-5, CK1(Tg)=0.710-5; CK2(Tp)=0.0310-9, CK2(Tg)=0.310-9.Local pinion fault: RMS=0.12, CK1(Tp)=1.610-5, CK1(Tg)=7.510-5; CK2(Tp)=0.410-9, CK2(Tg)=7.310-9.

Healthy condition: RMS=0.06, CK1(Tp)=1.310-5, CK1(Tg)=1.910-5; CK2(Tp)=0.710-9, CK2(Tg)=0.510-9.Local pinion fault: RMS=0.13, CK1(Tp)=23.310-5, CK1(Tg)=2.710-5; CK2(Tp)=63.910-9, CK2(Tg)=1.110-9.

Fig. 13. Residual signals after Fast DTW: (a) the raw residual signals rz after Fast DTW, and (b) the residual signals z after the proposed time resample.

Fig. 14. Dynamic gear mesh model of planetary gear-set.

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2175Healthy:

Faulty:

Time t

Faulty gear toothmeshing period

KLoss

Kmax

Kmin

Gea

r mes

hing

stiff

ness

K(t)

Fig. 15. (a) Error function and (b) gear meshing stiffness under healthy and faulty case.where the ith planet pinion has rotation i. xi and yi are the translational displacements of the ith planet. If spi and apio0,their values are compulsorily set to 0 as the teeth lose contact and the resulting spring force will be equal to zero. The backcollisions of the teeth are usually not taken into account in planetary gear-set models, because the gear backlash values of aplanetary gear-set are considerably larger than those of fixed-axis gear pair in order to ensure easy assembly and preventcontact on both flanks of the gear teeth [31]. The global equation of motion for the gearbox that can be expressed in matrixform as

M Q tCCb _Q tKtKbQ t Ft (15)

where M is the mass matrix, C is the damping matrix, K(t) is the time-varying gear meshing stiffness matrix, Cb and Kb arethe bearing damping and stiffness matrices, F(t) is the externally applied torques vector, and Q(t) is the degrees of freedomvector that contains two coordinates for translational vibration and a coordinate for torsional motion for each gear in theplane containing the gear. Fig. 15(b) illustrates a square waveform used to describe the time-varying gear meshing stiffnessbetween a gear pair that form the corresponding element in matrix K(t). The degradation of a gearbox as a fault progressesresults in a degradation of the gear mesh stiffness over the life of a gearbox [32,33]. In this work, the common local gearfaults are modeled in dynamic simulation study by assuming a local drop a general squared wave form gear meshingstiffness function (Fig. 15(b)). The rationale of this approach is based on the investigation on the variation in the gearmeshing stiffness studied in [34], wherein it was concluded that such faults are always accompanied by a local reduction inthe gear meshing stiffness. In this paper, the maximum and minimum values of gear meshing stiffness are assumed to beKmax5108 N/m, Kmin3108 N/m, respectively. The 25 and 50 percent local stiffness loss Kloss is used to simulate the

Healthy case: RMS = 310-7, CK1(Tsunf) = 1.510-5, CK1(Tringf) = 1.510-5, CK1(Tplanetf) = 1.310-5;Moderate fault: RMS= 410-7, CK1(Tsunf) = 2.3 10-5, CK1(Tringf)=1.3 10-5, CK1(Tplanetf) = 1.3 10-5;Severe fault: RMS= 710-7, CK1(Tsunf) = 2.5 10-5, CK1(Tringf)=1.5 10-5, CK1(Tplanetf) = 1.6 10-5;White noise: CK1(Tsunf) = 1.1 10-5, CK1(Tringf)=1.1 10-5, CK1(Tplanetf) = 1.1 10-5.

Fig. 16. Simulated residual vibration signals from the planetary gear-set under healthy and local sun gear faulted cases after application of the proposeddiagnosis approach.

moderate and severe local sun gear faulted cases respectively. The dynamic equations of the lumped parameter of gear-setsare numerically integrated in MATLAB/Simulink environment. A constant input torque 300 Nm is exerted on the input sideand k_

2output is used as a load torque. The parameter details of the investigated gear-set are provided in Appendix A.

3.3. Detection and location of a gear defect in planetary gearbox

In this simulation study, a gear-set having four planets in equally spaced configuration, with sun gear and planets' carrieracting as input and output respectively is investigated. Number of ring gear teeth Nr100, number of sun gear teeth Ns28,and number of planet gear teeth Np36. The dynamic response under healthy conditions and a local gear tooth defect in thesun gear of the planetary gear-set is simulated using the previously described lumped parameter model. A 15 dB wide bandwhite noise is added to the dynamic responses to simulate the practical measured signal. The nominal rotational frequencyfor sun fs, carrier fc and planets fp are 69.56 Hz, 15.21 Hz and 27.05 Hz under steady operating conditions, respectively. Thesimulated signal is then pre-processed using a band-pass filter with the central frequency around the gear mesh frequencyfm fcNr1521 Hz. For this planetary gear-set at the given simulated operating conditions, the local sun gear fault frequencycan be evaluated as fsunf fm/Ns(fsfc)54.35 Hz [23,35]. Therefore, for evaluation of correlated kurtosis, T correspondingto local sun gear fault is Tsunf(1/fsunf)/dt. Similarly, T corresponding to local ring gear fault is Tringf(1/fringf)/dt, and T valuecorresponding to local planet gear fault is Tplanetf(1/fplanetf)/dt, respectively. Fig. 16 presents the simulated residualvibration signals under healthy and local sun gear tooth faulted conditions. The evaluated CK1 with TTsunf, Tringf,and Tplanetf,and rms values of the residual signals are shown for different fault severity levels. Additionally, the CK values of a whitenoise are also given as a reference in Fig. 16. It can be observed that the rms values of the residual signals increase when thefault severity increases. CK1(Tsunf) is found to have significant increase in presence of faults that indicates the gear fault

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 216421802176Fig. 17. (a) Schematic of the test rig, (b) planetary gearbox test rig and (c) Seeded gear fault on the internal ring gear of the planetary gear-set.

happens at the sun gear. However, CK1(Tringf) and CK1(Tplanetf) are still remaining at smaller values comparable to the whitenoise case, which increase or decrease somewhat under the local sun gear tooth fault conditions. Comparing these CK1values in Fig. 16, it implies that strong periodic components resulting from local gear fault happen and only happen at thelocal sun gear fault frequency in the residual signal after the proposed fault diagnosis algorithm. Consequently, a selection ofappropriate thresholds on the rms and CK1 value can allow for detection of gear faults and its position.

Thus, from simulations of fixed axis as well as planetary gear-sets, it can be seen that a selection of appropriatethresholds on the rms and CKM values can enable detection of gear faults and its position using the proposed algorithm.

4. Experimental results

The proposed Fast DTW algorithm along with fault identification using correlated kurtosis was applied on experimentallymeasured vibration signals from a planetary gearbox test rig shown in Fig. 17. The input shaft of the gearbox is connected toa 3-phase AC motor (4 kW 4 pole) controlled using a standard industrial drive. A generator, with a resistive load bank, isdriven by the output shaft of the gearbox. The two tested gearboxes have back-to-back planetary gear-sets, each having fourplanets in an equally spaced configuration with number of ring gear teeth Nr84, number of sun gear teeth Ns28, andnumber of planet gear teeth Np28. As a result of the back-to-back scheme, the overall gear ratio of the gearbox equals toG1/G41/41, where G represents the gear ratio of a single planetary gear-set. The vibration signals are measured at a

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2177Fig. 18. The pre-defined window function for the planetary gearbox on the test rig as a sum of sinusoidal signals that are periodic at carrier rotationharmonics (a) discrete magnitude spectrum of Wj and (b) discrete phase spectrum of j.sampling rate of 20 kHz using an accelerometer attached to the gearbox housing outside the ring gear. A seeded spalled geartooth fault was introduced to one of the ring gear teeth (Fig. 17(c)) using electro-discharge machining (EDM).

The vibration transmission function from the gear mesh to sensor location described in Step 2 in Section 2.4 can beexperimentally obtained through demodulation of the measured vibration signal under healthy condition with controlledconstant speed to determine the amplitude modulation (AM) function as illustrated in Fig. 8. The magnitude of this AMfunction is then normalized from 0 to 1. Fig. 18 shows the pre-defined AM function of the tested planetary gearboxes that isestimated from the envelope signal of the measured vibration of the test rig under presumed healthy condition. Sinceaccording to Eq. (10) the jth component of window function is located at jth harmonic of carrier frequency, therefore, thecarrier order instead of frequency is shown on the x-axis of Fig. 18. Further note that, this pre-defined AM function containsthe vibration contribution from the manufacturing errors that may exist under the presumed healthy conditions at thebeginning of the test.

The vibration signal is measured with motor operating under open-loop speed control mode with nominal speed of1400 rev/min. However, small deviations and fluctuations of the operational speed could be observed as evident from themeasured spectra shown in Fig. 19. Fig. 19 illustrates the measured vibration spectrum around the nominal gear meshfrequency fm1400/60/(GNr)490 Hz for both healthy as well as ring gear fault conditions. It can be observed that themeasured peak of gear mesh frequency deviates from its nominal value and the spectra are blurred due to the inherentspeed fluctuations during the test. Afterwards the proposed time domain fault diagnosis approach described in Section 2.4 isapplied to the pre-processed data filtered with a FIR band-pass filter with the central frequency 490 Hz and bandwidth70 Hz used to extract the signal around the gear mesh frequency. The characteristic local ring gear fault frequency isfringf fc fm/Nr5.83 Hz, a corresponding Tringf equals to (1/fringf)/dt is chosen to evaluate the CKM value. As Ns equals to Npin this planetary gear-set, Tsunf also equals to Tplanetf. The residual signals obtained after application of Fast DTW arepresented in Fig. 20(a). This residual signal clearly highlights the differences between the measured vibration signal and thereference signal obtained during the healthy and ring gear fault conditions. The rms, increase from 1.2104 under thehealthy condition to 4.9104 under the ring gear faulty condition. CK1(Tringf) and CK2(Tringf) also increases from 7.9105

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 216421802178and 6.8109 under the healthy condition to compared large values 19.7105 and 47.7109, which imply that a localpinion gear fault exists. On the other hand, while CK1(Tsunf)CK1(Tplanetf) and CK2(Tsunf)CK2(Tplanetf) also increase to8.2105 and 8.9109, they still remains at comparatively small values. The effectiveness of proposed algorithm for faultdetection from practical measured signals is further demonstrated in Fig. 20(b), which presents the residual signal obtainedfrom measured vibration signal and reference signal without transforming either signal using the warp path obtained usingFast DTW. It can be observed that the residual signal obtained for healthy and a ring gear fault condition does not containany statistical differences, and it is difficult to detect gear faults without application of Fast DTW. Thus, it can be concludedthat Fast DTW processing optimally matches the measured vibration signal to reference signal that results in more accuratefeature extraction using the proposed approach.

Fig. 20. The residual signal between the measured vibration signal and reference signal (a) after the proposed diagnosis approach processing (b) only afterinitial phase matching.

Fig. 19. Measured vibration spectrum around gear mesh frequency.

include both practical fixed axis as well as epicyclic gearboxes.

L. Hong, J.S. Dhupia / Journal of Sound and Vibration 333 (2014) 21642180 2179Acknowledgment

The authors are pleased to acknowledge the financial support of Maritime Port Authority of Singapore (Grant number:M4060926).

Appendix A

See Table A1.

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Table A1System parameters for the example planetary gear system.

Sun Planet Ring Carrier

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A time domain approach to diagnose gearbox fault based on measured vibration signalsIntroductionProposed algorithm for gearbox fault diagnosis in time-domainDynamic time warpingFast dynamic time warpingCorrelated kurtosisProposed fault detection algorithm

Simulations based investigation of proposed algorithm's performanceDetection and location of a gear defect in single stage fixed axis gearboxDynamic model of planetary gear transmissionDetection and location of a gear defect in planetary gearbox

Experimental resultsConclusionsAcknowledgmentReferences